Algebraic geometry studies geometric objects defined by polynomial equations. Rather than using calculus to understand smooth manifolds, we employ commutative algebra and ring theory to investigate the structure of varieties — the solution sets of polynomial systems. This approach reveals deep connections between algebra and geometry: algebraic properties of polynomial rings encode geometric information about the associated varieties, while geometric intuition illuminates abstract algebraic facts. The field has become central to modern mathematics, with applications spanning number theory, cryptography, string theory, and computational methods. The course begins by anchoring the subject in classical motivation and then develops the fundamental correspondence between ideals and varieties. We start with affine varieties, which are subsets of affine space cut out by polynomial equations, and learn how the Nullstellensatz establishes a bijection between radical ideals and varieties. We then move to projective varieties, where we work in projective space to avoid points at infinity and gain projective compactness — a crucial advantage. Armed with these basic objects, we study the local and global structure of varieties through tangent spaces, smoothness, and dimension, discovering that dimension is governed by Krull's height theorem. This foundation naturally leads to algebraic curves, the one-dimensional varieties, where classical results like Bézout's theorem and the Riemann–Roch theorem showcase the power of the algebraic perspective. The later chapters deepen the theory and hint at further developments. We explore divisors and differentials on curves, tools that package geometric information into tractable algebraic objects, ultimately proving Riemann–Roch — a theorem that elegantly connects topology, algebra, and geometry. The bonus chapters venture beyond the classical setting: the Weil Conjectures introduce zeta functions and the Frobenius map, exposing the deep interplay between geometry over finite fields and topology over the complex numbers, while the chapter on schemes reveals why we must sometimes allow nilpotent elements in our rings to capture full geometric reality. Throughout, the guiding principle is that varieties are fundamentally dual to their coordinate rings, and understanding this duality unlocks powerful invariants and classification tools. # 1. Introduction This chapter sets the stage for the entire course by introducing the central objects of algebraic geometry through a single motivating example. The guiding idea is simple: take a polynomial, look at where it vanishes, and ask what kind of mathematical structure that vanishing set carries. The answers to this question will occupy us for the rest of the course. ## A Motivating Example [motivation] ### The Polynomial and Its Vanishing Set Algebraic geometry studies the geometry of solution sets of polynomial equations. To see what this means concretely, consider the polynomial \begin{align*} f(X, Y) = Y^2 - X(X-1)(X+1) \in \mathbb{Z}[X, Y]. \end{align*} Given a field $k$, its **vanishing set** is \begin{align*} V(f) = \{(z_1, z_2) \in k^2 : f(z_1, z_2) = 0\}. \end{align*} When $k = \mathbb{R}$, we can plot $V(f) \subset \mathbb{R}^2$ and observe something immediately interesting: a generic line meets the curve at three points. This is not a coincidence — it reflects the fact that $f$ has degree $3$, and over an algebraically closed field, Bézout's theorem (developed in Chapters 5 and 6) will make this precise. ### The Choice of Field Matters Working over $\mathbb{R}$ has a serious deficiency: some polynomials, such as $X^2 + 1$, have no real roots at all, making their vanishing sets empty or otherwise poorly behaved. For this reason, classical algebraic geometry works primarily over **algebraically closed fields** such as $\mathbb{C}$. Over non-algebraically-closed fields, the theory is still rich and important — Fermat's Last Theorem is a statement about $\mathbb{Q}$-points on a specific curve — but is considerably harder to develop. ### Geometry over $\mathbb{C}$ When $k = \mathbb{C}$, the vanishing set $V(f) \subset \mathbb{C}^2$ cannot be drawn directly (it is a real two-dimensional surface sitting in a real four-dimensional space), but we can still understand its topology. It turns out that $V(f)$, topologically, is a punctured torus: a torus with a finite number of points removed. This already suggests that $V(f)$ carries rich geometry. [illustration:ag-real-cubic-three-intersections] ### Polynomial Functions on Vanishing Sets Having identified $V(f)$ as a geometric object, the next question is: what are the natural functions on it? Any polynomial $g \in \mathbb{C}[X, Y]$ restricts to a function $V(f) \to \mathbb{C}$. The collection of all such restrictions is a commutative ring in which $\mathbb{C}$ embeds — this ring is called the **coordinate ring** of $V(f)$. A subtlety arises immediately: different polynomials in $\mathbb{C}[X, Y]$ can give the same function on $V(f)$. For instance, the zero function on $V(f)$ is the restriction of both $0$ and $Y^2 - X(X-1)(X+1)$. More generally, if $g_1, g_2 \in \mathbb{C}[X, Y]$ agree on $V(f)$, then $g_1 - g_2$ vanishes on $V(f)$, which in particular holds whenever $f \mid g_1 - g_2$. At minimum, the polynomial functions on $V(f)$ are captured by the quotient ring $\mathbb{C}[X, Y]/(f)$. ### The Three Main Characters These observations bring into focus the three central objects that algebraic geometry studies simultaneously: 1. **Vanishing sets** — the subsets $V(S) \subset k^n$ of affine $n$-space cut out by a collection of polynomials $S \subset k[X_1, \ldots, X_n]$. 2. **Coordinate rings** — quotient rings of the form $k[X_1, \ldots, X_n]/I$, where $I$ is an ideal generated by the polynomials in question. 3. **Ideals** — the ideal $I \subset k[X_1, \ldots, X_n]$ itself, which encodes the algebraic data of the equations. The deep content of algebraic geometry lies in the interplay between these three objects. Geometric properties of $V(S)$ (irreducibility, dimension, smoothness) correspond to algebraic properties of $I$ and $k[X_1,\ldots,X_n]/I$ (primeness, Krull dimension, regularity), and vice versa. Making this dictionary precise is the primary task of the course. [/motivation] ## The Course Roadmap The course proceeds in several stages, each introducing new geometry to overcome limitations of the previous setting. We begin in Chapter 2 with **affine varieties** — vanishing sets in $k^n$ for an algebraically closed field $k$. Here we develop the fundamental correspondences between geometry (vanishing sets) and algebra (ideals and coordinate rings), culminating in Hilbert's Nullstellensatz, which is the precise version of the statement that geometry and algebra see the same information. However, affine space has an intrinsic deficiency: parallel lines never meet, so intersection theory (counting how many times two curves cross) does not work as cleanly as degree considerations would suggest. Chapter 3 remedies this by introducing **projective varieties**, where we compactify $k^n$ by adding "points at infinity." In projective space, any two distinct lines in the plane meet in exactly one point, and Bezout's theorem counts intersections correctly. With projective varieties in hand, we turn to local structure in Chapter 4: **tangent spaces, smoothness, and dimension**. Tangent spaces are defined by linearising the polynomial equations, and a variety is smooth at a point if this linearisation captures the full local geometry. Dimension is defined geometrically and then shown to agree with the algebraic notion of Krull dimension. Chapter 5 specialises to **algebraic curves** — varieties of dimension one. In this setting, the theory becomes especially rich and concrete. Bezout's theorem gives a precise count of intersections, local coordinates allow us to study the curve near each point, and the notion of degree and ramification controls morphisms between curves. Chapter 6 treats **divisors on curves**, which provide a framework for studying rational functions (their zeros and poles) systematically. This leads to the **Riemann-Roch theorem**, one of the deepest results of the course, which relates the geometry of a curve (its genus) to the algebra of functions on it. The course concludes with two bonus lectures. The first introduces the **Weil conjectures**, which show how counting solutions over finite fields $\mathbb{F}_{p^n}$ is governed by the topology of the associated complex variety — a stunning connection between arithmetic, algebra, and topology. The second motivates the passage to **schemes**, explaining why the classical theory of varieties over algebraically closed fields is insufficient for modern number-theoretic applications. [remark: Base Field] Throughout, we work over an algebraically closed field $k$ unless stated otherwise. The algebraic closure assumption is essential for many of the fundamental theorems (Nullstellensatz, Bezout) and ensures that the dictionary between geometry and algebra is as clean as possible. The most important example is $k = \mathbb{C}$. [/remark] Having identified the three central characters of algebraic geometry—vanishing sets, coordinate rings, and ideals—we now make this correspondence precise by developing the foundational theory of affine varieties. Chapter 2 formalises vanishing sets as subsets of affine space and establishes the algebraic-geometric dictionary that will govern the entire course. # 2. Affine Varieties This chapter develops the foundational theory of affine varieties — the basic geometric objects of algebraic geometry. We begin by setting up the affine space and the correspondence between polynomial ideals and geometric loci, then explore the topology and decomposition theory of varieties, and finally build the algebraic machinery of coordinate rings, morphisms, and local rings that encode the geometry algebraically. The preceding introduction motivated the study of vanishing sets of polynomials; here we make that study precise. ## The Affine Space and Affine Varieties Our goal is to formalise the idea of a "vanishing set of polynomials" as a geometric object. Two pieces of data are involved: an ambient set in which the points live, and a system of polynomials whose simultaneous zeros we want to single out. Both matter — the same polynomial can have wildly different zero sets over different fields (consider $X^2 + 1$ over $\mathbb{R}$ versus $\mathbb{C}$), and the polynomial ring controls which functions count as "regular" on the resulting locus. To pin this down cleanly, we work over a fixed algebraically closed field $k$. The algebraic closure assumption is essential: it ensures that non-trivial polynomial equations have solutions, which underlies the geometric meaning of every construction that follows. [definition: Affine Space] The **affine space of dimension $n$ over $k$**, denoted $\mathbb{A}^n_k$, is the set $k^n$ equipped with the ring $k[X_1, \ldots, X_n]$ of polynomial functions. [/definition] The affine space is the ambient universe in which affine varieties live. Unlike a bare vector space, $\mathbb{A}^n_k$ remembers the algebra of polynomial functions — this interplay between geometry (the point set $k^n$) and algebra (the ring $k[X_1, \ldots, X_n]$) is the core theme of the chapter. [definition: Affine Variety] An **affine variety** is any subset of $\mathbb{A}^n_k$ of the form \begin{align*} V(S) = \{p \in \mathbb{A}^n_k : f(p) = 0 \text{ for all } f \in S\} \end{align*} where $S \subset k[X_1, \ldots, X_n]$. [/definition] An affine variety is thus the simultaneous zero locus of a collection of polynomials. We record several immediate observations that are used repeatedly: [remark: Basic Properties of V] 1. If $S$ contains a non-zero element of $k$, then $V(S) = \varnothing$. 2. If $S \subset S'$, then $V(S) \supset V(S')$ — more equations cut out a smaller locus. 3. If $I_S$ is the ideal generated by $S$, then $V(I_S) = V(S)$, since any $p$ vanishing on $S$ also vanishes on every element of $I_S$. [/remark] Observation (3) means we lose nothing by assuming $S$ is already an ideal. This connects the geometry to commutative algebra in a precise way. The ring $k[X_1, \ldots, X_n]$ is both a unique factorisation domain and, by the Hilbert Basis Theorem, a Noetherian ring. Noetherianness forces every ideal to be finitely generated. Geometrically, this means that every variety is an intersection of finitely many hypersurfaces: \begin{align*} V(S) = V(I_S) = V(f_1, \ldots, f_r) = \bigcap_{i=1}^r V(f_i) \end{align*} for some $f_1, \ldots, f_r \in I_S$. So no matter how many polynomial conditions we impose, the resulting variety is already cut out by finitely many. [example: Lines and Their Unions] Take $\ell_1(X,Y) = a_1 X + b_1 Y + c_1 \in k[X,Y]$. Then $L_1 = V(\ell_1) \subset \mathbb{A}^2_k$ is a line. If $\ell_2(X,Y) = a_2 X + b_2 Y + c_2$ is another linear polynomial, then $L_1 \cup L_2 = V(\ell_1 \ell_2)$, since a point vanishes on the product $\ell_1 \ell_2$ if and only if it vanishes on at least one factor. More generally, for any linear polynomials $\ell_1, \ldots, \ell_d \in k[X,Y]$, the variety $V(\ell_1 \cdots \ell_d)$ is the union of the lines $V(\ell_i)$. With generic coefficient choices, any two of these lines meet at exactly one point and no three of them share a common point. [/example] The previous example shows how varieties can be built from simpler pieces by varying the polynomial data. The next example illustrates what happens when we allow parameters to vary and watch the geometry change continuously. [example: A Degeneration — The Nodal Cubic] Consider the family of curves $V(Y^2 - X(X-\lambda)(X+1)) \subset \mathbb{A}^2_k$ over $k = \mathbb{C}$, parametrised by $\lambda$. At $\lambda = 1$ this gives the smooth elliptic curve $Y^2 = X(X-1)(X+1)$. As $\lambda \to 0$, the two roots $X = 0$ and $X = \lambda$ collide, and the curve degenerates. Over $\mathbb{R}$, the two separate ovals are drawn toward each other and meet at a singularity; over $\mathbb{C}$, one circular section of the torus shrinks to a point. This is a concrete instance of a geometric degeneration within a family of varieties. [/example] [illustration:ag-cubic-degeneration-family] ## Ideal–Variety Correspondence How does the algebraic structure of ideals interact with the geometry of varieties? Concretely: if we know ideals $I$ and $J$ separately, can we compute the variety of their intersection or sum purely in terms of $V(I)$ and $V(J)$? The following proposition answers this completely. [quotetheorem:2121] [citeproof:2121] The first part extends readily to arbitrary sums of ideals, so arbitrary intersections of varieties are again varieties. What the theorem does not say is that the variety of a product $IJ$ is the same as $V(I) \cup V(J)$: in general $V(IJ)$ could be larger because elements of $IJ$ need not vanish wherever every element of $I$ or $J$ does. The correct relationship is $V(IJ) = V(I \cap J) = V(I) \cup V(J)$ when $I$ and $J$ are radical ideals, a point the Nullstellensatz will make precise. ## Zariski Topology and Irreducibility Having set up the dictionary between ideals and varieties, we ask: does this correspondence carry topological information? The class of affine varieties is closed under arbitrary intersections (sums of ideals) and under finite unions (intersections of ideals), and it contains both the empty set $V(1) = \varnothing$ and the whole ambient space $V(0) = \mathbb{A}^n_k$. These are exactly the axioms for a system of closed sets. So the polynomial vanishing loci themselves declare what the natural topology should be — there is essentially no choice in the matter. [definition: Zariski Topology] The **Zariski topology** on $\mathbb{A}^n_k$ is the topology whose closed sets are the affine varieties $V(S)$ for $S \subset k[X_1, \ldots, X_n]$. [/definition] This is indeed a topology: the empty set $\varnothing = V(1)$ and the whole space $\mathbb{A}^n_k = V(0)$ are closed, arbitrary intersections of varieties are varieties (corresponding to sums of ideals), and finite unions of varieties are varieties (corresponding to products or intersections of ideals). The open sets are complements of varieties. In dimension one, the proper closed sets are finite sets of points, so the Zariski topology is much coarser than the Euclidean topology. A central concept is the irreducibility of a variety, which corresponds to the algebraic notion of a prime ideal. Irreducibility captures whether a variety has any "hidden factor structure" — whether it can be expressed as a union of two smaller varieties. [definition: Irreducibility] A variety $V \subset \mathbb{A}^n_k$ is **irreducible** if whenever $V = V_1 \cup V_2$ with $V_1, V_2$ closed, then $V_1 = V$ or $V_2 = V$. A variety is **reducible** if it is not irreducible. [/definition] The definition says that an irreducible variety cannot be split into two proper closed pieces. Before seeing examples, note the contrast with the topology: while every variety is closed, irreducibility is an additional structural condition. [example: A Reducible Variety] The variety $V(XYZ) \subset \mathbb{A}^3_k$ is reducible because \begin{align*} V(XYZ) = V(X) \cup V(Y) \cup V(Z), \end{align*} each factor giving a coordinate hyperplane. This decomposition cannot be further refined: each $V(X_i)$ is irreducible (corresponding to the prime ideal $(X_i)$). [/example] Reducible varieties always decompose into irreducible pieces, and the Noetherian property ensures this decomposition is finite: [quotetheorem:2122] [citeproof:2122] The Noetherian hypothesis is genuinely needed. In a non-Noetherian setting one can construct an infinite strictly descending chain of closed subsets that never terminates: for example, in the ring $k[X_1, X_2, X_3, \ldots]$ with infinitely many variables, the chain \begin{align*} V(X_1) \supsetneq V(X_1, X_2) \supsetneq V(X_1, X_2, X_3) \supsetneq \cdots \end{align*} is strictly descending (each step cuts out one more coordinate hyperplane) and never stabilises. Without finite generation of ideals, the recursive splitting argument above never reaches an irreducible piece. This is one of many places where the Hilbert Basis Theorem does crucial work in the background. Together, these results mean every affine variety has a canonical decomposition into irreducible components, exactly analogous to prime factorisation. The number of components and how they are arranged is the first geometric invariant of a variety; in Chapter 4 (see the section on Dimension via Transcendence Degree) we attach a precise *dimension* to each component, defined both as the length of the longest chain of irreducible subvarieties and as the transcendence degree of the function field, and show the two definitions agree. ## The Ideal of a Variety and Nullstellensatz We have mapped ideals to varieties via $I \mapsto V(I)$. The reverse direction assigns to each variety its ideal. [definition: Ideal of a Variety] Given a variety $V \subset \mathbb{A}^n_k$, its **ideal** is \begin{align*} I(V) = \{f \in k[X_1, \ldots, X_n] : f(p) = 0 \text{ for all } p \in V\}. \end{align*} [/definition] This is indeed an ideal (closed under addition and under multiplication by any polynomial). Several basic properties of this correspondence deserve notice: [remark: Properties of I(V)] 1. If $V = V(S)$, then $S \subset I(V)$. 2. $V(I(V)) = V$ for any variety $V$. 3. $V = W$ if and only if $I(V) = I(W)$ as subsets of $\mathbb{A}^n_k$. However, $I$ and $V$ are not exact inverses: for example, $I = (X^2) \subset k[X]$ gives $I(V(I)) = (X) \supsetneq I$. The passage $I \mapsto V(I) \mapsto I(V(I))$ produces a strictly larger ideal whenever $I$ is not a radical ideal. [/remark] The previous remark raises the key question: precisely which ideals arise as $I(V)$ for some variety $V$? The answer — that exactly the radical ideals do — is the content of Hilbert's Nullstellensatz, the central theorem connecting algebra and geometry. We first record the weak form, which captures the most basic meaning of algebraic closure. [quotetheorem:2123] [citeproof:2123] The weak Nullstellensatz genuinely requires algebraic closure. Over $\mathbb{R}$, the ideal $I = (X^2 + 1) \subset \mathbb{R}[X]$ is proper (since $X^2 + 1$ is irreducible over $\mathbb{R}$) yet $V(I) = \varnothing$: the polynomial $X^2 + 1$ has no real roots. The failure is exactly the failure of algebraic closure — $X^2 + 1$ does have roots in $\mathbb{C}$, and over $\mathbb{C}$ the variety $V(X^2 + 1) = \{i, -i\}$ is non-empty. This example shows the theorem is tight: without algebraic closure, proper ideals need not have geometric realisation. Before stating the strong form, we need the notion of a radical. [definition: Radical Ideal] The **radical** of an ideal $I \subset k[X_1, \ldots, X_n]$ is \begin{align*} \sqrt{I} = \{f \in k[X_1, \ldots, X_n] : \exists\, m \geq 1,\, f^m \in I\}. \end{align*} [/definition] The radical captures all polynomials that vanish "with multiplicity" on the zero set of $I$: if $f^m$ vanishes at every point where $I$ vanishes, then $f$ itself must vanish there too. This is exactly what $I(V(I))$ records — so one should expect $I(V(I)) = \sqrt{I}$, which is what the strong Nullstellensatz asserts. [quotetheorem:2124] [citeproof:2124] The Nullstellensatz is sharp in two directions. First, the algebraically closed hypothesis is necessary: over $\mathbb{R}$, the ideal $I = (X^2 + 1)$ satisfies $V(I) = \varnothing$ yet $I(V(I)) = I(\varnothing) = \mathbb{R}[X] \neq I$, so the equality $I(V(I)) = \sqrt{I}$ fails (here $\sqrt{I} = I$ since $X^2 + 1$ is already radical over $\mathbb{R}$, but $I(\varnothing) = k[X]$ is the whole ring). Second, the theorem does not say that every ideal is radical — there are many non-radical ideals. It says only that passing from an ideal $I$ to its variety and back always yields the radical $\sqrt{I}$. The Nullstellensatz completes the correspondence: the maps $V$ and $I$ set up a bijection between radical ideals in $k[X_1, \ldots, X_n]$ and affine varieties in $\mathbb{A}^n_k$. Under this bijection, the prime ideals correspond to irreducible varieties. A striking corollary characterises all maximal ideals of the polynomial ring: [quotetheorem:2125] [citeproof:2125] This provides a precise dictionary: points of $\mathbb{A}^n_k$ correspond bijectively to maximal ideals of $k[X_1, \ldots, X_n]$. The theorem fails over non-algebraically-closed fields: over $\mathbb{R}$, the ideal $(X^2 + 1)$ is maximal in $\mathbb{R}[X]$ (with quotient $\mathbb{C}$), yet it has no form $(X - a)$ for any $a \in \mathbb{R}$. Over $\mathbb{C}$, by contrast, every maximal ideal of $\mathbb{C}[X]$ is $(X - a)$ for some $a \in \mathbb{C}$, in accordance with the theorem. ## Irreducibility and Prime Ideals A variety can look irreducible just by inspection, but we need an algebraic criterion to prove irreducibility rigorously. The question is: what algebraic property of $I(V)$ captures whether $V$ can be split? The answer involves prime ideals, and the connection runs in both directions. [quotetheorem:2126] [citeproof:2126] The argument reveals the precise mechanism: a variety is reducible exactly when its ideal contains a product $fg$ without containing either factor — the algebraic definition of a non-prime ideal. The theorem connects seamlessly with the Nullstellensatz: under the bijection between radical ideals and varieties, prime ideals correspond to irreducible varieties and maximal ideals correspond to points (the "smallest" irreducible varieties). [example: Irreducible Varieties] The full affine space $\mathbb{A}^n_k$ is irreducible, since $I(\mathbb{A}^n_k) = (0)$ is prime. The cuspidal cubic $V(Y^2 - X^3) \subset \mathbb{A}^2_k$ is also irreducible: the ideal $(Y^2 - X^3)$ is prime because $Y^2 - X^3$ is irreducible in $k[X,Y]$, which is a UFD, so the ideal it generates is prime. [/example] The theorem gives a clean algebraic criterion for irreducibility, and also shows that the irreducibility of $V$ is really a property of the quotient ring $k[X_1, \ldots, X_n]/I(V)$: the variety is irreducible if and only if this quotient is an integral domain. This quotient is the central object of the next section. ## Coordinate Rings and Morphisms Given that the irreducibility of $V$ depends only on $k[X_1, \ldots, X_n]/I(V)$ as a ring, one is led to ask whether all intrinsic geometric properties of $V$ are encoded in this quotient — not just in the specific embedding $V \hookrightarrow \mathbb{A}^n_k$. To make this precise, we need to define what "intrinsic" means by specifying what the maps between varieties are. [definition: Coordinate Ring] For an affine variety $V \subset \mathbb{A}^n_k$, the **coordinate ring** is \begin{align*} \mathcal{O}_V = k[X_1, \ldots, X_n] / I(V). \end{align*} [/definition] The coordinate ring is simultaneously a $k$-algebra (a ring equipped with a compatible $k$-vector space structure) and captures all polynomial functions on $V$: an element $f \in \mathcal{O}_V$ is a well-defined function $f : V \to k$ obtained by evaluation (well-defined because we have quotiented out $I(V)$, the ideal of functions that vanish on $V$). The points of $V$ correspond to maximal ideals of $\mathcal{O}_V$. An important caveat: the isomorphism class of $\mathcal{O}_V$ as a $k$-algebra does not remember the embedding of $V$ into the ambient affine space. The embedding can be recovered only if we also specify the surjection $k[X_1, \ldots, X_n] \to \mathcal{O}_V$, which fixes a set of generators. But losing the embedding information turns out to be a feature, not a bug — we want to study the intrinsic geometry of $V$, not how it sits in $\mathbb{A}^n_k$. Before establishing the intrinsic characterisation, we need to formalise what it means for two varieties to be "the same". This requires the notion of a morphism. [definition: Morphism of Affine Varieties] A **morphism** from $V \subset \mathbb{A}^n_k$ to $\mathbb{A}^1_k$ is a map $V \to \mathbb{A}^1_k$ given by evaluation of some $f \in \mathcal{O}_V$. More generally, a morphism $V \to \mathbb{A}^m_k$ is a map given by evaluation of some $(f_1, \ldots, f_m) \in \mathcal{O}_V^m$. If $W \subset \mathbb{A}^m_k$ is another affine variety, a **morphism** $\varphi : V \to W$ is a morphism $V \to \mathbb{A}^m_k$ whose image is contained in $W$. An **isomorphism** is a morphism with a two-sided inverse. [/definition] [example: Morphisms Between Affine Varieties] The maps $\mathbb{A}^2_k \to \mathbb{A}^1_k$ given by $(x,y) \mapsto x$, $(x,y) \mapsto x^2$, and $(x,y) \mapsto x^2 y$ are all morphisms — each is given by evaluating a polynomial. The map $(x,y) \mapsto y$ restricts to a morphism $V(X) \to \mathbb{A}^1_k$, which is an isomorphism: the inverse sends $t \mapsto (0,t)$. More generally, for any linear coordinate hyperplane $V(X_1, \ldots, X_n) \subset \mathbb{A}^N_k$, the projection to the remaining coordinates gives an isomorphism $V(X_1, \ldots, X_n) \cong \mathbb{A}^{N-n}_k$. [/example] For morphisms $\varphi : V \to W$, we can construct a ring homomorphism in the opposite direction. Given $f \in \mathcal{O}_W$, the **pullback** $\varphi^*(f) = f \circ \varphi$ is an element of $\mathcal{O}_V$. This defines a $k$-algebra homomorphism $\varphi^* : \mathcal{O}_W \to \mathcal{O}_V$. Conversely, given a $k$-algebra homomorphism $\varphi^* : \mathcal{O}_W \to \mathcal{O}_V$, we can recover a morphism $\varphi : V \to W$ by $\varphi = (\varphi^*(X_1), \ldots, \varphi^*(X_m))$. This shows that morphisms of varieties and $k$-algebra homomorphisms of coordinate rings are in perfect correspondence, with the direction of the maps reversed. [quotetheorem:2127] [citeproof:2127] This theorem is the central result of the chapter: all intrinsic geometric information about an affine variety is encoded in its coordinate ring. Isomorphism classes of affine varieties correspond exactly to isomorphism classes of finitely generated, nilpotent-free $k$-algebras. The nilpotent-free condition is essential: if $\mathcal{O}_V$ had nilpotents, the variety would carry "phantom" algebraic structure not visible at the level of points. For example, the ideals $(X)$ and $(X^2)$ in $k[X]$ define the same geometric point $V(X) = V(X^2) = \{0\} \subset \mathbb{A}^1_k$, but the quotient $k[X]/(X^2)$ has the nilpotent element $X \bmod X^2$. Insisting on nilpotent-free ensures the coordinate ring faithfully reflects the underlying point set. Conversely, every nilpotent-free finitely generated $k$-algebra arises as a coordinate ring: given generators, take the quotient of the polynomial ring by the kernel of the surjection. [example: A Coordinate Ring That is Not a Polynomial Ring] Consider the $k$-algebra $k[X^2, X^3] \subset k[X]$. This algebra is not a polynomial ring — it is isomorphic to $k[Y_1, Y_2]/(Y_1^3 - Y_2^2)$, the coordinate ring of the cuspidal cubic $V(Y^2 - X^3) \subset \mathbb{A}^2_k$. The relation $Y_1^3 = Y_2^2$ reflects the defining equation of the cusp. [/example] ## Abstract Affine Varieties and Quasi-Affine Varieties The algebraic characterisation of affine varieties suggests defining them abstractly, without reference to an ambient affine space. [definition: Maximal Spectrum] The **maximal spectrum** $\mathrm{mSpec}(R)$ of a nilpotent-free finitely generated $k$-algebra $R$ is the set of maximal ideals of $R$, topologised by declaring the closed sets to be $\{V(I) : I \subset R \text{ an ideal}\}$. By the Nullstellensatz, $\mathrm{mSpec}(R)$ is naturally a space of points. [/definition] The maximal spectrum translates the algebraic data of $R$ back into a geometric object, using the Nullstellensatz to ensure that maximal ideals genuinely correspond to points. This is the key step that allows us to pass from algebra to geometry without fixing an ambient affine space. [definition: Abstract Affine Variety] An **abstract affine variety** is a pair $(V, R)$ where $V$ is a topological space, $R$ is a nilpotent-free finitely generated $k$-algebra, and $V \cong \mathrm{mSpec}(R)$. We write $\mathcal{O}_V = R$. A **closed subvariety** of $(V, R)$ is a subset $W \subset V$ of the form $V(I)$ for an ideal $I \subset R$. [/definition] This definition frees us from a specific embedding and makes precise the idea that varieties are geometric objects in their own right. The Zariski topology is coarser than classical topologies, but its open sets carry natural function theory. To understand functions on open sets, we need to allow denominators. [example: Functions on $\mathbb{A}^1_k \setminus \{0\}$] Consider $\mathbb{A}^1_k \setminus \{0\} \cong k^\times$. Every polynomial on $\mathbb{A}^1_k$ restricts to this open set. Among them, those that are nowhere vanishing on $k^\times$ are precisely the monomials $\lambda X^n$ for $\lambda \in k^\times$ and $n \geq 0$. These may be used as denominators, giving well-defined functions of the form $f(X)/X^n$ on $k^\times$. The natural ring of functions on $\mathbb{A}^1_k \setminus \{0\}$ is therefore $k[X, X^{-1}]$, the ring of Laurent polynomials. [/example] For an irreducible affine variety $V$, the coordinate ring $\mathcal{O}_V$ is an integral domain (since $I(V)$ is prime), and we may form its field of fractions: \begin{align*} \mathcal{O}_V(\eta) = \mathrm{FF}(\mathcal{O}_V). \end{align*} This is the **function field** of $V$. Any $f/g \in \mathcal{O}_V(\eta)$ defines a function on the open set $V \setminus V(g)$. Given an open set $U \subset V$, the ring of regular functions on $U$ is \begin{align*} \mathcal{O}_V(U) = \{f/g \in \mathcal{O}_V(\eta) : g(p) \neq 0 \text{ for all } p \in U\}. \end{align*} [definition: Quasi-Affine Variety] A **quasi-affine variety** is a pair $(U, R)$ where $U$ is a Zariski-open subset of some affine variety $V$, with $R \cong \mathcal{O}_V(U)$. [/definition] One might expect that removing a closed subvariety from an affine variety always enlarges the ring of regular functions. The following example shows this expectation is wrong in sufficiently high dimension. [example: The Complement of the Origin in $\mathbb{A}^2_k$] Take $U = \mathbb{A}^2_k \setminus \{(0,0)\}$. One might expect the ring of regular functions $\mathcal{O}(U)$ to be strictly larger than $k[X,Y]$, since $U$ is a proper open subset. In fact, $\mathcal{O}(U) = k[X,Y]$: every rational function in two variables that is regular everywhere on $U$ is already a polynomial. This is a consequence of Hartogs' theorem in the algebraic setting — removing a codimension-two subvariety does not create new regular functions. It contrasts with the one-variable case, where $\mathbb{A}^1_k \setminus \{0\}$ does have new regular functions ($X^{-1}$). [/example] Also note that $k[X, X^{-1}] \cong k[X,Y]/(XY-1)$, so the quasi-affine variety $\mathbb{A}^1_k \setminus \{0\}$ has the same ring of regular functions as the affine variety $V(XY - 1) \subset \mathbb{A}^2_k$. This is not a coincidence — it reflects that these two spaces are isomorphic as abstract varieties. ## Local Rings and the Cotangent Space While $\mathcal{O}_V(\eta)$ is defined globally (as a field), its elements are only regular on certain open sets. To capture the local behaviour of functions near a specific point, we pass to the local ring. [definition: Local Ring at a Point] Let $V$ be an irreducible affine variety and $p \in V$. An element $h \in \mathcal{O}_V(\eta)$ is **regular at $p$** if it has a representative $h = f/g$ with $g(p) \neq 0$. The **local ring** (or **localisation**) of $V$ at $p$ is \begin{align*} \mathcal{O}_{V,p} = \{h \in \mathcal{O}_V(\eta) : h \text{ is regular at } p\}. \end{align*} [/definition] The local ring has a natural maximal ideal $\mathfrak{m}_p = \{h \in \mathcal{O}_{V,p} : h(p) = 0\}$, and the quotient $\mathcal{O}_{V,p}/\mathfrak{m}_p \cong k$. Every element of $\mathcal{O}_{V,p} \setminus \mathfrak{m}_p$ is invertible (since if $h(p) \neq 0$, the function $1/h$ is regular near $p$). This makes $\mathcal{O}_{V,p}$ a local ring in the sense of commutative algebra. [definition: Local Ring (Commutative Algebra)] A commutative ring $R$ is **local** if it has a unique maximal ideal. [/definition] The definition is concise, but what makes it useful is the following algebraic criterion, which shows that "local" is equivalent to a very concrete property of the units. [quotetheorem:2128] [citeproof:2128] The criterion says: a ring is local precisely when its non-units form an ideal, which is then automatically the unique maximal ideal. In practice, this lets us verify locality by checking that $1 - xy$ is a unit whenever $x$ is a non-unit and $y$ is arbitrary — or equivalently, that the sum of two non-units is a non-unit. It is instructive to see what fails in a non-local ring. Take $R = k[X]$: this ring has infinitely many maximal ideals — by the Nullstellensatz (over algebraically closed $k$), every $a \in k$ gives a distinct maximal ideal $(X - a)$. The set of non-units of $k[X]$ is the set of non-constant polynomials together with $0$, and this set is *not* an ideal: for instance, $X$ and $1 - X$ are both non-units, but $X + (1 - X) = 1$ is a unit. So the criterion of the theorem fails, exactly because the non-units are spread across infinitely many distinct maximal ideals. This illustrates *why* localising at a point matters: passing from $k[X]$ to $\mathcal{O}_{\mathbb{A}^1_k, 0}$ collapses all the maximal ideals except $(X)$ into units, leaving exactly one maximal ideal — the one corresponding to the chosen point. Local rings are the natural home for studying a variety near a single point because they have stripped away all information about other points. [example: The Local Ring of $\mathbb{A}^1_k$ at the Origin] Take $V = \mathbb{A}^1_k$ and $p = 0$. The local ring is \begin{align*} \mathcal{O}_{\mathbb{A}^1_k, 0} = \{f/g : f,g \in k[X],\, g(0) \neq 0\}, \end{align*} that is, rational functions whose denominator does not vanish at $0$. For instance, $(X-1)^{-1} \in \mathcal{O}_{\mathbb{A}^1_k, 0}$, since $X - 1$ is non-vanishing at $0$. Thinking of elements as power series around $0$, an element $h$ belongs to the local ring precisely when its Laurent expansion has no negative powers of $X$; it is a unit if and only if its constant term is non-zero. The maximal ideal $\mathfrak{m}_0 = (X)$ consists of functions vanishing at $0$, and $\mathfrak{m}_0^2 = (X^2)$ consists of functions vanishing to order $2$. [/example] This analogy with power series motivates a key definition: the **cotangent space**, which linearises the maximal ideal. [definition: Zariski Cotangent Space] For a point $p \in V$, the **Zariski cotangent space** to $V$ at $p$ is \begin{align*} T^*_{V,p} = \mathfrak{m}_p / \mathfrak{m}_p^2, \end{align*} viewed as a $k$-vector space. [/definition] Geometrically, $T^*_{V,p}$ captures the linear parts of functions vanishing at $p$: modding out by $\mathfrak{m}_p^2$ kills all terms of order $2$ or higher, leaving only the linear (cotangent) information. The **Zariski tangent space** $T_{V,p}$ is the dual $k$-vector space $(T^*_{V,p})^*$. Before formalising smoothness, a word about *dimension*. So far we have used the term informally — "the cuspidal cubic is one-dimensional", "$V(XY)$ is the union of two lines, each of dimension one" — without a precise definition. Dimension theory is the subject of Chapter 4, where we develop two equivalent definitions: a geometric one in terms of chains of irreducible subvarieties, and an algebraic one (Krull dimension) in terms of chains of prime ideals in the coordinate ring. For the rest of this section we use $\dim V$ as an informal placeholder for this notion; the smoothness criterion below will be re-expressed in Chapter 4 as a Jacobian rank condition that does not require dimension to be defined first. For an irreducible variety $V$, the cotangent space dimension is bounded below by $\dim V$ at every point, and equals $\dim V$ exactly at the "generic" points where $V$ looks locally like affine space. At more degenerate points, the cotangent space jumps strictly higher — the tangent space becomes too big to fit inside the variety. This gives the correct pointwise notion of smoothness: [definition: Smoothness] Let $V$ be an affine variety, and let $p \in V$. Heuristically, $p$ is **smooth** (or **non-singular**) if \begin{align*} \dim_k T^*_{V,p} \le \text{the local dimension of } V \text{ at } p, \end{align*} where the local dimension is the dimension of the unique irreducible component of $V$ containing $p$ (when $p$ lies on exactly one component). When $V$ is irreducible of dimension $d$, this reduces to $\dim_k T^*_{V,p} = d$ (the inequality $\dim_k T^*_{V,p} \ge d$ is automatic, hence equality at smooth points). The variety $V$ is **smooth** if every point of $V$ is smooth. The notion of dimension used here is developed in Chapter 4. The equivalent Jacobian rank criterion will be proved there and serves as the working definition of smoothness in practice. [/definition] The dimension of $T^*_{V,p}$ is a local invariant that measures how complicated the variety is near $p$. At a smooth point, this dimension matches the local dimension of $V$; at a singular point, it is strictly larger. [example: A Singular Variety] Consider $V = V(XY) \subset \mathbb{A}^2_k$, the union of the two coordinate axes. Each component $V(X)$ and $V(Y)$ is a copy of $\mathbb{A}^1_k$, hence one-dimensional, so the local dimension at any point is $1$. At the smooth point $p = (1,0)$, which lies only on $V(Y)$, the local ring is determined by the single equation $Y = 0$. The maximal ideal of $\mathcal{O}_{V,p}$ is generated by $Y$ (the function $X - 1$ vanishes at $p$ but is invertible in $\mathcal{O}_{V,p}$ near $p$ once we work modulo $XY = 0$, since locally $X \neq 0$), and $\mathfrak{m}_p/\mathfrak{m}_p^2$ is one-dimensional with basis $\{Y \bmod \mathfrak{m}_p^2\}$. So $\dim_k T^*_{V,p} = 1$, matching the local dimension — the point is smooth. At the origin $p = 0$, the maximal ideal of $\mathcal{O}_{V,0}$ is generated by both $X$ and $Y$, subject to the relation $XY = 0$. Concretely, $\mathfrak{m}_0 = (X, Y)/(XY)$ and \begin{align*} \mathfrak{m}_0^2 = (X^2, XY, Y^2)/(XY) = (X^2, Y^2)/(XY), \end{align*} so the quotient $\mathfrak{m}_0/\mathfrak{m}_0^2$ has $\{X \bmod \mathfrak{m}_0^2,\ Y \bmod \mathfrak{m}_0^2\}$ as a basis — the relation $XY = 0$ does not produce a relation between $X$ and $Y$ modulo $\mathfrak{m}_0^2$, since $XY$ already lies in $(X^2, Y^2)$ (it is sent to $0$ in the quotient by virtue of $XY = 0$, and is in any case quadratic). Hence $\dim_k T^*_{V,0} = 2$. But the local dimension of $V$ at $0$ is still $1$ (each component through $0$ is one-dimensional), and $2 > 1$. The cotangent space is too big — algebraically, the two branches of $V$ pull the tangent space in two independent directions at the origin, while the variety itself has only one dimension's worth of room. The origin is a singular point, and $V(XY)$ is not smooth. [/example] The cotangent space definition of smoothness is purely algebraic and will be complemented in Chapter 4 by the geometric definition via the Jacobian criterion. The two definitions agree and together give the first tool for understanding the local structure of varieties. [illustration:ag-singular-v-xy-tangent] With the foundations of affine geometry in place, we confront a fundamental limitation: parallel lines never meet in affine space, which disrupts the count of intersections predicted by degree. Chapter 3 remedies this deficiency by adjoining points at infinity, constructing projective space where any two distinct lines meet exactly once. # 3. Projective Varieties Projective varieties enlarge the affine picture by adjoining "points at infinity" in a way that resolves several defects of affine geometry: parallel lines now meet, generic curves of degree $d_1$ and $d_2$ intersect in $d_1 d_2$ points (counted with multiplicity), and finite-type objects become proper. This chapter develops projective space, the projective Nullstellensatz, regular and rational maps in this setting, and how to form products inside the projective world. ## The Projective Space [motivation] ### The Problem with Affine Space Working exclusively in affine space $\mathbb{A}^n_k$ leads to an unsatisfying asymmetry: two distinct lines in $\mathbb{A}^2_k$ either intersect in exactly one point, or they are parallel and fail to meet at all. Whether two lines intersect depends not just on the lines themselves but on their relative slopes — a piece of data that feels extrinsic. We want a setting in which any two distinct lines meet, in exactly one point, without exception. The remedy is to add "points at infinity" — one for each slope — so that parallel lines meet there. The slope of a line in $\mathbb{A}^2_k$ is an element of $k \sqcup \{\infty\}$, and this set is exactly what we will call $\mathbb{P}^1_k$. We then form $\mathbb{P}^2_k = \mathbb{A}^2_k \sqcup \mathbb{P}^1_k$, where the "boundary" $\mathbb{P}^1_k$ encodes all intersections at infinity. Continuing inductively, $\mathbb{P}^3_k = \mathbb{A}^3_k \sqcup \mathbb{P}^2_k$, and so on. This inductive construction is intuitively clear, but rather awkward to work with. The cleaner — and ultimately more powerful — approach is to define projective space directly via an equivalence relation. [/motivation] ### Projective Space and Homogeneous Coordinates [definition: Projective Space] The **projective space** $\mathbb{P}^n_k$ of dimension $n$ over a field $k$ is the set \begin{align*} \mathbb{P}^n_k := \bigl(k^{n+1} \setminus \{(0,\ldots,0)\}\bigr) \big/ {\sim} \end{align*} where the equivalence relation is \begin{align*} (x_0, \ldots, x_n) \sim (\lambda x_0, \ldots, \lambda x_n) \qquad \text{for all } \lambda \in k^\times. \end{align*} [/definition] Geometrically, $\mathbb{P}^n_k$ is the set of all lines through the origin in $k^{n+1}$: each equivalence class $[x_0 : \cdots : x_n]$ consists of all nonzero scalar multiples of the vector $(x_0, \ldots, x_n)$. [definition: Homogeneous Coordinates] A point of $\mathbb{P}^n_k$ is specified by a tuple $(a_0, \ldots, a_n) \neq (0,\ldots,0)$, which determines the equivalence class. We write this class as $[a_0 : \cdots : a_n]$ and call this the **homogeneous coordinates** of the point. The representation is not unique: $[a_0 : \cdots : a_n] = [\lambda a_0 : \cdots : \lambda a_n]$ for any $\lambda \in k^\times$. [/definition] [remark: Well-definedness of Vanishing Conditions] Although the coordinates of a point in $\mathbb{P}^n_k$ are only defined up to a common scalar, the condition $x_i = 0$ is well-defined: if $(x_0,\ldots,x_n) \sim (\lambda x_0, \ldots, \lambda x_n)$, then $x_i = 0$ if and only if $\lambda x_i = 0$. This means subsets of the form $\{x_i = 0\}$ are well-defined subsets of $\mathbb{P}^n_k$, a fact we exploit immediately. [/remark] ### Affine Charts The link between $\mathbb{P}^n_k$ and the familiar affine space $\mathbb{A}^n_k$ is made precise through affine charts (also called affine patches). For each $0 \leq i \leq n$, define the open subset \begin{align*} U_i := \{[x_0 : \cdots : x_n] \in \mathbb{P}^n_k : x_i \neq 0\}. \end{align*} On $U_i$, every point has a unique representative with $x_i = 1$: namely \begin{align*} [x_0 : \cdots : x_n] = \left[\frac{x_0}{x_i} : \cdots : 1 : \cdots : \frac{x_n}{x_i}\right], \end{align*} where the $1$ occupies position $i$. This gives a bijection \begin{align*} U_i &\xrightarrow{\;\sim\;} \mathbb{A}^n_k, \qquad [x_0 : \cdots : x_n] \mapsto \left(\frac{x_0}{x_i}, \ldots, \widehat{\frac{x_i}{x_i}}, \ldots, \frac{x_n}{x_i}\right), \end{align*} (omitting the $i$-th coordinate, which equals $1$). Each $U_i$ is therefore a copy of $\mathbb{A}^n_k$ sitting inside $\mathbb{P}^n_k$. [quotetheorem:2129] [citeproof:2129] This cover is the rigorous version of the inductive construction: the complement of $U_0$ is $\{x_0 = 0\} \cong \mathbb{P}^{n-1}_k$, giving the decomposition \begin{align*} \mathbb{P}^n_k = U_0 \sqcup \{x_0 = 0\} \cong \mathbb{A}^n_k \sqcup \mathbb{P}^{n-1}_k. \end{align*} The piece $\{x_0 = 0\}$ is called the **hyperplane at infinity** relative to the chart $U_0$, and its points are the "points at infinity" that parallel lines in $U_0 \cong \mathbb{A}^n_k$ share. [illustration:ag-p2-affine-decomposition] [example: The Affine Line in Projective Space] Take $n = 1$. Then $\mathbb{P}^1_k$ has two affine charts: $U_0 = \{[x_0 : x_1] : x_0 \neq 0\} \cong \mathbb{A}^1_k$ via $[x_0:x_1] \mapsto x_1/x_0$, and $U_1 = \{[x_0:x_1] : x_1 \neq 0\} \cong \mathbb{A}^1_k$ via $[x_0:x_1] \mapsto x_0/x_1$. The complement of $U_0$ is the single point $[0:1]$, which is "the point at infinity" of the affine line $U_0$. Thus $\mathbb{P}^1_k = \mathbb{A}^1_k \sqcup \{\infty\}$, realising the familiar Riemann sphere picture over $k = \mathbb{C}$. [/example] ### Lines and Hyperplanes at Infinity The language of "points at infinity" depends on which affine chart we choose to call the "finite part." Different choices of chart $U_i$ give different hyperplanes at infinity, namely $\{x_i = 0\}$. There is nothing intrinsically infinite about any of these hyperplanes: the distinction between "affine" and "infinity" is a matter of perspective, not of the geometry of $\mathbb{P}^n_k$ itself. [example: Parallel Lines Meet at Infinity] Work in $\mathbb{P}^2_k$ with affine chart $U_0 = \{[1:x:y]\} \cong \mathbb{A}^2_k$. Consider two parallel lines in $\mathbb{A}^2_k$: \begin{align*} L_1: y = mx + c_1, \qquad L_2: y = mx + c_2, \quad c_1 \neq c_2. \end{align*} In homogeneous coordinates $[X_0 : X_1 : X_2]$ (where $x = X_1/X_0$, $y = X_2/X_0$), these become \begin{align*} L_1: X_2 = mX_1 + c_1 X_0, \qquad L_2: X_2 = mX_1 + c_2 X_0. \end{align*} Subtracting gives $(c_1 - c_2)X_0 = 0$, i.e. $X_0 = 0$. Substituting back yields $X_2 = mX_1$, so the unique intersection point is $[0 : 1 : m]$, which lies on the hyperplane at infinity $\{X_0 = 0\} \cong \mathbb{P}^1_k$. Two parallel lines with slope $m$ meet at exactly the point $[0:1:m] \in \mathbb{P}^1_k$. [/example] ### Limits in Projective Space and Algebraic Compactness One of the key virtues of projective space is that certain "limits" that fail to exist in $\mathbb{A}^n_k$ become well-defined in $\mathbb{P}^n_k$. [example: A Limit in Projective Space] In $\mathbb{A}^2_k$, consider the parametric curve $(t^2, t^3)$ as $t$ "tends to infinity." This does not converge to any point of $\mathbb{A}^2_k$. But in $\mathbb{P}^2_k$, embed $\mathbb{A}^2_k$ as $U_0 = \{[1:x:y]\}$, so the curve becomes $[1:t^2:t^3]$. Rescaling by $1/t^3$ (valid for $t \neq 0$): \begin{align*} [1 : t^2 : t^3] = [1/t^3 : 1/t : 1] \xrightarrow{t \to \infty} [0 : 0 : 1]. \end{align*} The point $[0:0:1]$ is a well-defined element of $\mathbb{P}^2_k$, lying at infinity relative to the chart $U_0$. Projective space has "absorbed" the limit. [/example] This phenomenon generalises: over $k = \mathbb{C}$, the quotient topology on $\mathbb{P}^n_\mathbb{C}$ induced from the standard Euclidean topology on $\mathbb{C}^{n+1} \setminus \{0\}$ is compact (by a standard compactness argument: the unit sphere in $\mathbb{C}^{n+1}$ surjects onto $\mathbb{P}^n_\mathbb{C}$). This compactness is the geometric underpinning of many finiteness theorems in algebraic geometry. ### The Zariski Topology on Projective Space Having described the affine charts, there are two natural ways to put a Zariski topology on $\mathbb{P}^n_k$. The first is via the affine cover. We declare a set $C \subset \mathbb{P}^n_k$ to be **closed** if and only if $C \cap U_i$ is Zariski closed in $U_i \cong \mathbb{A}^n_k$ for every $0 \leq i \leq n$. One checks that this defines a valid topology. However, this approach feels indirect — it defines the topology on $\mathbb{P}^n_k$ by reducing to the affine situation we already understand. The second approach uses polynomials directly, and requires the notion of a homogeneous polynomial. [definition: Homogeneous Polynomial] A polynomial $f \in k[X_0, \ldots, X_n]$ is **homogeneous of degree $d$** if \begin{align*} f(\lambda X_0, \ldots, \lambda X_n) = \lambda^d f(X_0, \ldots, X_n) \end{align*} for all $\lambda \neq 0$ (equivalently, every monomial in $f$ has total degree exactly $d$). [/definition] A general polynomial $f \in k[X_0, \ldots, X_n]$ does not define a function on $\mathbb{P}^n_k$: the value $f(x_0, \ldots, x_n)$ changes when we replace $(x_0, \ldots, x_n)$ by $(\lambda x_0, \ldots, \lambda x_n)$. But if $f$ is homogeneous of degree $d$, then $f(\lambda x_0, \ldots, \lambda x_n) = \lambda^d f(x_0, \ldots, x_n)$, so although the value is not well-defined, its vanishing is: $f(x_0, \ldots, x_n) = 0$ if and only if $f(\lambda x_0, \ldots, \lambda x_n) = 0$. The **vanishing locus** of a homogeneous polynomial is therefore a well-defined subset of $\mathbb{P}^n_k$. [example: Homogeneous vs Non-Homogeneous Polynomials] In $k[X_0, X_1, X_2]$: the polynomials $X_0$, $X_0 X_1$, $X_0 X_1 + X_2^2$, and $X_0 + X_1 + X_2$ are all homogeneous (of degrees $1$, $2$, $2$, and $1$ respectively). In contrast, $X_0^2 + X_1$ is not homogeneous (it mixes degrees $2$ and $1$), and neither is $X_0 + 1$ (which mixes degrees $1$ and $0$). One verifies: $f(\lambda X_0, \lambda X_1, \lambda X_2) = \lambda^2 X_0^2 + \lambda X_1 \neq \lambda^d(X_0^2 + X_1)$ for any fixed $d$. [/example] [definition: Projective Variety] A **projective variety** in $\mathbb{P}^n_k$ is the vanishing locus \begin{align*} V(S) := \{[x_0 : \cdots : x_n] \in \mathbb{P}^n_k : f(x_0, \ldots, x_n) = 0 \text{ for all } f \in S\} \end{align*} of a set $S \subset k[X_0, \ldots, X_n]$ consisting entirely of homogeneous polynomials. [/definition] [remark: The Ideal Generated by S] The ideal $(S) \subset k[X_0, \ldots, X_n]$ generated by a set $S$ of homogeneous polynomials may contain non-homogeneous elements, but its homogeneous generators are all that matter for the vanishing locus. One can also verify that the Zariski topology on $\mathbb{P}^n_k$ defined by declaring projective varieties to be closed sets coincides with the quotient topology descended from $\mathbb{A}^{n+1}_k = k^{n+1}$ (viewed with its Zariski topology). [/remark] ### Symmetries and First Examples $\mathbb{P}^n_k$ carries a large group of symmetries. The general linear group $\mathrm{GL}_{n+1}(k)$ acts on $k^{n+1}$, and since scalar matrices act as the identity on $\mathbb{P}^n_k$, this descends to a faithful action of the projective general linear group \begin{align*} \mathrm{PGL}_{n+1}(k) := \mathrm{GL}_{n+1}(k) / k^\times \end{align*} on $\mathbb{P}^n_k$. This action is transitive (any point can be sent to $[1:0:\cdots:0]$) and is highly transitive: any $(n+2)$ points in **general position** can be mapped to the standard points \begin{align*} [1:0:\cdots:0],\; [0:1:0:\cdots:0],\; \ldots,\; [0:\cdots:0:1],\; [1:1:\cdots:1]. \end{align*} **Linear subvarieties.** A projective subspace $\mathbb{P}^m \hookrightarrow \mathbb{P}^n$ is obtained by choosing an $(m+1)$-dimensional linear subspace $L \subset k^{n+1}$ and forming $(L \setminus \{0\})/k^\times \cong \mathbb{P}^m$. This is the vanishing locus of the $(n-m)$ independent linear homogeneous equations defining $L$, so $\mathbb{P}^m$ is a projective variety of dimension $m$. **Varieties in $\mathbb{P}^1_k$.** Let $f \in k[X_0, X_1]$ be homogeneous of degree $d$. Then $V(f) \subset \mathbb{P}^1_k$ consists of at most $d$ points. In fact, since $k$ is algebraically closed, $f$ factors completely as a product of $d$ linear factors, and counting with multiplicity we obtain exactly $d$ points. [definition: Multiplicity in $\mathbb{P}^1_k$] Let $f \in k[X_0, X_1]$ be homogeneous of degree $d$ and let $p \in V(f) \subset \mathbb{P}^1_k$. Suppose $p \in U_0 = \mathbb{P}^1_k \setminus \{X_0 = 0\}$, identified with $\mathbb{A}^1_k$ via $[a_0:a_1] \mapsto a_1/a_0$. Under this identification $f(X_0, X_1) = X_0^d \cdot f(1, X_1/X_0)$, so $f$ becomes a degree-$d$ polynomial in the variable $t = X_1/X_0$. The **multiplicity** $\mathrm{mult}_p(f)$ is the multiplicity of the root $t = a_1/a_0$ of this polynomial. One checks this is independent of the choice of chart, and \begin{align*} \sum_{p \in V(f)} \mathrm{mult}_p(f) = d. \end{align*} [/definition] **Quadric hypersurfaces.** Suppose $\mathrm{char}(k) \neq 2$ and let $f \in k[X_0, \ldots, X_n]$ be a homogeneous polynomial of degree $2$ (a quadratic form). Any quadratic form can be diagonalised over an algebraically closed field. [quotetheorem:2130] The course quotes this result without proof; it follows from the classification of non-degenerate quadratic forms over algebraically closed fields, where every such form is equivalent to $X_0^2 + \cdots + X_n^2$. [example: Quadrics in Low Dimensions] In $\mathbb{P}^1_k$: the polynomial $f = X_0^2 + X_1^2$ factors as $(X_0 - \lambda X_1)(X_0 + \lambda X_1)$ where $\lambda^2 = -1$ (possible since $k$ is algebraically closed), giving two distinct points. In $\mathbb{P}^2_k$: likewise, $V(X_0^2 + X_1^2) = V\bigl((X_0 - \lambda X_1)(X_0 + \lambda X_1)\bigr)$ is a union of two lines. On the other hand, $V(X_0^2 + X_1^2 + X_2^2) \subset \mathbb{P}^2_k$ is a non-degenerate conic, the simplest example of a smooth projective curve of degree $2$. [/example] Beyond quadrics, the geometry of higher-degree hypersurfaces grows rapidly more complex. Cubic surfaces in $\mathbb{P}^3_k$ are already rich objects (they contain exactly $27$ lines over an algebraically closed field). Quartic threefolds in $\mathbb{P}^4_k$ were only understood in the latter part of the twentieth century, and the geometry of quintic fourfolds in $\mathbb{P}^5_k$ remains an active research area. ### Bezout's Theorem (Preview) One of the central results of algebraic geometry, whose proof is developed later in the course, is Bezout's theorem. It illustrates precisely the power of working in projective space rather than affine space: in affine space, intersection counts can fail due to components escaping to infinity or lines being parallel. [quotetheorem:2131] The weighted version of Bézout's theorem for smooth plane curves is proved in Chapter 6 (see the section on Bézout's Theorem for Smooth Curves there); the special cases of lines and conics are established in Chapter 5. For now, note that the conclusion is a statement about $\mathbb{P}^2_k$: in $\mathbb{A}^2_k$, two parallel lines (degree $1$ each) would give $0$ intersection points, violating $d_1 d_2 = 1$. Projective space restores the count by supplying the missing intersection point at infinity. ## Homogeneous Ideals In Section 2.1 we built projective space $\mathbb{P}^n_k$ and observed that a polynomial $f \in k[X_0, \ldots, X_n]$ does not have a well-defined value at a point $[a_0 : \cdots : a_n] \in \mathbb{P}^n_k$, because scaling the homogeneous coordinates by $\lambda \neq 0$ scales $f$ by $\lambda^d$. What is well-defined, however, is whether $f$ vanishes there — as long as $f$ is homogeneous, since then $f(\lambda a_0, \ldots, \lambda a_n) = \lambda^d f(a_0, \ldots, a_n)$. The zeros of $f$ are therefore a well-defined subset of $\mathbb{P}^n_k$. This forces us to rethink what an "ideal" should mean in the projective setting. The correct notion is not "an ideal consisting entirely of homogeneous polynomials" — most ideals do not have that property — but rather an ideal generated by homogeneous polynomials. ### Homogeneous Ideals and Vanishing Sets [definition: Homogeneous Ideal] An ideal $I \subset k[X_0, \ldots, X_n]$ is **homogeneous** if it is generated by homogeneous polynomials. [/definition] The key structural fact about homogeneous ideals is that they are closed under taking homogeneous parts. Recall that any polynomial can be written uniquely as a finite sum of homogeneous polynomials: for $f \in k[X_0, \ldots, X_n]$ write \begin{align*} f = \sum_{i \geq 0} f_{[i]}, \end{align*} where $f_{[i]}$ is either zero or a homogeneous polynomial of degree $i$, called the **degree-$i$ part** of $f$. [quotetheorem:2132] [citeproof:2132] This characterisation tells us exactly when we can meaningfully speak of the vanishing locus of an ideal in projective space. [definition: Projective Vanishing Set] For a polynomial $f \in k[X_0, \ldots, X_n]$ and a point $p \in \mathbb{P}^n_k$, we say $f(p) = 0$ if $f_{[i]}(p) = 0$ for all $i$. For a homogeneous ideal $I$, the **projective vanishing set** is \begin{align*} V(I) = \{ p \in \mathbb{P}^n_k : f(p) = 0 \text{ for all } f \in I \}. \end{align*} [/definition] The definition extends naturally to arbitrary subsets: for any $S \subset k[X_0, \ldots, X_n]$, set $V(S) = V(I_S)$ where $I_S$ is the ideal generated by $S$. [quotetheorem:2133] [citeproof:2133] Going in the other direction, from a projective subvariety to an ideal, we define \begin{align*} I(V)^h = \langle f \in k[X_0, \ldots, X_n] : f \text{ homogeneous}, \, f(p) = 0 \text{ for all } p \in V \rangle. \end{align*} This is the **homogeneous ideal of $V$**, and it is homogeneous by construction. ### The Projective Nullstellensatz In the affine setting, Hilbert's Nullstellensatz gives a complete dictionary between radical ideals and varieties. Projective space has its own version, but with one important wrinkle: the ideal $(X_0, \ldots, X_n)$ — the **irrelevant ideal** — vanishes nowhere in $\mathbb{P}^n_k$ because a point $[a_0 : \cdots : a_n]$ must have at least one nonzero coordinate. [quotetheorem:2134] The theorem states that the only homogeneous ideals with empty projective vanishing set are those that swallow the irrelevant ideal. The full version recovers the affine pattern. [quotetheorem:2135] Both theorems are proved using the affine Nullstellensatz applied to the **cone** over $V$: the affine variety $\tilde{V} \subset \mathbb{A}^{n+1}_k$ obtained by viewing $V(I)$ as a subset of $k^{n+1} \setminus \{0\}$ and taking its closure. ### The Zariski Topology on Projective Space [definition: Zariski Topology on Projective Space] The **Zariski topology** on $\mathbb{P}^n_k$ is defined by declaring the closed sets to be exactly the sets of the form $V(I)$ for homogeneous ideals $I \subset k[X_0, \ldots, X_n]$. A **projective variety** is an irreducible closed subset of $\mathbb{P}^n_k$. [/definition] One checks that this is indeed a topology: arbitrary intersections and finite unions of projective vanishing sets are again of this form, using the sum and product of ideals respectively. A key observation is that this definition is intrinsic to $\mathbb{P}^n_k$ in the following sense: a linear change of coordinates on $k^{n+1}$ permutes the homogeneous ideals and hence leaves the Zariski topology invariant. This topology is equivalent to the one obtained by gluing the affine charts $U_i = \{X_i \neq 0\} \cong \mathbb{A}^n_k$: a subset of $\mathbb{P}^n_k$ is Zariski closed in the glued topology if and only if its intersection with each $U_i$ is Zariski closed in $\mathbb{A}^n_k$, and this coincides with being of the form $V(I)$ for a homogeneous ideal. ### Irreducibility in Projective Space The notion of irreducibility carries over from the affine case without change. [definition: Irreducible Projective Variety] A nonempty closed subset $V \subset \mathbb{P}^n_k$ is **irreducible** if whenever $V = V_1 \cup V_2$ with $V_1, V_2$ closed, then $V = V_1$ or $V = V_2$. [/definition] Just as in the affine case, irreducibility is characterised algebraically. [quotetheorem:2136] The proof follows the same argument as in the affine case: if $fg \in I(V)^h$ with $f, g$ homogeneous, decompose $V = (V \cap V(f)) \cup (V \cap V(g))$ and use irreducibility. Irreducibility has a striking consequence in projective space: proper closed subsets are genuinely small. [quotetheorem:2137] [citeproof:2137] [remark: Affine Charts Are Dense] As an immediate corollary, each affine chart $U_i = \{X_i \neq 0\} \subset \mathbb{P}^n_k$ is Zariski dense in $\mathbb{P}^n_k$. Indeed, $U_i$ is the complement of the hyperplane $V(X_i) \subsetneq \mathbb{P}^n_k$, and $\mathbb{P}^n_k$ is irreducible (its ideal is the zero ideal, which is prime). The density theorem applies. [/remark] This density is geometrically meaningful: it says that projective space cannot be recovered from any single affine chart, but is nonetheless "almost entirely" covered by each one. ### Affine Pieces of Projective Varieties The Zariski topology on $\mathbb{P}^n_k$ is compatible with the affine charts. Under the identification $U_i \leftrightarrow \mathbb{A}^n_k$ (setting $X_i = 1$), the intersection of a projective variety with an affine chart is an affine variety. [quotetheorem:2138] [citeproof:2138] [quotetheorem:2139] Since $\mathbb{P}^n_k = U_0 \cup \cdots \cup U_n$, the covering is immediate from the previous theorem. [example: The Three-Line Triangle] Let $V = V(X_0 X_1 X_2) \subset \mathbb{P}^2_k$ and consider the affine chart $U_0 = \{X_0 \neq 0\} \cong \mathbb{A}^2_k$ with coordinates $(Y_1, Y_2) = (X_1/X_0, X_2/X_0)$. Setting $X_0 = 1$ in $X_0 X_1 X_2$ gives $Y_1 Y_2$. Therefore \begin{align*} V \cap U_0 = V(Y_1 Y_2) \subset \mathbb{A}^2_k, \end{align*} which is the union of the two coordinate axes in $\mathbb{A}^2_k$. Projectively, $V(X_0 X_1 X_2)$ is the union of three lines: $V(X_0)$, $V(X_1)$, and $V(X_2)$. These three lines form a triangle in $\mathbb{P}^2_k$, meeting pairwise at the three coordinate vertices $[1:0:0]$, $[0:1:0]$, $[0:0:1]$. The affine piece $V \cap U_0$ sees only two of the three lines (the ones through the point $[0:0:1]$ and $[0:1:0]$ that meet $U_0$) while the third line $V(X_0)$ is the "line at infinity" relative to $U_0$. Note that the Zariski closure in $\mathbb{P}^2_k$ of $V \cap U_0 = V(Y_1 Y_2)$ is not $V$ itself: the closure is $V(X_1 X_2) = V(X_1) \cup V(X_2)$, which is only two of the three lines. [/example] [illustration:ag-projective-triangle] ### Homogenisation and Projective Closure The relation between affine and projective varieties is made precise by homogenisation, which allows us to take the projective closure of an affine variety. [definition: Homogenisation of a Polynomial] For $f \in k[Y_1, \ldots, Y_n]$ of total degree $d$, the **homogenisation** of $f$ with respect to $X_0$ is \begin{align*} f^h(X_0, \ldots, X_n) = X_0^d \, f\!\left(\frac{X_1}{X_0}, \ldots, \frac{X_n}{X_0}\right) \in k[X_0, \ldots, X_n]. \end{align*} This is a homogeneous polynomial of degree $d$. The reverse operation is **dehomogenisation**: for a homogeneous $g \in k[X_0, \ldots, X_n]$, its dehomogenisation is $g(1, Y_1, \ldots, Y_n) \in k[Y_1, \ldots, Y_n]$. [/definition] [example: Homogenising a Cubic] Let $f(Y_1, Y_2) = Y_1^3 + Y_1 Y_2 + Y_2^2$, which has total degree $3$. Then \begin{align*} f^h(X_0, X_1, X_2) = X_0^3 f\!\left(\frac{X_1}{X_0}, \frac{X_2}{X_0}\right) = X_1^3 + X_0 X_1 X_2 + X_0 X_2^2. \end{align*} One checks directly that $f^h$ is homogeneous of degree $3$, and that $f^h(1, Y_1, Y_2) = f(Y_1, Y_2)$. [/example] The relationship between a homogeneous polynomial and the homogenisation of its dehomogenisation is clean. [quotetheorem:2140] [citeproof:2140] Homogenisation of polynomials extends to ideals. [definition: Homogenisation of an Ideal] For an ideal $I \subset k[Y_1, \ldots, Y_n]$, its **homogenisation** is \begin{align*} I^h = \langle f^h : f \in I \rangle \subset k[X_0, \ldots, X_n]. \end{align*} [/definition] [remark: Homogenisation of Ideals Is Not Naive] If $I$ is principal and generated by $f$, then $I^h$ is principal and generated by $f^h$. However, for a general ideal $I = \langle f_1, \ldots, f_r \rangle$ it is not true that $I^h = \langle f_1^h, \ldots, f_r^h \rangle$. For example, take the twisted cubic $V = \{(t, t^2, t^3) : t \in k\} \subset \mathbb{A}^3_k$ with ideal $I(V) = \langle Y_1^3 - Y_3, Y_1^2 - Y_2 \rangle$. The ideal generated by the individual homogenisations $\langle X_1^3 - X_3 X_0^2, X_1^2 - X_0 X_2 \rangle$ is strictly smaller than $I(V)^h$. The correct notion for computing $I^h$ from generators involves Gröbner bases. [/remark] The main theorem on projective closure now follows. [quotetheorem:2141] [citeproof:2141] [example: Two Non-Isomorphic Affine Varieties with Isomorphic Projective Closures] The affine varieties $V(Y_1 Y_2 - 1)$ (a hyperbola) and $V(Y_2 - Y_1^2)$ (a parabola) in $\mathbb{A}^2_k$ are not isomorphic as affine varieties: their coordinate rings $k[Y_1, Y_2]/\langle Y_1 Y_2 - 1 \rangle \cong k[t, t^{-1}]$ and $k[Y_1, Y_2]/\langle Y_2 - Y_1^2 \rangle \cong k[t]$ are non-isomorphic (the first is not a polynomial ring, while the second is). However, their projective closures in $\mathbb{P}^2_k$ are isomorphic. The homogenisation of $Y_1 Y_2 - 1$ is $X_1 X_2 - X_0^2$, and the homogenisation of $Y_2 - Y_1^2$ is $X_0 X_2 - X_1^2$. Both define smooth conics in $\mathbb{P}^2_k$, and any two smooth conics in $\mathbb{P}^2_k$ are isomorphic (they are all isomorphic to $\mathbb{P}^1_k$ via a rational parametrisation). This illustrates that the projective closure can "repair" the geometry lost at infinity. [/example] ## Rational Maps on Projective Varieties One of the first things we noticed about projective space $\mathbb{P}^n_k$ is that a homogeneous polynomial $F \in k[X_0, \ldots, X_n]$ is not a well-defined function on $\mathbb{P}^n_k$: the value $F(\lambda p)$ depends on the choice of representative $\lambda p$ for the point $p$. This creates a fundamental tension — we want to do algebra, but our ambient space resists the most basic algebraic operation of evaluation. The resolution is to work with *ratios* of homogeneous polynomials of the same degree, which are genuinely well-defined. This section develops the theory of such functions and the maps they define, culminating in the notion of birational equivalence, the coarsest interesting equivalence relation in algebraic geometry. ### Regular Functions and the Function Field We begin by making precise what it means for a function to be algebraically defined on a projective variety. [definition: Homogeneous Coordinate Ring] Let $V \subset \mathbb{P}^n_k$ be a projective variety with vanishing ideal $I(V) \subset k[X_0, \ldots, X_n]$. The **homogeneous coordinate ring** of $V$ is \begin{align*} S(V) = k[X_0, \ldots, X_n] / I(V). \end{align*} [/definition] The homogeneous coordinate ring is the coordinate ring of the **affine cone** of $V$: the vanishing locus of $I(V)$ inside $\mathbb{A}^{n+1}_k$. Elements of $S(V)$ are functions on the affine cone, but not on $V$ itself — they depend on the choice of representative for a projective point. This is not a defect of the definition; it is precisely the reason we introduce the function field separately. [definition: Function Field of a Projective Variety] Let $V \subset \mathbb{P}^n_k$ be an irreducible projective variety. The **function field** (or field of rational functions) of $V$ is \begin{align*} \mathcal{O}_V(\eta) = \left\{ \frac{\bar{P}}{\bar{Q}} : P, Q \in k[X_0, \ldots, X_n] \text{ homogeneous}, \deg P = \deg Q, Q \notin I(V) \right\} \end{align*} where $\bar{P}, \bar{Q}$ denote the images of $P, Q$ in $S(V)$. [/definition] The equal-degree condition is what makes $P/Q$ well-defined on $\mathbb{P}^n_k$: if $p = [\lambda a_0 : \cdots : \lambda a_n]$, then $P(\lambda a)/Q(\lambda a) = \lambda^d P(a) / \lambda^d Q(a) = P(a)/Q(a)$. The irreducibility of $V$ ensures that $S(V)$ is a domain, so localization at the nonzero elements is possible. The entire construction is forced on us by the key observation: [remark: No Global Functions on Projective Space] $\mathbb{P}^n_k$ admits no nonconstant globally defined algebraic functions. Any algebraic function on all of $\mathbb{P}^n_k$ must be constant. This is the projective analogue of Liouville's theorem, and it is why the function field — defined only on dense open subsets — is the right object to study. [/remark] The function field is a finitely generated field extension of $k$, as the following proposition shows. [quotetheorem:2142] [citeproof:2142] [example: Function Field of Projective Space] If $V = \mathbb{P}^n_k$, then $\mathcal{O}_V(\eta) \cong \mathcal{O}_{\mathbb{A}^n_k}(\eta) \cong k(Y_1, \ldots, Y_n)$, the purely transcendental extension of $k$ in $n$ variables. This follows immediately from the proof above, taking the affine chart $U_0 \cong \mathbb{A}^n_k$. [/example] ### Regularity and Local Rings Given an irreducible projective variety $V$ and a rational function $h \in \mathcal{O}_V(\eta)$, we can ask where $h$ is actually defined. [definition: Regular Point of a Rational Function] Let $V \subset \mathbb{P}^n_k$ be irreducible and $h \in \mathcal{O}_V(\eta)$. We say $h$ is **regular at a point** $p \in V$ if there exists a representation $h = P/Q$ with $P, Q \in k[X_0, \ldots, X_n]$ homogeneous of the same degree and $Q(p) \neq 0$. The **local ring** of $V$ at $p$ is \begin{align*} \mathcal{O}_{V,p} = \{ h \in \mathcal{O}_V(\eta) : h \text{ is regular at } p \}. \end{align*} [/definition] A rational function $h$ regular at $p$ determines a well-defined value in $k$ at $p$, and hence defines a genuine algebraic function on the Zariski open set $\{p \in V : h \text{ regular at } p\}$. The local ring $\mathcal{O}_{V,p}$ is indeed a local ring: its unique maximal ideal consists of those $h$ that vanish at $p$. ### Rational Maps between Projective Varieties Now we turn to maps. Suppose we want to define an algebraic map $\mathbb{P}^n_k \to \mathbb{P}^m_k$. Since a point in $\mathbb{P}^m_k$ is a ratio $[b_0 : \cdots : b_m]$, a natural attempt is to pick homogeneous polynomials $F_0, \ldots, F_m \in k[X_0, \ldots, X_n]$ all of the same degree $d$ and define \begin{align*} p \mapsto [F_0(p) : \cdots : F_m(p)]. \end{align*} This is well-defined wherever $(F_0(p), \ldots, F_m(p)) \neq (0, \ldots, 0)$, and the equal-degree condition ensures it is independent of the choice of homogeneous representative for $p$. If we multiply all the $F_i$ by another homogeneous polynomial $G$, the tuple $[F_0 G : \cdots : F_m G]$ defines the same map but on a smaller domain. Since $k[X_0, \ldots, X_n]$ is a UFD, there is a "best" representative with the largest possible domain, obtained by cancelling common factors. [definition: Rational Map between Projective Varieties] Let $V \subset \mathbb{P}^n_k$ be an irreducible projective variety. A **rational map** $\varphi : V \dashrightarrow \mathbb{P}^m_k$ is an equivalence class of tuples $[F_0 : \cdots : F_m]$ of homogeneous polynomials of equal degree, where the base locus $\bigcap_i V(F_i)$ does not contain $V$. Two tuples $[F_0 : \cdots : F_m]$ and $[G_0 : \cdots : G_m]$ determine the **same rational map** if $F_i G_j - F_j G_i \in I(V)^h$ for all $i, j$. A **rational map** $\varphi : V \dashrightarrow W$ into a projective variety $W \subset \mathbb{P}^m_k$ is a rational map $V \dashrightarrow \mathbb{P}^m_k$ such that $\varphi(\mathrm{dom}\, \varphi) \subset W$. [/definition] [illustration:ag-rational-map-domain] [definition: Base Locus and Morphism] Given a rational map $\varphi : \mathbb{P}^n_k \dashrightarrow \mathbb{P}^m_k$ represented by $(F_0, \ldots, F_m)$ without common factors, its **base locus** is $\bigcap_i V(F_i)$. A **point** $p \in V$ is **regular for $\varphi$** if some representative $[F_0 : \cdots : F_m]$ satisfies $F_i(p) \neq 0$ for some $i$. The **domain** of $\varphi$ is $\mathrm{dom}\, \varphi = \{p \in V : \varphi \text{ is regular at } p\}$. The rational map is a **morphism** if $\mathrm{dom}\, \varphi = V$, i.e., the base locus does not meet $V$. [/definition] [definition: Dominant Rational Map] A rational map $\varphi : V \dashrightarrow W$ is **dominant** if $\varphi(\mathrm{dom}\, \varphi)$ is dense in $W$. [/definition] Dominance is the correct condition to impose when composing rational maps: if $\varphi : V \dashrightarrow W$ is dominant and $\psi : W \dashrightarrow Z$ is any rational map, then the composition $\psi \circ \varphi$ is defined, because $\varphi(\mathrm{dom}\, \varphi)$ is dense in $W$ and hence meets $\mathrm{dom}\, \psi$. Without dominance, the image of $\varphi$ might miss $\mathrm{dom}\, \psi$ entirely, making composition impossible. ### Examples: Projections, the Veronese, and the Cremona Transformation [example: Projection from a Point] Fix homogeneous coordinates $[X_0 : \cdots : X_n]$ on $\mathbb{P}^n_k$. The **projection from the point** $p_0 = [0 : \cdots : 0 : 1]$ is the rational map \begin{align*} \pi : \mathbb{P}^n_k \dashrightarrow \mathbb{P}^{n-1}_k, \quad [a_0 : \cdots : a_n] \mapsto [a_0 : \cdots : a_{n-1}]. \end{align*} Its base locus is $\{p_0\}$, which is exactly the point we are projecting from. For any other point $p \in \mathbb{P}^n_k \setminus \{p_0\}$, the map sends $p$ to the intersection of the line $\overline{p_0 p}$ with the hyperplane $\{X_n = 0\} \cong \mathbb{P}^{n-1}_k$. More generally, to project from an arbitrary point $p \in \mathbb{P}^n_k$, one first applies a linear change of coordinates on $k^{n+1}$ sending $p$ to $[0 : \cdots : 0 : 1]$. [/example] [example: The Veronese Embedding] There are $N = \binom{n+d}{d}$ monomials of degree $d$ in $n+1$ variables. Label them $F_0, F_1, \ldots, F_{N-1}$. The **$d$-th Veronese map** is \begin{align*} \nu_d : \mathbb{P}^n_k \dashrightarrow \mathbb{P}^{N-1}_k, \quad p \mapsto [F_0(p) : F_1(p) : \cdots : F_{N-1}(p)]. \end{align*} Since the monomials of degree $d$ cannot simultaneously vanish at any point of $\mathbb{P}^n_k$ (for any $[a_0 : \cdots : a_n] \neq 0$, the monomial $a_i^d$ is nonzero for some $i$), the base locus is empty. Therefore $\nu_d$ is a morphism, defined everywhere. For the simplest case $n = 1$, $d = 2$: the monomials are $X_0^2, X_0 X_1, X_1^2$, giving $\nu_2 : \mathbb{P}^1_k \to \mathbb{P}^2_k$, $[a_0 : a_1] \mapsto [a_0^2 : a_0 a_1 : a_1^2]$. The image is the conic $V(Y_0 Y_2 - Y_1^2) \subset \mathbb{P}^2_k$, and $\nu_2$ is an isomorphism onto this conic. In general, $\nu_d$ embeds $\mathbb{P}^n_k$ as a projective variety in $\mathbb{P}^{N-1}_k$. [/example] [example: The Cremona Transformation] The **Cremona transformation** $\mathbb{P}^n_k \dashrightarrow \mathbb{P}^n_k$ is defined by \begin{align*} [X_0 : X_1 : \cdots : X_n] \mapsto \left[\frac{1}{X_0} : \frac{1}{X_1} : \cdots : \frac{1}{X_n}\right] = [X_1 X_2 \cdots X_n : X_0 X_2 \cdots X_n : \cdots : X_0 X_1 \cdots X_{n-1}]. \end{align*} The base locus consists of the $n+1$ coordinate hyperplanes $\{X_i = 0\}$ pairwise intersected, i.e., the union of coordinate subspaces $\{X_i = X_j = 0\}$. This rational map is dominant and is its own inverse: applying it twice returns the identity on the complement of the base locus. [/example] [example: Base Locus and Morphism Distinguished] Consider $\varphi : \mathbb{P}^2_k \dashrightarrow \mathbb{P}^1_k$ given by $[X_0 : X_1 : X_2] \mapsto [X_0 : X_1]$. This is the projection from $[0:0:1]$, and its base locus is $\{[0:0:1]\}$. So $\varphi$ is a rational map but not a morphism. By contrast, $\psi : \mathbb{P}^1_k \to \mathbb{P}^2_k$ given by $[X_0 : X_1] \mapsto [X_0^2 : X_1^2 : X_0 X_1]$ has empty base locus (since $X_0^2, X_1^2, X_0 X_1$ cannot all vanish simultaneously on $\mathbb{P}^1_k$), so $\psi$ is a morphism. [/example] ### The Veronese and Affine Complements of Hypersurfaces The Veronese embedding has a striking geometric consequence: complements of hypersurfaces in projective space are affine. [quotetheorem:2143] [citeproof:2143] [remark: Morphisms between Projective Spaces] This is proved in Chapter 4 using Krull's Height Theorem: all morphisms $\mathbb{P}^n_k \to \mathbb{P}^m_k$ with $n > m$ are constant (see the section on Krull's Height Theorem and Consequences there). The case $n = 2$, $m = 1$ can be seen directly: any two homogeneous polynomials $F_0, F_1 \in k[X_0, X_1, X_2]$ of the same degree have $V(F_0) \cap V(F_1) \neq \varnothing$ by Bezout's theorem (since both define curves in $\mathbb{P}^2_k$ and two curves in $\mathbb{P}^2_k$ always intersect), so the base locus of any such pair is nonempty. This explains the earlier observation about projection from a point: some information is always lost. [/remark] ### Birational Equivalence If $\varphi : V \dashrightarrow W$ is a dominant rational map, then pullback along $\varphi$ gives a field homomorphism $\varphi^* : \mathcal{O}_W(\eta) \to \mathcal{O}_V(\eta)$ of field extensions of $k$. This makes function fields into a contravariant functor on dominant rational maps. The invertible case is the most important. [definition: Birational Equivalence] Irreducible projective varieties $V$ and $W$ are **birational** (written $V \simeq_{\mathrm{bir}} W$) if there exist rational maps $\varphi : V \dashrightarrow W$ and $\psi : W \dashrightarrow V$ such that $\varphi \circ \psi$ and $\psi \circ \varphi$ are both defined and equal to the identity. Equivalently, $V$ and $W$ share a common dense Zariski open subset. [/definition] [example: Affine and Projective Space are Birational] $\mathbb{A}^n_k \simeq_{\mathrm{bir}} \mathbb{P}^n_k$. The standard affine chart $U_0 = \{X_0 \neq 0\} \cong \mathbb{A}^n_k$ is a dense open subset of $\mathbb{P}^n_k$, which simultaneously serves as a dense open of $\mathbb{A}^n_k$ itself. The rational maps are the inclusion $\mathbb{A}^n_k \hookrightarrow \mathbb{P}^n_k$ (sending $(y_1, \ldots, y_n) \mapsto [1 : y_1 : \cdots : y_n]$) and the inverse $[X_0 : \cdots : X_n] \mapsto (X_1/X_0, \ldots, X_n/X_0)$, both defined on dense opens. [/example] [example: Affine Variety is Birational to its Projective Closure] Any affine variety $V \subset \mathbb{A}^n_k$ is birationally equivalent to its projective closure $\bar{V} \subset \mathbb{P}^n_k$. The projective closure is obtained by homogenising the defining ideal of $V$, and the inclusion $V \hookrightarrow \bar{V}$ identifies $V$ with a dense open subset of $\bar{V}$. [/example] The fundamental theorem of birational geometry for projective varieties relates this geometric notion to algebra: [quotetheorem:2144] [citeproof:2144] [explanation: Significance of Birational Classification] The theorem reframes the geometric classification problem — "when are two varieties the same up to birational equivalence?" — as a purely algebraic one: "when are two finitely generated field extensions of $k$ isomorphic?" This is an enormous simplification in principle, though field extensions themselves can be highly complex. The birational classification of varieties is one of the central programmes of modern algebraic geometry, including the Minimal Model Programme (MMP) in higher dimensions. A key feature of birational equivalence is that it is much coarser than isomorphism: two varieties can be birationally equivalent even if they differ significantly at finitely many points (or along lower-dimensional subvarieties). For curves, however, birational equivalence and isomorphism coincide — every smooth projective curve is determined up to isomorphism by its function field. For surfaces and higher-dimensional varieties, birationally equivalent varieties can be quite different, and understanding the "minimal" or "canonical" representative in each birational class is the central challenge. [/explanation] [remark: Birational Equivalence on Affine Varieties] The definition of birational equivalence extends straightforwardly to affine varieties: $V$ and $W$ are birational if they share a common dense Zariski open subset, or equivalently if their coordinate rings have isomorphic fields of fractions as $k$-extensions. This is compatible with the projective definition, since an affine variety and its projective closure are always birational. [/remark] ## Product of Varieties Having developed rational maps and morphisms between projective varieties in the preceding sections, a natural question arises: given two projective varieties $V \subset \mathbb{P}^n_k$ and $W \subset \mathbb{P}^m_k$, how do we make their Cartesian product $V \times W$ into a projective variety? The answer is more delicate than it might first appear, because the naive approach fails. ### Why the Naive Product Fails One might hope that the product topology on $\mathbb{P}^n_k \times \mathbb{P}^m_k$ — where each factor is given the Zariski topology — would itself be a Zariski topology for some ambient projective space. This hope is misplaced. [remark: Failure of the Naive Product] The Zariski topology on $\mathbb{P}^n_k \times \mathbb{P}^m_k$ formed by taking the product of the two Zariski topologies is strictly coarser than the Zariski topology that $\mathbb{P}^n_k \times \mathbb{P}^m_k$ should carry as a projective variety. In the product topology, closed sets are finite unions of sets of the form $A \times B$ where $A \subset \mathbb{P}^n_k$ and $B \subset \mathbb{P}^m_k$ are Zariski closed. But the diagonal $\Delta = \{(p, p) : p \in \mathbb{P}^n_k\} \subset \mathbb{P}^n_k \times \mathbb{P}^n_k$ is not of this form, yet it should be closed in any reasonable product variety structure. The correct topology on the product must be defined extrinsically, by embedding the product into a larger projective space. [/remark] The resolution is the **Segre embedding**, which maps $\mathbb{P}^n_k \times \mathbb{P}^m_k$ injectively into a higher-dimensional projective space and thereby endows the product with the structure of a projective variety. ### The Segre Embedding The idea behind the Segre embedding is to encode the pair of homogeneous coordinate vectors $[x_0 : \cdots : x_n]$ and $[y_0 : \cdots : y_m]$ as their tensor product — a matrix whose $(i,j)$-entry is $x_i y_j$. The entries of this matrix then serve as homogeneous coordinates in a larger projective space. [definition: Segre Embedding] Let $k$ be an algebraically closed field. The **Segre embedding** is the map \begin{align*} \Sigma_{n,m} : \mathbb{P}^n_k \times \mathbb{P}^m_k &\longrightarrow \mathbb{P}^{(n+1)(m+1)-1}_k \\ ([x_0 : \cdots : x_n],\, [y_0 : \cdots : y_m]) &\longmapsto [x_i y_j]_{0 \le i \le n,\, 0 \le j \le m} \end{align*} where the target coordinates $[Z_{ij}]$ are indexed by pairs $(i,j)$ with $0 \le i \le n$ and $0 \le j \le m$, so $Z_{ij} = x_i y_j$. [/definition] The map $\Sigma_{n,m}$ is well-defined: scaling $[x_0 : \cdots : x_n]$ by $\lambda$ and $[y_0 : \cdots : y_m]$ by $\mu$ scales every entry $x_i y_j$ by $\lambda \mu$, which does not change the homogeneous coordinates $[Z_{ij}]$. The key content of the following theorem is that this map is injective, and that its image is a projective variety cut out by explicit polynomial equations. [quotetheorem:2145] [citeproof:2145] [example: Segre Embedding of $\mathbb{P}^1_k \times \mathbb{P}^1_k$] Take $n = m = 1$. Then $(n+1)(m+1) - 1 = 3$, and the Segre embedding maps \begin{align*} \Sigma_{1,1} : \mathbb{P}^1_k \times \mathbb{P}^1_k &\longrightarrow \mathbb{P}^3_k \\ ([x_0 : x_1], [y_0 : y_1]) &\longmapsto [x_0 y_0 : x_0 y_1 : x_1 y_0 : x_1 y_1]. \end{align*} Writing the target coordinates as $[Z_{00} : Z_{01} : Z_{10} : Z_{11}]$, the image is $V(Z_{00} Z_{11} - Z_{01} Z_{10})$. This is a smooth quadric surface in $\mathbb{P}^3_k$. The two families of lines on this quadric correspond precisely to the two projection maps: fixing a point in $\mathbb{P}^1_k$ on the first factor gives a line on the quadric, and similarly for the second factor. [/example] [illustration:ag-segre-p1-p1-quadric] ### The Product as a Projective Variety Having embedded $\mathbb{P}^n_k \times \mathbb{P}^m_k$ into projective space as a closed subvariety, we can use the Segre embedding to define the product of arbitrary projective varieties. The first step is to verify that the two coordinate projections are morphisms. [quotetheorem:2146] This is verified on the standard affine open cover. On the affine chart $\{Z_{ij} \ne 0\}$ of the Segre variety, the coordinates $x_s/x_i$ and $y_t/y_j$ are recovered as $Z_{sj}/Z_{ij}$ and $Z_{it}/Z_{ij}$ respectively, which are regular functions. The projections are then given by explicit regular formulas on each chart. The projections being morphisms unlocks the main corollary of the section. [quotetheorem:2147] [citeproof:2147] [explanation: What the Product Topology Looks Like] The Segre construction tells us explicitly what the closed sets of $V \times W$ are. A subset $Z \subset V \times W$ is closed if and only if it is the vanishing locus of a collection of polynomials in the coordinates $Z_{ij} = x_i y_j$ — equivalently, in the products $x_i y_j$. Such polynomials are not arbitrary polynomials in $x$ and $y$ separately, but rather polynomials in the **bilinear monomials** $x_i y_j$. This connects to the intrinsic description of the product topology, which we describe in the next subsection: the closed sets should be defined by **bihomogeneous** polynomial equations in the two sets of variables separately. The Segre embedding realises this concretely by packaging bihomogeneous data into honest homogeneous coordinates on the target projective space. [/explanation] ### The Intrinsic Description The Segre embedding gives an extrinsic way to define the product variety by embedding it into a larger projective space. There is also a more intrinsic description directly in terms of polynomial equations on $\mathbb{P}^n_k \times \mathbb{P}^m_k$. [definition: Bihomogeneous Polynomial] A polynomial $F(X_0, \ldots, X_n, Y_0, \ldots, Y_m) \in k[X_0, \ldots, X_n, Y_0, \ldots, Y_m]$ is **bihomogeneous of bidegree $(d, e)$** if it is homogeneous of degree $d$ in the $X$-variables and homogeneous of degree $e$ in the $Y$-variables. That is, \begin{align*} F(\lambda X_0, \ldots, \lambda X_n, \mu Y_0, \ldots, \mu Y_m) = \lambda^d \mu^e\, F(X_0, \ldots, X_n, Y_0, \ldots, Y_m) \end{align*} for all $\lambda, \mu \in k$. [/definition] The vanishing locus of a bihomogeneous polynomial is well-defined on $\mathbb{P}^n_k \times \mathbb{P}^m_k$, since scaling the $X$-coordinates by $\lambda$ and the $Y$-coordinates by $\mu$ scales $F$ by $\lambda^d \mu^e \ne 0$. We can therefore define a topology on $\mathbb{P}^n_k \times \mathbb{P}^m_k$ by declaring the closed sets to be finite intersections and arbitrary unions of vanishing loci of bihomogeneous polynomials. [quotetheorem:2148] The key observation is that a polynomial in the $Z_{ij}$-coordinates restricts to a bihomogeneous polynomial in the $x_i$ and $y_j$ coordinates via the substitution $Z_{ij} = x_i y_j$, and conversely, any bihomogeneous polynomial in $x_i, y_j$ can be expressed in terms of the monomials $x_i y_j = Z_{ij}$ (possibly after introducing extra variables for higher bidegrees, using the Segre relations). ### A Consequence: The Veronese via Segre The Segre embedding is not merely a tool for constructing product varieties — it also furnishes a systematic way to produce new morphisms between projective spaces. [remark: The Veronese as a Restriction of Segre] Consider the Segre embedding $\Sigma_{n,n} : \mathbb{P}^n_k \times \mathbb{P}^n_k \to \mathbb{P}^{(n+1)^2 - 1}_k$. The diagonal $\Delta = \{(p, p) : p \in \mathbb{P}^n_k\}$ is a closed subvariety of $\mathbb{P}^n_k \times \mathbb{P}^n_k$ (it is the vanishing locus of the bihomogeneous equations $X_i Y_j - X_j Y_i = 0$). Restricting $\Sigma_{n,n}$ to $\Delta$ gives a morphism \begin{align*} \mathbb{P}^n_k \to \mathbb{P}^{(n+1)^2 - 1}_k, \qquad [x_0 : \cdots : x_n] \mapsto [x_i x_j]_{0 \le i, j \le n}. \end{align*} The image coordinates $x_i x_j$ are exactly the degree-2 monomials in $x_0, \ldots, x_n$ (with $x_i x_j = x_j x_i$ giving symmetry). This is the **degree-2 Veronese embedding** of $\mathbb{P}^n_k$. Thus the Veronese arises as the restriction of a Segre map to the diagonal — a clean illustration of how the two constructions interact. [/remark] Having built projective varieties and their morphisms through embeddings like Segre and Veronese, we now zoom in to understand local geometry. Chapter 4 develops the theory of tangent spaces and dimension by linearising the defining equations at each point, creating an invariant that distinguishes smooth points from singular ones. # 4. Tangent Space, Smoothness, and Dimension This chapter develops two of the most fundamental invariants of an algebraic variety: its tangent space and its dimension. The two approaches — geometric linearisation and field-theoretic transcendence degree — turn out to give the same answer, and the proof of their equivalence is a highlight of the course. The chapter also introduces the notion of smoothness, identifies the locus of singular points as a proper closed subset, and establishes Krull's Height Theorem, from which several striking corollaries about projective varieties follow immediately. ## Tangent Spaces via Linearisation The guiding philosophy is to study a variety by approximating it with a linear object at each point. In analysis, this would mean taking a derivative; in algebraic geometry, it means differentiating the defining polynomials. [motivation] ### Why linearise? Suppose $V = V(f) \subset \mathbb{A}^n_k$ is an affine hypersurface and $\ell = \{a + bt : b \in k^n \setminus \{0\}\}$ is a line through the point $p = (a_1, \ldots, a_n) \in V$. To measure how $\ell$ meets $V$ near $p$, substitute into $f$: \begin{align*} g(t) = f(a_1 + b_1 t, \ldots, a_n + b_n t) = c_0 + c_1 t + c_2 t^2 + \cdots \end{align*} The line passes through $p$ when $c_0 = f(a) = 0$, which holds since $p \in V$. The line is tangent to $V$ at $p$ precisely when $c_1 = 0$ as well — that is, when it touches $V$ to first order rather than merely crossing it. Computing $c_1$ by differentiating, one finds $c_1 = \sum_i \frac{\partial f}{\partial X_i}(p)\, b_i$. The condition $c_1 = 0$ is a single linear equation in the direction vector $b$, cutting out a hyperplane. The tangent space is exactly this hyperplane, together with the basepoint $p$. [/motivation] Working out the Taylor expansion at $p$, the condition $c_1 = 0$ is equivalent to requiring $\ell \subset V\!\left(\sum_i \frac{\partial f}{\partial X_i}(p)(X_i - a_i)\right)$, i.e. that the direction vector $b$ lies in the zero locus of the linear polynomial $\sum_i \frac{\partial f}{\partial X_i}(p)(X_i - a_i)$. Note that the partial derivatives here are symbolic differentiations of polynomials, not analytic limits. [definition: Affine Tangent Space to a Hypersurface] Let $V = V(f) \subset \mathbb{A}^n_k$. The **affine tangent space** to $V$ at $p \in V$ is the affine variety \begin{align*} T^{\mathrm{aff}}_{V,p} = V\!\left(\sum_i \frac{\partial f}{\partial X_i}(p)(X_i - a_i)\right) \subset \mathbb{A}^n_k. \end{align*} [/definition] This is an affine hyperplane (of dimension $n-1$) unless all partial derivatives of $f$ vanish at $p$, in which case $T^{\mathrm{aff}}_{V,p} = \mathbb{A}^n_k$ has dimension $n$. [definition: Smoothness for Hypersurfaces] An affine hypersurface $V = V(f) \subset \mathbb{A}^n_k$ is **smooth** if $\dim T^{\mathrm{aff}}_{V,p}$ is independent of $p \in V$. [/definition] For a hypersurface, the only possible values of $\dim T^{\mathrm{aff}}_{V,p}$ are $n-1$ (when some $\frac{\partial f}{\partial X_i}(p) \neq 0$) or $n$ (when all partial derivatives vanish at $p$). A hypersurface fails to be smooth precisely at points where the gradient of $f$ vanishes — the singular points. [example: The Node $V(Y^2 - X^2(X+1))$] Let $V = V(Y^2 - X^2(X+1)) \subset \mathbb{A}^2_k$. The partial derivatives of $f = Y^2 - X^2(X+1)$ are \begin{align*} \frac{\partial f}{\partial X} = -3X^2 - 2X = -X(3X+2), \quad \frac{\partial f}{\partial Y} = 2Y. \end{align*} Both vanish simultaneously iff $Y = 0$ and $X \in \{0, -2/3\}$. Among these candidates, only $(0,0)$ lies on $V$: indeed $(-2/3, 0)$ would require $0 = (-2/3)^2((-2/3)+1) = (4/9)(1/3) = 4/27 \neq 0$, so it is not on $V$. Therefore the gradient vanishes on $V$ only at $(0,0)$, giving $T^{\mathrm{aff}}_{V,(0,0)} = \mathbb{A}^2_k$ of dimension $2$. At any other point of $V$ at least one partial derivative is nonzero, so $\dim T^{\mathrm{aff}}_{V,p} = 1$. Since the dimension of the tangent space is not constant, $V$ is not smooth. The origin is a node — a self-intersection point where two branches of the curve cross. [/example] [illustration:ag-nodal-cubic-tangents] ## Tangent Space of a General Variety The hypersurface definition cuts out the tangent space by a single linear equation in $k^n$. For a variety defined by several polynomials $f_1, \ldots, f_r$, the natural question is: which linear subspace captures all tangent directions? The answer turns out to be the intersection of the individual hyperplane conditions — equivalently, the kernel of the Jacobian matrix of all the $f_i$ at $p$. One must check that this coincides with the "correct" geometric notion at smooth points, and indeed it does. [definition: Tangent Space of an Affine Variety] Let $V \subset \mathbb{A}^n_k$ be an affine variety and $p \in V$. The **tangent space** to $V$ at $p$ is the linear subspace \begin{align*} T_{V,p} = \left\{ v \in k^n : \forall f \in I(V),\ \sum_{i=1}^n \frac{\partial f}{\partial X_i}(p)\, v_i = 0 \right\}. \end{align*} [/definition] In matrix terms, if $I(V) = \langle f_1, \ldots, f_m \rangle$, then $T_{V,p} = \ker\!\left(\frac{\partial f_j}{\partial X_i}(p)\right)_{j,i}$, the kernel of the $m \times n$ Jacobian matrix of the generators evaluated at $p$. [remark: Affine versus Linear] The affine tangent space $T^{\mathrm{aff}}_{V,p}$ and the (linear) tangent space $T_{V,p}$ encode the same information: $T^{\mathrm{aff}}_{V,p} = p + T_{V,p}$ as a translate. Working with $T_{V,p}$ as a vector space is more convenient for general varieties. [/remark] [definition: Tangent Space of a Projective Variety] If $V \subset \mathbb{P}^n_k$ is a projective variety and $p \in V$, the **tangent space** $T_{V,p}$ is defined as the tangent space to $p$ in any nonempty affine open subset of $\mathbb{P}^n_k$ that contains $p$. [/definition] One checks that this is independent of the choice of affine chart, so the definition is well-posed. ## Smoothness and the Dimension of the Tangent Space From the Node example, the tangent space at the singular origin jumped from dimension $1$ (expected for a curve) to dimension $2$. Is this the universal behaviour of singular points? The answer is yes for irreducible varieties: a smooth point is precisely one where the tangent space has the minimum possible dimension, and such points form the generic case. The singular locus — where the dimension jumps — is always a proper closed subset. [definition: Smooth Point and Smooth Variety] A variety $V$ (affine or projective) is **smooth** if $\dim T_{V,p}$ is independent of $p \in V$. A point $p \in V$ is **smooth** if $\dim T_{V,p}$ equals the dimension of the union of all irreducible components of $V$ that contain $p$; otherwise $p$ is **singular**. [/definition] The relationship between tangent space dimension and the geometric dimension of the variety is the key invariant: for an irreducible variety $V$, one always has $\dim V = \min_{p \in V} \dim T_{V,p}$ (this is proved below via the algebraic approach to dimension). Singular points are precisely those where the tangent space is strictly larger than expected. For a general (possibly reducible) variety, the dimension is $\dim V = \max_i \dim V_i$ where $V_i$ are the irreducible components. [quotetheorem:2149] [citeproof:2149] The characteristic-zero hypothesis genuinely matters here. In characteristic $p > 0$, the Frobenius morphism $x \mapsto x^p$ is everywhere inseparable, and varieties such as $V(Y^2 - X^p)$ over $\mathbb{F}_p$ have $\partial f/\partial Y = 2Y$ and $\partial f/\partial X = 0$ identically (since $pX^{p-1} = 0$ in characteristic $p$), so the Jacobian criterion gives the wrong answer and every point appears singular even though the variety is geometrically a smooth curve. The correct replacement uses the notion of geometric regularity. [remark: Jumping on Closed Sets] The dimension of $T_{V,p}$ can only jump upward on Zariski-closed sets. For an irreducible variety, the locus where $\dim T_{V,p} = \dim V$ (i.e. the smooth locus) is Zariski-open and dense. [/remark] ## The Differential of a Rational Map Tangent spaces assemble into a linearisation of the variety, and morphisms (and rational maps) between varieties induce linear maps between tangent spaces, exactly as in differential geometry. Let $\varphi: V \dashrightarrow W$ be a rational map and $p \in \mathrm{dom}(\varphi)$. Writing $\varphi = (f_1, \ldots, f_m)$ in local coordinates, define: [definition: Differential of a Rational Map] The **differential** of $\varphi$ at $p$ is the linear map \begin{align*} d\varphi_p : T_{V,p} \longrightarrow T_{W,\varphi(p)} \end{align*} defined by $d\varphi_p(v) = \left(\frac{\partial f_j}{\partial X_i}(p)\right)_{j,i} v$. [/definition] [quotetheorem:2150] [citeproof:2150] The hypotheses here are mild — the chain rule holds for any rational map at a point in its domain. What the theorem does not say is that $d\varphi_p$ is injective or surjective; those are additional properties of specific maps. For instance, a constant map has $d\varphi_p = 0$ everywhere, while an isomorphism has $d\varphi_p$ an isomorphism at every smooth point. The chain rule also has no content at points outside the domain of definition: if $p \notin \mathrm{dom}(\varphi)$, the differential is simply not defined there. The chain rule has an immediate and important consequence. [quotetheorem:2151] [citeproof:2151] The hypothesis that smooth points exist — guaranteed in characteristic zero — is what makes the differential argument work: at a smooth point, $\dim T_{V,p} = \dim V$, so the isomorphism of tangent spaces translates directly into equality of dimensions. In positive characteristic, if the smooth locus were empty (which cannot happen over perfect fields, but illustrates the logical structure), the argument would break down. The theorem also only guarantees equality of dimension, not isomorphism: $\mathbb{P}^1_k \times \mathbb{P}^1_k$ and $\mathbb{P}^2_k$ are birational but not isomorphic, as we prove at the end of this chapter. This is the key motivating result for the algebraic approach to dimension: if dimension is a birational invariant, and if every irreducible variety is birational to something arising from a finitely generated field extension, then dimension should be computable from field theory alone. ## Dimension via Transcendence Degree The birational invariance of dimension suggests that the dimension of $V$ should equal a purely field-theoretic invariant of the function field $\mathcal{O}_V(\eta) = \mathrm{FF}(k[V])$. [definition: Purely Transcendental Extension] A finitely generated field extension $K/k$ is **purely transcendental** if $K \cong k(x_1, \ldots, x_n)$ for some algebraically independent $x_1, \ldots, x_n \in K$. [/definition] [example: The Function Field of $\mathbb{P}^n_k$] The function field $\mathcal{O}_{\mathbb{P}^n_k}(\eta) \cong k(t_1, \ldots, t_n)$ is purely transcendental over $k$. Here $t_i = X_i/X_0$ are the affine coordinate functions on the standard chart $\{X_0 \neq 0\} \cong \mathbb{A}^n_k$, and they are algebraically independent because any polynomial relation among them would give a relation among the $X_i$ in $k[X_0, \ldots, X_n]/(X_0)$, which is impossible. [/example] [quotetheorem:2152] [citeproof:2152] The separability clause in characteristic zero is automatic: every algebraic extension of a field of characteristic zero is separable. In characteristic $p > 0$, however, a finite extension can be purely inseparable — for instance, $\mathbb{F}_p(t^{1/p})/\mathbb{F}_p(t)$ is purely inseparable of degree $p$. The theorem still holds in that generality (one can always find a purely transcendental $K_0$ with $K/K_0$ finite, even if not separable), but the separability is needed in the next step when we apply the Primitive Element Theorem. [definition: Transcendence Degree] The **transcendence degree** of a finitely generated field extension $K/k$, written $\mathrm{trdeg}_k(K)$, is the cardinality of any maximal algebraically independent subset of $K$ over $k$. By the above theorem, this equals $n$ where $K_0 = k(x_1, \ldots, x_n)$ is the purely transcendental part. [/definition] The transcendence degree is well-defined (any two maximal independent subsets have the same size, by a standard exchange argument analogous to dimension of a vector space). The following proposition shows that the function field of any irreducible variety is generated by a single algebraic element over a purely transcendental subfield. [quotetheorem:2153] [citeproof:2153] The Primitive Element Theorem requires separability: in characteristic $p > 0$, a purely inseparable extension $K/K_0$ of degree $> 1$ cannot be generated by a single element that is the root of a separable polynomial, so the statement fails as stated. For instance, $\mathbb{F}_p(s^{1/p}, t^{1/p})/\mathbb{F}_p(s, t)$ has degree $p^2$ but cannot be generated by one algebraic element. In characteristic zero this is never an issue. This theorem has a geometric counterpart: every irreducible variety is birational to a hypersurface. [quotetheorem:2154] [citeproof:2154] The hypersurface produced here is not unique: different choices of primitive generator $y$ give different (but birational) hypersurfaces. What the theorem guarantees is that some hypersurface model exists, not that there is a canonical one. In particular, the hypersurface $V(f)$ may be singular even when $V$ itself is smooth; Hironaka's Resolution of Singularities (mentioned in the remark below) produces a smooth model, but that requires considerably more work. [remark: Resolution of Singularities] A related but much deeper result (Hironaka's Resolution of Singularities) states that in characteristic zero, every irreducible variety is birational to a smooth projective variety. This requires the full machinery of blowing up, developed in more advanced treatments. [/remark] ## Dimension Equals Transcendence Degree We now have two competing definitions of dimension: the geometric one (the minimum tangent space dimension at smooth points) and the algebraic one (the transcendence degree of the function field). A priori these could differ — the tangent space is a local analytic object, while the transcendence degree is a global field-theoretic one. That they agree is the content of the next theorem, and its proof is a clean illustration of how birational geometry mediates between the two perspectives. [quotetheorem:2155] [remark: Well-Definedness] This theorem in particular establishes that the transcendence degree is a birational invariant, i.e. it does not depend on the choice of generators for the function field. [/remark] [proof] Since dimension is a birational invariant (by the differential/chain rule argument above), and since $V$ is birational to the hypersurface $V(f) \subset \mathbb{A}^{n+1}_k$ constructed in the previous theorem, it suffices to verify the equality for $V = V(f)$. There, the function field is $\mathrm{FF}(k[W_1, \ldots, W_{n+1}]/(f))$, which has transcendence degree $n$ over $k$ (since $W_1, \ldots, W_n$ are algebraically independent modulo $(f)$, and $W_{n+1}$ is algebraic over them). On the other hand, $V(f)$ is a hypersurface in $\mathbb{A}^{n+1}_k$, which has dimension $n$ by the geometric definition of dimension for hypersurfaces. The two agree. [/proof] The theorem fails in the expected way when irreducibility is dropped: the variety $V(XY) \subset \mathbb{A}^2_k$ is a union of two lines, each of dimension $1$ and transcendence degree $1$, but its coordinate ring $k[X,Y]/(XY)$ has zero-divisors and the function field is not even well-defined as a domain. For reducible varieties, one works component by component. An important subtlety: the theorem identifies $\dim V$ with $\mathrm{trdeg}_k\,\mathcal{O}_V(\eta)$, but it does not say that every value of transcendence degree is realised by a smooth variety in any fixed ambient space. That is an embedding question, not a birational one. ## Krull's Height Theorem and Consequences We have seen that $n$ equations should cut dimension by at most $n$. The intuition in $\mathbb{A}^n_k$ is clear: each additional equation is a hypersurface condition, and hypersurfaces lower dimension by one. But does a hypersurface always cut dimension by exactly one? Could it lower dimension by more, or could the intersection be empty? Krull's Height Theorem gives a sharp algebraic answer and yields, as immediate consequences, several surprising rigidity results for projective space. [definition: Height of a Prime Ideal] Let $R$ be a finitely generated nilpotent-free $k$-algebra and $\mathfrak{p} \subset R$ a prime ideal. The **height** (or codimension) of $\mathfrak{p}$ is \begin{align*} \mathrm{ht}(\mathfrak{p}) = \sup\{n \geq 0 : \exists\ \mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n = \mathfrak{p}\}, \end{align*} the supremum of lengths of chains of prime ideals terminating at $\mathfrak{p}$. [/definition] [example: Height in polynomial rings] In $k[X,Y,Z]$, the maximal ideal $\mathfrak{m} = (X,Y,Z)$ has height $3$, witnessed by the chain $(0) \subsetneq (X) \subsetneq (X,Y) \subsetneq (X,Y,Z)$. No longer chain exists because $k[X,Y,Z]$ has Krull dimension $3$. For a more structural example, consider $\mathfrak{p} = (Y^2 - X^3) \subset k[X,Y]$. Since $Y^2 - X^3$ is irreducible, $\mathfrak{p}$ is a prime of height $1$ (one step up from $(0)$). However, in the quotient ring $R = k[X,Y]/(Y^2-X^3)$, the images $\bar{X}$ and $\bar{Y}$ satisfy $\bar{Y}^2 = \bar{X}^3$, so $\bar{X}$ and $\bar{Y}$ are not independent; the maximal ideal $(\bar{X}, \bar{Y})$ of $R$ has height $1$ in $R$, yet it requires two generators. This shows that Krull's bound (height $\leq$ number of generators) is not always attained: a prime of height $1$ can require two generators in the quotient. [/example] [definition: Minimal Prime over an Ideal] Let $I \subset R$ be an ideal. A **minimal prime over $I$** is a minimal element (under inclusion) among all prime ideals $\mathfrak{p} \supset I$. [/definition] Minimal primes over $I$ are not unique in general — for instance, if $I = (XY) \subset k[X,Y]$, then $(X)$ and $(Y)$ are both minimal primes over $I$. [quotetheorem:2156] Krull's theorem says that the height of a prime — i.e. the codimension of the corresponding variety — is bounded by the number of generators needed to cut it out. This is a sharp algebraic form of the geometric principle that $n$ equations cut out a variety of codimension at most $n$. The bound in Krull's theorem is sharp: the prime $(X_1, \ldots, X_n)$ in $k[X_1, \ldots, X_n]$ has height exactly $n$ and is generated by exactly $n$ elements. However, the theorem says nothing about what happens when the bound is not achieved: a prime of height $1$ might be generated by $1$ element (a principal prime) or might require more (as in the cuspidal example above). The theorem is also one-sided — it bounds height from above by generator count, not from below. The theorem has several striking consequences for projective varieties. [quotetheorem:2157] [citeproof:2157] The hypothesis $n \geq 2$ is essential: for $n = 1$, two distinct points in $\mathbb{P}^1_k$ are disjoint hypersurfaces (zero-dimensional), so the result fails. The theorem also requires projective space — two parallel lines in $\mathbb{A}^2_k$ are disjoint hypersurfaces, demonstrating that the projective setting is what forces non-empty intersection. The result applies to any two hypersurfaces regardless of degree; in particular, two curves in $\mathbb{P}^2_k$ always meet (Bezout's theorem gives a sharper count of the intersection points with multiplicity). [quotetheorem:2158] [citeproof:2158] This result has no analogue in differential topology: there exist non-constant smooth maps $S^{2n} \to S^2$ (e.g. the Hopf fibration for $n = 1$). The algebraic setting is far more rigid. Crucially, the hypothesis $n \geq 2$ cannot be dropped: the identity map $\mathrm{id}: \mathbb{P}^1_k \to \mathbb{P}^1_k$ is a non-constant morphism when $n = 1$. [quotetheorem:2159] [citeproof:2159] Again, the dimension hypothesis $n > m$ is sharp: for $n = m$, the identity map is non-constant. The theorem says that projective spaces of lower dimension are "too small" to receive non-constant maps from higher-dimensional projective spaces — a purely algebraic rigidity with no smooth analogue. Combined with the previous results, we see that $\mathbb{P}^n_k$ is, in a sense, "maximally rigid" among smooth projective varieties of its dimension. [quotetheorem:2160] [citeproof:2160] This example is a prototype for the general phenomenon that birational equivalence is far coarser than isomorphism. The two surfaces $\mathbb{P}^1_k \times \mathbb{P}^1_k$ and $\mathbb{P}^2_k$ are distinguished not by dimension, function field, or even many cohomological invariants (both are rational surfaces), but by the structure of their morphisms to lower-dimensional projective spaces. A hypersurface $C \subset \mathbb{P}^1_k \times \mathbb{P}^1_k$ has a bidegree $(a,b)$ measuring how it wraps around each factor; no such bidegree exists in $\mathbb{P}^2_k$, where every curve is cut out by a single homogeneous polynomial. This richer structure of curve classes will reappear when we study divisors. [remark: Geometry of $\mathbb{P}^1 \times \mathbb{P}^1$] Despite the non-isomorphism, $\mathbb{P}^1_k \times \mathbb{P}^1_k$ and $\mathbb{P}^2_k$ share many numerical invariants (they are both rational surfaces of dimension 2). Their difference becomes visible in the structure of their curve classes: in $\mathbb{P}^2_k$, all curves are cut out by a single homogeneous polynomial; in $\mathbb{P}^1_k \times \mathbb{P}^1_k$, curves have a bidegree $(a,b)$ measuring how they wrap around each factor. This richer structure will reappear when we study divisors. [/remark] Armed with tangent spaces and the full machinery of dimension theory, we focus on the most tractable and richest class of varieties: smooth projective irreducible curves. Chapter 5 applies Bézout's theorem to count intersections precisely, introduces local coordinates to study curve geometry at individual points, and establishes the degree-ramification relationship that governs all morphisms between curves. # 5. Algebraic Curves Having developed the general theory of projective varieties, tangent spaces, and dimension, we now restrict attention to the most tractable class of varieties: smooth projective irreducible curves, meaning projective varieties of dimension 1. This chapter introduces three interrelated themes — Bézout's theorem and intersection theory, the local structure of curves via local coordinates, and the global relationship between the degree of a morphism and the ramification of its fibers. Together, these prepare the ground for the study of divisors and the Riemann–Roch theorem in the next chapter. ## Bézout's Theorem and Intersections of Plane Curves How many points does a degree-$c$ plane curve meet a degree-$d$ plane curve in? Two distinct lines in the affine plane either meet in one point or are parallel — but in $\mathbb{P}^2_k$ they always meet in exactly one point, since parallelism is eliminated by the projective closure. Does this extend: does a curve of degree $c$ always meet a curve of degree $d$ in exactly $cd$ points? If not, what conditions force the count to equal $cd$ exactly, and what invariant captures the geometry at points of higher-order tangency? Before stating the answer, we note a subtlety: what does it even mean to speak of a projective curve "abstractly," independent of any embedding? Isomorphism class is the right answer in principle, but we will see that this is best understood through the local and global invariants developed below. [quotetheorem:2161] The hypothesis that $C$ and $D$ share no irreducible component is essential: if $F = G$, then $C = D$ and the intersection is the entire curve — infinitely many points, not $cd$. Equally important is working in projective space: two parallel lines in $\mathbb{A}^2_k$ have empty intersection, whereas Bézout requires $cd = 1$ for two lines. Projectivity supplies the "missing" intersection at infinity. Note also what Bézout does not provide at this stage: a construction of the multiplicities $m_p$. Their proper definition requires intersection-theoretic machinery developed in later algebraic geometry, and we proceed here with the weighted count as a black box to be unpacked in subsequent work. [remark: Intersection Multiplicities] The integers $m_p$ carry the notion of intersection multiplicity at $p$. A proper treatment of these multiplicities requires intersection-theoretic machinery developed in later algebraic geometry, so we state Bézout's theorem in this weighted form and explore its consequences. [/remark] We do not prove Bézout's theorem in full generality yet, but we establish it in the key special cases of low degree, which already yield nontrivial geometry. [quotetheorem:2162] [citeproof:2162] This special case already shows why algebraic closure is indispensable: over $\mathbb{R}$, the line $V(X_2)$ and the conic $V(X_0^2 + X_1^2 + X_2^2)$ meet in no real points, but over $\mathbb{C}$ the count comes out to $2 = 1 \cdot 2$ as required. [quotetheorem:2163] [citeproof:2163] [remark: Lines and Conics Are Isomorphic] This proposition shows that lines and smooth conics in $\mathbb{P}^2_k$ are isomorphic as abstract varieties. Later we will see that varieties $V(F)$ and $V(G)$ with $\deg F \geq 3$ and $\deg F \neq \deg G$ are never isomorphic, so degree is a genuine invariant at degree $\geq 3$. [/remark] [quotetheorem:2164] [citeproof:2164] Notice that the bound $4$ here matches the Bézout bound $cd = 2 \cdot 2 = 4$ exactly for two conics — the theorem says no more than this can happen, and Bézout guarantees equality (with multiplicities) when the curves share no component. The situation changes dramatically at higher degree: for two cubics, Bézout gives $cd = 9$, and the nine base points of a pencil of cubics form the foundation of the Cayley–Bacharach theorem, a story for a later chapter. The next result is a beautiful illustration of how linear algebra and intersection theory interact. [quotetheorem:2165] [citeproof:2165] The "no three collinear" hypothesis is necessary for uniqueness, not merely a technicality: if three of the five points lie on a line $\ell$, then for any line $\ell'$ the union $\ell \cup \ell'$ is a reducible conic through those three points (and we can choose $\ell'$ to pass through the remaining two), so uniqueness fails and infinitely many degenerate conics appear. The hypothesis rules out this degenerate case. [explanation: Two Families of Curves] The course singles out two important families. In $\mathbb{P}^2_k$, we have the hypersurfaces $V(F_d)$ for $F_d$ a degree-$d$ homogeneous polynomial. In $\mathbb{P}^1_k \times \mathbb{P}^1_k$, we have the bihomogeneous loci $V(G_{d_1, d_2})$ for $G_{d_1, d_2}$ of bidegree $(d_1, d_2)$. These two families are not isomorphic in general, and understanding their differences motivates much of what follows. From this point on, a **curve** means an irreducible projective variety of dimension $1$. We will shortly add smoothness to this list of standing hypotheses. [/explanation] [quotetheorem:2166] [citeproof:2166] [remark: A Weak Form of Bézout] As an immediate consequence, if $C$ is not contained in any hyperplane, then for any hyperplane $H \subset \mathbb{P}^n_k$, the intersection $H \cap C$ is a finite collection of points. This is a very weak version of Bézout's theorem; the full version pins down the count with multiplicity. [/remark] ## Local Coordinates on Smooth Curves On an affine variety $V \subset \mathbb{A}^n_k$, the maximal ideal at a point $p$ is generated by the $n$ coordinate functions $x_1 - p_1, \ldots, x_n - p_n$. On a smooth curve — a variety of dimension $1$ — we expect only one degree of freedom at each point. Should the maximal ideal $\mathfrak{m}_p$ of the local ring therefore be principal, generated by a single element? And if so, what does such a generator look like, and how uniquely is it determined? We now develop the local theory at a smooth point $p$ of a curve $C$. Recall that at any point $p$, the local ring $\mathcal{O}_{C,p} = \{f \in \mathcal{O}_C(\eta) : f \text{ regular at } p\}$ has a unique maximal ideal $\mathfrak{m}_p = \{f \in \mathcal{O}_{C,p} : f(p) = 0\}$. The key structural theorem is that smoothness forces this maximal ideal to be principal. [quotetheorem:2167] Note that this fails when $C$ has dimension greater than $1$ or when $p$ is a singular point — in either case $\mathfrak{m}_p$ need not be principal. A concrete example: on the nodal cubic $C = V(Y^2 - X^2(X+1))$, the node at the origin $p = (0, 0)$ has $\mathfrak{m}_p$ generated by both $x$ and $y$, and no single element generates $\mathfrak{m}_p$ in $\mathcal{O}_{C,p}$. Smoothness is what collapses the $n$-generator situation to a $1$-generator one. [definition: Local Coordinate] A generator $t_p$ of $\mathfrak{m}_p$ (i.e., $\mathfrak{m}_p = (t_p)$) is called a **local coordinate** or **local parameter** at $p$. It is unique up to multiplication by a unit $\lambda \in \mathcal{O}_{C,p}^\times$. [/definition] [remark: Tangent Space Interpretation] The tangent space $T_p C$ is naturally isomorphic to the linear dual of the Zariski cotangent space $\mathfrak{m}_p / \mathfrak{m}_p^2$. In particular, $\dim_k(\mathfrak{m}_p / \mathfrak{m}_p^2) = 1$ if and only if $C$ is smooth at $p$. Smoothness is precisely the condition that $t_p$ generates $\mathfrak{m}_p$. [/remark] To prove the theorem, we first establish a key algebraic lemma. [quotetheorem:2168] [citeproof:2168] Nakayama's lemma is a surprisingly powerful tool for detecting when modules vanish or when generating sets can be trimmed. The next result applies it directly to show that almost any element of $\mathfrak{m}_p$ generates it, provided we avoid the "higher-order" part $\mathfrak{m}_p^2$. [quotetheorem:2169] [citeproof:2169] The theorem on principality of $\mathfrak{m}_p$ now follows immediately: pick any $t \in \mathfrak{m}_p \setminus \mathfrak{m}_p^2$, which exists because at a smooth point $\mathfrak{m}_p \neq \mathfrak{m}_p^2$ (the cotangent space is one-dimensional and nonzero), and apply the result above. The quotients $\mathcal{O}_{C,p} / \mathfrak{m}_p^n$ carry the information of the Taylor series expansion of functions at $p$: $\mathcal{O}_{C,p} / \mathfrak{m}_p^n$ is a $k$-vector space of dimension $n$, and the image of $f \in \mathcal{O}_{C,p}$ in this quotient records the expansion of $f$ up to order $n-1$ in the local parameter. The maximal ideal $\mathfrak{m}_p$ is exactly the set of elements whose constant term vanishes. [example: Local Coordinate on the Affine Line] Take $C = \mathbb{A}^1_k$ and $p = 0$. Then $\mathcal{O}_{C,p} = \{f/g : f, g \in k[t],\, g(0) \neq 0\}$ and $\mathfrak{m}_p = (t)$. The element $1/(1 - t^2) \in \mathcal{O}_{C,p}$ "feels like" the formal power series $1 + t^2 + t^4 + \cdots$, which is invertible in $\mathcal{O}_{C,p}$ because its constant term is $1 \neq 0$. [/example] [example: Smooth vs. Singular Local Ring on a Cubic] Consider the nodal cubic $C = V(Y^2 - X^2(X+1)) \subset \mathbb{A}^2_k$. At a smooth point such as $p = (3, 6)$, the element $x - 3$ lies in $\mathfrak{m}_p \setminus \mathfrak{m}_p^2$ (its class in $\mathfrak{m}_p / \mathfrak{m}_p^2$ is nonzero, since the tangent direction at $p$ is not vertical), so $t_p = x - 3$ is a valid local parameter. At the node $p = (0, 0)$, the situation is different: both $x$ and $y$ vanish at $p$ and are needed to generate $\mathfrak{m}_p$ in $\mathcal{O}_{C,p}$. One can check directly that neither $x$ nor $y$ alone generates $\mathfrak{m}_p$ — any element of $\mathfrak{m}_p \setminus \mathfrak{m}_p^2$ would need to capture both "branches" of the node, but $\mathfrak{m}_p / \mathfrak{m}_p^2$ is two-dimensional at a node, so no single generator exists. This is precisely why the principal-maximal-ideal theorem requires smoothness. [/example] ### The Order of Vanishing For a polynomial $f \in k[X]$ in one variable, the order of vanishing at $0$ is simply the multiplicity of $X$ as a factor: $f(X) = X^n g(X)$ with $g(0) \neq 0$ gives $\operatorname{ord}_0(f) = n$. On a smooth curve, the maximal ideal $\mathfrak{m}_p$ is principal — so every regular function $f$ at $p$ factors as $f = t_p^n \cdot u$ for some unit $u \in \mathcal{O}_{C,p}^\times$ and $n \geq 0$. Can we therefore assign a well-defined integer to every element of the function field $\mathcal{O}_C(\eta)^\times$, extending this factorisation to rational functions with poles? And does the resulting integer depend on the choice of local parameter $t_p$? The principality of $\mathfrak{m}_p$ has a crucial corollary: every nonzero rational function $f \in \mathcal{O}_C(\eta)^\times$ has a well-defined order of vanishing at each smooth point $p$. [definition: Order of Vanishing] Let $p \in C$ be a smooth point with local parameter $t_p$. For $f \in \mathcal{O}_C(\eta)^\times$, define \begin{align*} \operatorname{ord}_p(f) = \max\{n \geq 0 : f \in \mathfrak{m}_p^n\} \end{align*} when $f \in \mathcal{O}_{C,p}$, and extend to the fraction field by $\operatorname{ord}_p(f/g) = \operatorname{ord}_p(f) - \operatorname{ord}_p(g)$. [/definition] The order is independent of the choice of local parameter: if $t_p$ and $t_p'$ both generate $\mathfrak{m}_p$, then $t_p' = \lambda t_p$ for a unit $\lambda \in \mathcal{O}_{C,p}^\times$, and this does not change the exponent in $f = t_p^n u$. The following theorem collects the key algebraic properties. [quotetheorem:2170] [citeproof:2170] The fact that $\operatorname{ord}_p$ is a group homomorphism $\mathcal{O}_C(\eta)^\times \to \mathbb{Z}$ whose kernel is $\mathcal{O}_{C,p}^\times$ and whose "positive part" is $\mathfrak{m}_p$ means exactly that $\mathcal{O}_{C,p}$ is a **discrete valuation ring** — a local PID with a unique nonzero prime ideal. This is a central structural fact: the local rings of a smooth curve are among the simplest possible Noetherian local rings, and the whole theory of divisors in the next chapter is built on this foundation. [example: Order on the Affine Line] Take $C = \mathbb{A}^1_k$, $p = 0$. Then $\mathcal{O}_{\mathbb{A}^1_k}(\eta) = k(t)$ and $\mathcal{O}_{\mathbb{A}^1_k, 0} = \{f/g : f, g \in k[t],\, g(0) \neq 0\}$. For the rational function \begin{align*} h(t) = \frac{t^2 + 8t^3}{1 + t^4}, \end{align*} the numerator has $t^2$ as its lowest-order term and the denominator is a unit at $0$, so $\operatorname{ord}_0(h) = 2$. [/example] [definition: Zeros and Poles] Let $f \in \mathcal{O}_C(\eta)$ and $p \in C$ smooth. If $\operatorname{ord}_p(f) = n \neq 0$, we say $f$ has a **zero of order $n$** at $p$ when $n > 0$, and a **pole of order $-n$** at $p$ when $n < 0$. [/definition] Two further corollaries illuminate the structure of rational functions on smooth curves. [quotetheorem:2171] This is a one-dimensional phenomenon: it fails in higher dimensions. For instance, $f = X/Y$ in the function field of $k[X, Y]$ satisfies $\operatorname{ord}_0(f)$ being undefined in the usual sense — neither $f$ nor $f^{-1}$ is regular at the origin in $\mathbb{A}^2_k$. The result for curves is a direct consequence of $\operatorname{ord}_p(f)$ being a well-defined integer: if $\operatorname{ord}_p(f) \geq 0$ then $f$ is regular; if $\operatorname{ord}_p(f) < 0$ then $\operatorname{ord}_p(f^{-1}) = -\operatorname{ord}_p(f) > 0$, so $f^{-1}$ is regular (and even vanishes). The smooth-curve hypothesis is what gives us $\operatorname{ord}_p$ as a genuine integer-valued homomorphism. [quotetheorem:2172] [citeproof:2172] The extension theorem depends crucially on smoothness. On a cuspidal cubic $C = V(Y^2 - X^3)$, the rational map from $\mathbb{P}^1_k$ to $C$ defined by $t \mapsto (t^2, t^3)$ is a bijection of varieties but its inverse $[x:y:1] \mapsto [y:x]$ (for $x \neq 0$) fails to extend to a morphism at the cusp, precisely because the local ring at the cusp is not a discrete valuation ring. Thus the theorem genuinely requires a smooth source curve. [remark: Projection from a Smooth Conic] Given a smooth conic $C \subset \mathbb{P}^2_k$ and $p \in C$, the projection from $p$ is a rational map $\pi: \mathbb{P}^2_k \dashrightarrow \mathbb{P}^1_k$. Its restriction to $C$ is necessarily a morphism by the theorem above — a fact we also established directly when we parametrised conics. [/remark] ## Degree and Ramification of Morphisms Between Curves Consider the map $z \mapsto z^n$ from $\mathbb{C}^\times$ to itself. Over a generic point $w \neq 0$ in the target, there are exactly $n$ preimages — the $n$-th roots of $w$. Yet the structure of those $n$ preimages can "collapse": at $w = 0$, all $n$ roots coincide at $0$, and the fiber is a single point with higher-order contact. How does this "generic fiber size equals $n$, but special fibers collapse" pattern generalise to morphisms between algebraic curves? What algebraic invariant captures $n$, and what local invariant captures the degree of collapsing? Having established the local structure, we now turn to the global relationship between two smooth projective irreducible curves connected by a morphism. This theory parallels the theory of holomorphic maps between Riemann surfaces. Let $f: C \to D$ be a nonconstant morphism of curves. By the theorem that the image of a closed set under a projective morphism is closed, $f(C)$ is closed in $D$; since $f$ is nonconstant, $f(C)$ is not a finite set of points, so $f(C) = D$. Thus $f$ is surjective. [quotetheorem:2173] [citeproof:2173] The finiteness of the field extension $[\mathcal{O}_C(\eta) : \mathcal{O}_D(\eta)]$ is the key numerical invariant of the morphism. [definition: Degree of a Morphism] For a nonconstant morphism $f: C \to D$ between curves, the **degree** of $f$ is \begin{align*} \deg f = [\mathcal{O}_C(\eta) : \mathcal{O}_D(\eta)]. \end{align*} [/definition] [remark: Algebro-Geometric Degree vs. Topological Degree] This is the algebro-geometric analogue of the degree of a continuous map between topological surfaces. The next theorem makes the analogy precise by relating $\deg f$ to preimage counts. [/remark] [example: Squaring Map on $\mathbb{P}^1_k$] The morphism $\mathbb{P}^1_k \to \mathbb{P}^1_k$ sending $t \mapsto t^2$ (on the affine chart $\mathbb{A}^1_k \subset \mathbb{P}^1_k$) has degree $2$, since the induced extension $k(t^2) \hookrightarrow k(t)$ has degree $2$. [/example] To relate the degree to fiber sizes, we introduce ramification. [definition: Ramification Index] Let $\varphi: C \to D$ be a nonconstant morphism of smooth projective irreducible curves, $p \in C$, and $q = \varphi(p) \in D$. The **ramification index** of $\varphi$ at $p$ is \begin{align*} e_p = \operatorname{ord}_p(\varphi^* t_q), \end{align*} where $t_q$ is a local coordinate at $q$ and $\varphi^*: \mathcal{O}_{D,q} \to \mathcal{O}_{C,p}$ is the pullback. [/definition] The ramification index $e_p$ measures how many times the map "wraps" around $q$ locally at $p$: $e_p = 1$ means the map is locally an isomorphism, while $e_p > 1$ means the map is locally $e_p$-to-one in the strongest sense, with $p$ being a genuine ramification point. [example: Ramification of the Squaring Map] For $\varphi: \mathbb{A}^1_k \to \mathbb{A}^1_k$, $z \mapsto z^2$, with $p = 0$ and $q = 0$: the local coordinate at $q$ is $t_q = z$, and $\varphi^* t_q = z^2$, so $e_0 = \operatorname{ord}_0(z^2) = 2$. At any $p \neq 0$, the map is locally a unit times a linear term, so $e_p = 1$. [/example] [quotetheorem:2174] [citeproof:2174] This theorem requires both projectivity and smoothness in a strong way. On an affine piece, the fiber count can miss points "at infinity" and the formula fails. On a singular curve, $e_p$ is not even well-defined at singular points, since $\mathcal{O}_{C,p}$ is not a DVR and $\operatorname{ord}_p$ does not exist as an integer-valued homomorphism. The projective-plus-smooth setting is what makes both the degree and the ramification index well-defined and globally consistent. [quotetheorem:2175] When $k$ has characteristic $0$, all finite extensions are separable, so ramification is always finite. The theorem fails in characteristic $p$: the Frobenius morphism $t \mapsto t^p$ is purely inseparable of degree $p$, and every point has ramification index $p$. [example: Ramification on Elliptic Curves] Consider the affine curve $C^{\text{aff}} = V(Y^2 - (X-a)(X-b)(X-c))$ for distinct $a, b, c \in k$, and the projection $\pi: C^{\text{aff}} \to \mathbb{A}^1_k$ onto the $X$-coordinate. At a generic value $x \neq a, b, c$, there are exactly two preimages (corresponding to $\pm\sqrt{(x-a)(x-b)(x-c)}$), so $e_p = 1$ for both. At $x = a$ (and similarly $b$ or $c$), there is a unique point on $C^{\text{aff}}$, so $e_p = 2$ — this is a ramification point. The total degree is $2$, consistent with $e_p + e_{p'} = 1 + 1 = 2$ at a generic fiber. [/example] [illustration:ag-squaring-map-p1] ### Global Consequences: Zeros and Poles Sum to Zero Consider the affine line $\mathbb{A}^1_k$: the function $X$ vanishes at $0$ and has no poles, so the total zero-minus-pole count is $1$, not $0$. On the projective line $\mathbb{P}^1_k$, by contrast, $X$ has a zero at $[0:1]$ and a pole at $[1:0]$ (the "point at infinity"), and the count becomes $1 - 1 = 0$. This is not a coincidence: on any smooth projective irreducible curve, zeros and poles must balance exactly. Why? And what does this say about rational functions on more general curves? The degree-ramification theorem has an important global consequence for rational functions. [quotetheorem:2176] [citeproof:2176] Both hypotheses — projectivity and irreducibility — are essential. On the affine line, $X$ has a zero at $0$ and no poles, so the sum of orders is $1 \neq 0$: the projective closure supplies the pole at infinity that restores balance. On a reducible curve such as the union of the $x$-axis and the $y$-axis in $\mathbb{A}^2_k$, the function $x/y$ does not have a well-defined order at the intersection point $(0,0)$, and the global count breaks down. [remark: Morphisms to $\mathbb{P}^1_k$ and Rational Functions] The gist of this proof is that morphisms from a smooth curve to $\mathbb{P}^1_k$ correspond to rational functions. In higher dimensions, in order to construct morphisms $C \to \mathbb{P}^N_k$ we will need collections of rational functions satisfying certain regularity conditions. This observation motivates the theory of **divisors**, which organises the infinite-dimensional $k$-vector space $\mathcal{O}_C(\eta)$ into finite-dimensional subspaces by tracking zeros and poles. That theory is developed in the next chapter. [/remark] The final section of Chapter 5 reveals a profound constraint: rational functions on a smooth projective curve must have equal numbers of zeros and poles, so their zero-pole data is not freely chosen. Chapter 6 systematises this data by introducing divisors—formal finite combinations of points—and develops the Riemann–Roch theorem, which pins down how many rational functions have prescribed pole bounds. # 6. Divisors on Curves The previous chapter ended with a global identity: on any smooth projective irreducible curve, every rational function has the same number of zeros as poles, counted with multiplicity. This hints that zero-pole data is a genuine structural invariant of the curve. In this chapter we formalise that data as a divisor, study divisors up to the equivalence induced by principal divisors, introduce the canonical class coming from differential forms, and culminate in the Riemann–Roch theorem, which pins down the dimension of meromorphic functions with prescribed pole bounds. ## Basic Constructions In the previous chapter, we established a fundamental constraint on rational functions on a smooth projective curve $C$: for any nonzero $f \in \mathcal{O}_C(\eta)$, the set of points where $f$ is not regular is finite, and the integers $\mathrm{ord}_p(f)$ satisfy \begin{align*} \sum_{p \in C} \mathrm{ord}_p(f) = 0. \end{align*} This is not merely a curiosity. It tells us that zeros and poles of rational functions are never freely chosen — they are constrained to cancel in total. The question is: how do we encode this cancellation algebraically, and what does it buy us? If we want to construct a morphism $C \to \mathbb{P}^N_k$, we need to produce $N$ rational functions that are simultaneously regular on a dense open set and have controlled, comparable pole behaviour at the remaining points. The right framework for organising this data — tracking which points appear, with what multiplicity, and which configurations are realisable by rational functions — is the theory of divisors. [motivation] ### From rational maps to collections of functions Suppose $\varphi : C \to \mathbb{P}^N_k$ is a nonconstant morphism whose image is not contained in the hyperplane $\{X_0 = 0\}$. Then there is an open subset $C^\circ \subset C$ that maps into the affine patch $\mathbb{A}^N_k$, and each coordinate function $f_i = X_i/X_0$ restricts to a rational function on $C$ that is regular on $C^\circ$. At the finitely many points $q \in C \setminus C^\circ$, all $f_i$ have poles, and crucially $\mathrm{ord}_q(f_i) = \mathrm{ord}_q(f_j)$ for all $i, j$ — the pole orders at excluded points agree across all coordinate functions. This observation motivates studying not individual rational functions but *collections* of rational functions sharing a common regularity profile. The right way to specify such a profile is to declare, at each point of $C$, an integer bound on the order of vanishing or the order of the pole. A divisor does exactly this: it is a formal assignment of an integer to every point of $C$, vanishing at all but finitely many points. [/motivation] ### The Divisor Group [definition: Divisor] Let $C$ be a smooth projective irreducible curve over an algebraically closed field $k$. A **divisor** on $C$ is an element of the free abelian group \begin{align*} \mathrm{Div}(C) = \bigoplus_{p \in C} \mathbb{Z}[p], \end{align*} that is, a formal finite $\mathbb{Z}$-linear combination $D = \sum_{p \in C} a_p [p]$ where $a_p \in \mathbb{Z}$ and $a_p = 0$ for all but finitely many $p$. [/definition] The group $\mathrm{Div}(C)$ is abelian under the natural addition of coefficients: $(D + E)_p = a_p + b_p$. The zero divisor has all coefficients equal to zero. [definition: Degree and Effectivity] Given a divisor $D = \sum_i a_i [p_i]$, its **degree** is \begin{align*} \deg D = \sum_i a_i \in \mathbb{Z}. \end{align*} We say $D$ is **effective** (written $D \geq 0$) if $a_i \geq 0$ for all $i$. More generally, $D \geq E$ means $D - E$ is effective. [/definition] The degree map $\deg : \mathrm{Div}(C) \to \mathbb{Z}$ is a group homomorphism. Effective divisors should be thought of as specifying a finite set of points with non-negative multiplicities — they arise naturally as intersection loci. ### Principal Divisors and Linear Equivalence The bridge between divisors and rational functions is the following construction. [definition: Divisor of a Rational Function] Let $f \in \mathcal{O}_C(\eta)^\times$ be a nonzero rational function on $C$. The **divisor of $f$** is \begin{align*} \operatorname{div}(f) = \sum_{p \in C} \mathrm{ord}_p(f)\, [p]. \end{align*} A divisor $D$ is called **principal** if $D = \operatorname{div}(f)$ for some $f \in \mathcal{O}_C(\eta)^\times$. [/definition] Since $f$ has only finitely many zeros and poles, $\operatorname{div}(f)$ is indeed an element of $\mathrm{Div}(C)$. The map $f \mapsto \operatorname{div}(f)$ is a group homomorphism from $\mathcal{O}_C(\eta)^\times$ to $\mathrm{Div}(C)$: we have $\operatorname{div}(fg) = \operatorname{div}(f) + \operatorname{div}(g)$ and $\operatorname{div}(f^{-1}) = -\operatorname{div}(f)$, both following directly from the additivity of $\mathrm{ord}_p$. [quotetheorem:2177] [citeproof:2177] The converse — that every degree-zero divisor is principal — fails in general. It holds for $\mathbb{P}^1_k$: given any degree-zero divisor $D = \sum n_i[a_i]$ with $\sum n_i = 0$, the rational function $f = \prod_i (x - a_i)^{n_i}$ has $\operatorname{div}(f) = D$, where affine coordinates are chosen so $a_i \neq [1:0]$. For curves of higher genus the converse fails, and that failure is exactly what the Picard group measures. [definition: Linear Equivalence] Two divisors $D, E \in \mathrm{Div}(C)$ are **linearly equivalent**, written $D \sim E$, if $D - E = \operatorname{div}(f)$ for some $f \in \mathcal{O}_C(\eta)^\times$. [/definition] Linear equivalence is an equivalence relation (reflexive by $\operatorname{div}(1) = 0$, symmetric by $\operatorname{div}(f^{-1}) = -\operatorname{div}(f)$, transitive by $\operatorname{div}(fg) = \operatorname{div}(f) + \operatorname{div}(g)$). The principal divisors form a subgroup of $\mathrm{Div}(C)$, and linear equivalence is exactly the equivalence relation generated by this subgroup. ### The Picard Group [definition: Picard Group] The **Picard group** of $C$ is the quotient \begin{align*} \mathrm{Pic}(C) = \mathrm{Div}(C) \,/\, \{\text{principal divisors}\}. \end{align*} Its elements, called **divisor classes**, are linear equivalence classes $[D]$. The **degree-zero Picard group** is \begin{align*} \mathrm{Pic}^0(C) = \ker\!\big(\deg : \mathrm{Pic}(C) \to \mathbb{Z}\big), \end{align*} the subgroup consisting of classes of degree-zero divisors. [/definition] Since every principal divisor has degree zero, the degree map descends to a well-defined homomorphism $\deg : \mathrm{Pic}(C) \to \mathbb{Z}$, and $\mathrm{Pic}^0(C)$ is its kernel. The quotient $\mathrm{Pic}(C)/\mathrm{Pic}^0(C)$ is isomorphic to the image of $\deg$ in $\mathbb{Z}$; for a connected curve this image is all of $\mathbb{Z}$, so there is a short exact sequence \begin{align*} 0 \to \mathrm{Pic}^0(C) \to \mathrm{Pic}(C) \xrightarrow{\deg} \mathbb{Z} \to 0. \end{align*} The group $\mathrm{Pic}^0(C)$ is the algebro-geometric incarnation of the Jacobian variety of $C$. For $C = \mathbb{P}^1_k$, the converse of the degree-zero theorem holds, so $\mathrm{Pic}^0(\mathbb{P}^1_k) = 0$ and $\mathrm{Pic}(\mathbb{P}^1_k) \cong \mathbb{Z}$, generated by the class of any point. For an elliptic curve, $\mathrm{Pic}^0(C)$ is isomorphic to the curve itself as an algebraic group — a manifestation of the deep interaction between the curve's geometry and its function theory. ### Hyperplane Sections and Divisors from Polynomials There is a second natural source of divisors on $C$, coming from intersecting with hypersurfaces in a projective embedding. [definition: Divisor of a Linear Form] Suppose $C \subset \mathbb{P}^N_k$ and $L$ is a linear form with $C \not\subset V(L)$. The **hyperplane section divisor** of $L$ is \begin{align*} \operatorname{div}(L) = \sum_{p \in C} n_p\, [p], \end{align*} where $n_p = \mathrm{ord}_p(L/X_i)$ for any index $i$ with $X_i(p) \neq 0$. [/definition] The independence of the choice of $i$ follows from the fact that $X_i/X_j$ is a unit in $\mathcal{O}_{C,p}$ whenever both $X_i(p) \neq 0$ and $X_j(p) \neq 0$, so changing the index does not alter the order of vanishing. A parallel definition applies to homogeneous polynomials of higher degree. [definition: Divisor of a Homogeneous Polynomial] Let $G$ be a homogeneous polynomial of degree $m$ with $C \not\subset V(G)$. The **divisor of $G$** is \begin{align*} \operatorname{div}(G) = \sum_{p \in C} n_p\, [p], \qquad n_p = \mathrm{ord}_p(G/X_i^m) \end{align*} for any $i$ with $X_i(p) \neq 0$. [/definition] [quotetheorem:2178] [citeproof:2178] This common degree is called the **degree of $C$** in the embedding $C \subset \mathbb{P}^n_k$. Any two hyperplane section divisors are therefore linearly equivalent. As an application, we recover a form of Bézout's theorem. [quotetheorem:2179] [citeproof:2179] This bound is sharp when $C$ and $C'$ meet transversally: if every intersection point is a simple zero of both defining polynomials, then $|C \cap C'| = \deg(C)\deg(C')$ exactly. The inequality in general reflects the contribution of higher-order tangencies, which inflate the local intersection multiplicity at a point beyond 1. In particular, a line meets a smooth curve of degree $d$ in exactly $d$ points counted with multiplicity — this is the content of the degree formula for $C' = V(L)$ with $\deg L = 1$. ### Pullback of Divisors Divisors interact naturally with morphisms between curves. [definition: Pullback of a Divisor] Let $\varphi : C \to D$ be a nonconstant morphism of smooth projective irreducible curves and let $E = \sum_q b_q [q] \in \mathrm{Div}(D)$. The **pullback** of $E$ along $\varphi$ is \begin{align*} \varphi^* E = \sum_{q \in D} b_q \sum_{\varphi(p) = q} e_p\, [p] \in \mathrm{Div}(C), \end{align*} where $e_p = \mathrm{ord}_p(\varphi^* t_q)$ is the ramification index of $\varphi$ at $p$, with $t_q$ a local coordinate at $q$. [/definition] The pullback $\varphi^* : \mathrm{Div}(D) \to \mathrm{Div}(C)$ is a group homomorphism. For a single point $q \in D$, $\varphi^*[q] = \sum_{\varphi(p) = q} e_p [p]$, which has degree $\sum_{\varphi(p) = q} e_p = \deg \varphi$ by the Degree Equals Sum of Ramification Indices theorem from Chapter 5. In particular, pullback multiplies degrees by $\deg \varphi$: \begin{align*} \deg(\varphi^* E) = (\deg \varphi)(\deg E). \end{align*} Moreover, $\varphi^*$ is compatible with principal divisors: $\varphi^*(\operatorname{div}(f)) = \operatorname{div}(f \circ \varphi)$ for $f \in \mathcal{O}_D(\eta)^\times$. Consequently, pullback descends to a group homomorphism $\varphi^* : \mathrm{Pic}(D) \to \mathrm{Pic}(C)$. ### The Riemann–Roch Space We now arrive at the central construction that ties together divisors and linear algebra. [definition: Riemann–Roch Space] For a divisor $D \in \mathrm{Div}(C)$, the **Riemann–Roch space** of $D$ is \begin{align*} \mathcal{L}(D) = \{f \in \mathcal{O}_C(\eta)^\times : \operatorname{div}(f) + D \geq 0\} \cup \{0\}. \end{align*} [/definition] Concretely, if $D = \sum_p a_p [p]$, then $f \in \mathcal{L}(D)$ means that $\mathrm{ord}_p(f) \geq -a_p$ for all $p \in C$. In words: $f$ is allowed to have a pole at $p$ of order at most $a_p$ if $a_p > 0$, and is forced to vanish at $p$ to order at least $-a_p$ if $a_p < 0$. The set $\mathcal{L}(D)$ is closed under addition and scalar multiplication by $k^\times$ (since $\mathrm{ord}_p$ is a valuation), so it is a $k$-vector space. It is always finite-dimensional. [example: Riemann–Roch Space on $\mathbb{P}^1$] Take $C = \mathbb{P}^1_k$ and $D = n[1:0]$ for some $n \geq 0$, where $[1:0]$ is the point at infinity in the standard affine chart $t = X_1/X_0$. For $f \in \mathcal{L}(D)$, the condition $\operatorname{div}(f) + D \geq 0$ requires $\mathrm{ord}_p(f) \geq 0$ for all $p \neq [1:0]$ and $\mathrm{ord}_{[1:0]}(f) \geq -n$. On the affine line $\mathbb{A}^1_k \subset \mathbb{P}^1_k$, the rational functions regular away from $[1:0]$ are exactly the polynomial functions in $t$, and $\mathrm{ord}_{[1:0]}(t^j) = -j$. The condition $\mathrm{ord}_{[1:0]}(f) \geq -n$ then forces $f$ to be a polynomial of degree at most $n$ in $t$. Therefore $\mathcal{L}(n[1:0]) = \mathrm{Span}_k\{1, t, t^2, \ldots, t^n\}$, which has dimension $n+1$. In particular, for $n = 1$ the case $D = [1:0]$ gives $\mathcal{L}(D) = \mathrm{Span}_k\{1, t\}$, the linear functions on the affine patch, of dimension 2. [/example] [quotetheorem:2180] [citeproof:2180] [remark: Dimension and Equivalence] The dimension $\ell(D) := \dim_k \mathcal{L}(D)$ is preserved under linear equivalence: if $D \sim E$, then $\mathcal{L}(D) \cong \mathcal{L}(E)$ as $k$-vector spaces via the isomorphism $g \mapsto g/f$ where $D - E = \operatorname{div}(f)$. The proof that this is an isomorphism is direct: $g \in \mathcal{L}(D)$ iff $\operatorname{div}(g) + D \geq 0$ iff $\operatorname{div}(g/f) + E = \operatorname{div}(g) - \operatorname{div}(f) + E = \operatorname{div}(g) + D - \operatorname{div}(f) + E - D + \operatorname{div}(f) \geq 0$ — unravelling the definitions shows $\operatorname{div}(g/f) + E \geq 0$ iff $\operatorname{div}(g) + D \geq 0$, so the map is an isomorphism. This justifies working with divisors only up to linear equivalence: the space $\mathcal{L}(D)$ depends only on the class $[D] \in \mathrm{Pic}(C)$. [/remark] The Riemann–Roch space $\mathcal{L}(D)$ organises the infinite-dimensional $k$-vector space $\mathcal{O}_C(\eta)$ into manageable finite-dimensional pieces. Each choice of basis $\{f_0, \ldots, f_N\}$ for some $\mathcal{L}(D)$ determines a rational map $C \dashrightarrow \mathbb{P}^N_k$ by $p \mapsto [f_0(p) : \cdots : f_N(p)]$, which is in fact a morphism since $C$ is smooth. The central problem of the theory is to determine $\ell(D) = \dim_k \mathcal{L}(D)$ — and this is precisely the problem that the Riemann–Roch theorem, treated in the next section, solves. ## Differentials [motivation] ### The gap in our toolkit Section 5.1 gave us rational functions on a curve, their orders at points, and the divisor language to package this data. But there is a canonical class of geometric objects on a smooth curve that rational functions do not capture: differential 1-forms. In differential geometry, a 1-form on a Riemann surface is something you can pull back along maps, integrate along paths, and whose zeros and poles reflect the topology of the surface. The sum of the orders of any meromorphic 1-form on a compact Riemann surface of genus $g$ equals $2g - 2$ — a purely topological invariant. ### The algebraic question In an algebraic setting we have no analysis: no limits, no integration in the classical sense. Yet the data of a differential 1-form — specifically, how it transforms under coordinate changes and what orders it has at points — is entirely algebraic. The challenge is to construct, from ring-theoretic data alone, a module that plays the role of the space of 1-forms, and to show that the divisor of any nonzero element has degree $2g - 2$. The key insight is that the Leibniz rule $d(fg) = f\,dg + g\,df$ and $k$-linearity of $d$ determine everything. We formalise this via the module of Kähler differentials. [/motivation] ### Kähler Differentials for a Field Extension What purely algebraic object behaves like "the space of $df$'s" on the function field of a curve? On a smooth manifold, a 1-form assigns to each point a cotangent vector; on an algebraic curve, we lack any notion of limits to define tangent vectors directly. The question is whether the Leibniz rule $d(fg) = f\,dg + g\,df$ and $k$-linearity alone are enough to pin down a useful algebraic replacement. It turns out they are: the following universal construction recovers exactly the right object. Let $k$ be an algebraically closed field of characteristic zero, and let $K/k$ be a transcendental field extension. [definition: k-linear Derivation] A **$k$-linear derivation** of $K$ into a $K$-vector space $U$ is a $k$-linear map $D: K \to U$ satisfying the Leibniz rule: \begin{align*} D(xy) = x\,D(y) + y\,D(x) \quad \text{for all } x, y \in K. \end{align*} [/definition] The Leibniz rule forces $D(\lambda) = 0$ for all $\lambda \in k$ (since $D$ is $k$-linear and $D(1) = D(1 \cdot 1) = 2D(1)$ implies $D(1) = 0$). So a derivation measures how $K$ varies relative to $k$, not within $k$. [definition: Module of Kähler Differentials] The **module of Kähler differentials** $\Omega_{K/k}$ is the $K$-vector space \begin{align*} \Omega_{K/k} = \frac{\text{$K$-vector space spanned by symbols $\delta(f)$ for $f \in K$}}{\langle\, \delta(x+y) - \delta(x) - \delta(y),\; \delta(\lambda),\; \delta(xy) - x\,\delta(y) - y\,\delta(x) \mid x,y \in K,\, \lambda \in k\,\rangle} \end{align*} equipped with the **exterior derivative** $d: K \to \Omega_{K/k}$ defined by $df = [\delta(f)]$. [/definition] The construction packages exactly the Leibniz rule into a universal object. The crucial feature is the following universal property: for any $K$-vector space $U$ and any $k$-linear derivation $D: K \to U$, there exists a unique $K$-linear map $\lambda: \Omega_{K/k} \to U$ such that $D = \lambda \circ d$. In other words, $d: K \to \Omega_{K/k}$ is the universal $k$-linear derivation. [remark: Chain Rule from Universality] The universal property gives an algebraic chain rule for free. If $f \in k(W_1, \ldots, W_r)$ is a rational function and $y = f(x_1, \ldots, x_r)$ with $x_i \in K$, then \begin{align*} dy = \sum_{i=1}^{r} \frac{\partial f}{\partial W_i}(x_1, \ldots, x_r)\, dx_i. \end{align*} In particular, if $x_1, \ldots, x_r$ generate $K$ over $k$, then $dx_1, \ldots, dx_r$ span $\Omega_{K/k}$ over $K$. [/remark] The key structural result for function fields of curves is the following. [quotetheorem:2181] [citeproof:2181] This theorem tells us that the function field of a smooth curve has a one-dimensional space of "abstract differentials" over $K$, generated by any local coordinate. [remark: Characteristic Zero Hypothesis] In characteristic $p > 0$, the relation $d(x^p) = p x^{p-1}\,dx = 0$ can cause $\Omega_{K/k}$ to be larger or degenerate. The correct replacement for characteristic zero is the separability hypothesis used in the theorem. From here on we work over characteristic zero for simplicity, but separability is the essential condition. [/remark] ### Differentials on a Smooth Curve Given that $\Omega_{K/k}$ is a one-dimensional $K$-vector space, what does this module look like locally at a point $p \in C$? Near $p$ we have a local coordinate $t_p$, and the abstract generator $dt_p$ should generate the "local differentials." The real question is whether local freeness holds: is every rational differential that is regular at $p$ a multiple of $dt_p$ by a regular function? Without smoothness this can fail — at a cusp, for instance, the local ring is not a DVR and the module of differentials can acquire torsion. Smoothness is exactly the condition that ensures the following clean picture. Let $C$ be a smooth projective curve over $k$, with function field $K = \mathcal{O}_C(\eta)$ where $\eta$ is the generic point. [definition: Rational Differential] The **space of rational differentials** on $C$ is $\Omega_C(\eta) = \Omega_{\mathcal{O}_C(\eta)/k}$. A rational differential $\omega \in \Omega_C(\eta)$ is **regular at $p \in C$** if it can be expressed as $\omega = \sum_i f_i\, dg_i$ with $f_i, g_i \in \mathcal{O}_{C,p}$. The set of differentials regular at $p$ is denoted $\Omega_{C,p}$, which is naturally an $\mathcal{O}_{C,p}$-module. [/definition] The local structure of this module is completely determined by any local coordinate. [quotetheorem:2182] [citeproof:2182] This freeness is essential: it means every rational differential, when regular at $p$, looks like $f\,dt_p$ for a uniquely determined $f \in \mathcal{O}_{C,p}$ (up to the choice of $t_p$). When we change local coordinate from $t_p$ to $t_p'$, we have $dt_p = u\,dt_p'$ for some unit $u \in \mathcal{O}_{C,p}^\times$, so the generating element transforms by a unit — consistent with the order being well-defined. Smoothness is what makes this work. At a singular point such as the cusp of $y^2 = x^3$, the local ring is not a discrete valuation ring: the conductor ideal is nontrivial, and $\Omega_{C,p}$ can fail to be free. The local freeness theorem thus relies entirely on the assumption that $p$ is a smooth point, and it breaks down in characteristic zero exactly when the local ring fails to be regular. ### The Order of a Differential We have two pieces: any nonzero rational differential $\omega$ can be written locally as $f\,dt_p$, and we know how to measure the order of a rational function $f$ at $p$. How should we assign an order to the differential $\omega$ itself? The Leibniz rule forces $dt_p$ to transform by a unit under coordinate change, so the order of $f$ is independent of which local coordinate we pick. This motivates the following definition. [definition: Order of a Differential] Let $\omega \in \Omega_C(\eta)$ be a nonzero rational differential and $p \in C$ a point with local coordinate $t_p$. Write $\omega = f\,dt_p$ for a unique $f \in K^\times$. The **order of $\omega$ at $p$** is \begin{align*} \operatorname{ord}_p(\omega) = \operatorname{ord}_p(f). \end{align*} [/definition] This is well-defined: if $t_p'$ is another local coordinate then $dt_p = u\,dt_p'$ with $u \in \mathcal{O}_{C,p}^\times$, so $\omega = fu\,dt_p'$ and $\operatorname{ord}_p(fu) = \operatorname{ord}_p(f) + \operatorname{ord}_p(u) = \operatorname{ord}_p(f)$ since $u$ is a unit. Exactly as for rational functions, $\operatorname{ord}_p(\omega) \neq 0$ for only finitely many $p \in C$. The argument is essentially the same: on any affine patch $\omega$ is expressed using finitely many regular functions, and outside that patch one uses the explicit coordinate change. ### The Canonical Divisor Every nonzero rational differential $\omega$ has a well-defined order at each point, and these orders are zero away from finitely many points. What total degree does this force on the resulting divisor? On $\mathbb{P}^1$ the computation $\operatorname{div}(dt) = -2[\infty]$ shows the degree is $-2 = 2(0) - 2$. The canonical divisor makes this into a definition that applies to all smooth curves. [definition: Divisor of a Differential] Let $\omega \in \Omega_C(\eta)$ be a nonzero rational differential. The **divisor of $\omega$** is \begin{align*} \operatorname{div}(\omega) = \sum_{p \in C} \operatorname{ord}_p(\omega)\,[p]. \end{align*} A divisor $D$ on $C$ is called **canonical** if $D = \operatorname{div}(\omega)$ for some nonzero $\omega \in \Omega_C(\eta)$. [/definition] Different nonzero differentials produce linearly equivalent divisors: if $\omega, \varpi \in \Omega_C(\eta)$ are both nonzero, then $\omega/\varpi$ is a well-defined element of $K^\times$ (since $\Omega_{K/k}$ is one-dimensional over $K$), and $\operatorname{div}(\omega) - \operatorname{div}(\varpi) = \operatorname{div}(\omega/\varpi)$ is principal. Therefore all canonical divisors lie in the same linear equivalence class. [definition: Canonical Class] The **canonical class** $K_C$ is the class of $\operatorname{div}(\omega)$ for any nonzero $\omega \in \Omega_C(\eta)$ in $\operatorname{Pic}(C)$. The genus of $C$ is defined as $g(C) = \dim_k \mathcal{L}(K_C)$ where $K_C$ denotes any representative canonical divisor. [/definition] [remark: Canonical vs Principal] Canonical divisors and principal divisors are distinct classes in general. Both are linearly equivalent to each other within their respective families, but a canonical divisor is the divisor of a differential form, not a function. They transform differently under coordinate changes: a function transforms with no correction, while a differential picks up the Jacobian of the coordinate change, which is why $\operatorname{div}(\omega)$ is not generally principal. [/remark] ### The Canonical Divisor of $\mathbb{P}^1_k$ [example: Canonical Divisor of the Projective Line] Let $C = \mathbb{P}^1_k$ with homogeneous coordinates $[X_0 : X_1]$. Let $t = X_0/X_1$ be the standard local coordinate near $[0:1]$. We compute $\operatorname{div}(dt)$. For $p \in \mathbb{P}^1_k \setminus \{[1:0]\}$, the rational function $t - a$ (where $p = [a:1]$) is a local coordinate at $p$, and $dt = d(t-a)$, so $\operatorname{ord}_p(dt) = 0$. At $[1:0]$, the local coordinate is $s = X_1/X_0 = 1/t$. We compute $dt = d(1/s) = -s^{-2}\,ds = -t^2\,ds$, so $\omega = dt = (-t^2)\,ds$. At $[1:0]$ we have $\operatorname{ord}_{[1:0]}(t) = -1$, so $\operatorname{ord}_{[1:0]}(-t^2) = -2$. Therefore \begin{align*} \operatorname{div}(dt) = -2\,[1:0]. \end{align*} This canonical divisor has degree $-2 = 2g(\mathbb{P}^1_k) - 2 = 2(0) - 2$. Since the degree is negative, $\mathcal{L}(\operatorname{div}(dt)) = 0$, confirming $g(\mathbb{P}^1_k) = 0$. [/example] [illustration:ag-p1-coordinate-change] ### The Degree of the Canonical Divisor on a Plane Curve The example of $\mathbb{P}^1_k$ illustrates the general pattern. For a smooth plane curve, one can construct a differential by differentiating the defining equation. Let $C = V(F_d) \subset \mathbb{P}^2_k$ where $F_d$ is a homogeneous polynomial of degree $d$, and suppose $C$ is smooth. On the affine patch where $Z \neq 0$, set $f(x,y) = F(x,y,1)$. Smoothness guarantees that $f_x$ and $f_y$ do not simultaneously vanish on $C$. Wherever $f_y \neq 0$, the equation $f_x\,dx + f_y\,dy = 0$ on $C$ gives \begin{align*} \omega = \frac{dx}{f_y} = -\frac{dy}{f_x}, \end{align*} a rational differential that has no zeros or poles on the affine patch (since $f_y$ is a regular function that does not vanish where we use this expression, and the two expressions agree by the relation on $C$). Analysing what happens at infinity via coordinate changes establishes the following. [quotetheorem:2183] The proof is not covered in the course; the key computation is tracking the orders of $\omega = dx/f_y$ at the points at infinity using the coordinate change to patches where $X \neq 0$ or $Y \neq 0$. Since $H$ has degree $d$ (by Bézout's theorem: a line meets $C$ in $d$ points counted with multiplicity), we obtain: \begin{align*} \deg K_C = (d-3)\,d = d^2 - 3d = 2\cdot\frac{(d-1)(d-2)}{2} - 2 = 2g - 2 \end{align*} where $g = (d-1)(d-2)/2$ is the genus of a smooth plane curve of degree $d$. This confirms the degree formula $\deg K_C = 2g - 2$. [corollary: Genus of a Smooth Plane Cubic] A smooth plane curve of degree $3$ has genus $1$. [/corollary] [proof] Take $d = 3$. Then $K_C = (3-3)H = 0$, the zero divisor. So $\mathcal{L}(K_C) = \mathcal{L}(0) = k$, which has $k$-dimension $1$. Thus $g(C) = 1$. [/proof] [corollary: No Plane Curve of Degree $\geq 3$ is Isomorphic to $\mathbb{P}^1_k$] A smooth plane curve of degree $d \geq 3$ is not isomorphic to $\mathbb{P}^1_k$. [/corollary] [proof] For $d \geq 3$, the canonical divisor $K_C = (d-3)H$ has degree $(d-3)d \geq 0$. In particular $\mathcal{L}(K_C)$ is nonzero (it contains the constants when $d = 3$, and grows for $d > 3$), so $g(C) \geq 1$. But $g(\mathbb{P}^1_k) = 0$, and isomorphisms preserve the genus. [/proof] ### The Degree Formula $\deg K_C = 2g - 2$ The computation for plane curves is a special case of a fundamental theorem that holds for all smooth projective curves. [quotetheorem:2184] This theorem is most naturally a consequence of the Riemann–Roch theorem (Section 5.3), which gives $\ell(D) - \ell(K_C - D) = \deg D - g + 1$ for any divisor $D$. Applying Riemann–Roch to $D = K_C$ and using the fact that $\ell(0) = 1$ yields $\deg K_C = 2g - 2$. The degree formula $\deg K_C = 2g - 2$ is the algebraic counterpart of the Gauss–Bonnet theorem: on a compact Riemann surface of genus $g$, the Euler characteristic is $\chi = 2 - 2g$, and the divisor of any meromorphic 1-form has degree $-\chi = 2g - 2$. The algebraic proof recovers this topological invariant without any reference to analysis or topology, purely from the structure of the function field and the local freeness of $\Omega_{C,p}$. ## The Riemann–Roch Theorem [motivation] ### The problem: computing $\ell(D)$ Section 5.1 established that for any divisor $D$ on a smooth projective curve $C$ with $\deg D \geq 0$, the space $\mathcal{L}(D)$ is finite-dimensional. But knowing that $\ell(D) < \infty$ is only the beginning. The natural follow-up question is: can we compute $\ell(D)$ in terms of $\deg D$ and some intrinsic invariant of the curve? To calibrate our expectations, consider $C = \mathbb{P}^1_k$ and the divisor $D = n[P]$ for a point $P \in \mathbb{P}^1_k$ and $n \geq 0$. Using the local coordinate $t$ near $P$, the rational functions with at most a pole of order $n$ at $P$ and no other poles are exactly the polynomials of degree at most $n$ in $t$. More precisely, if we take $P = [\,0:1\,]$ and use the coordinate $t = X_1/X_0$, then \begin{align*} \mathcal{L}(n[P]) = \langle 1,\, t,\, t^2,\, \ldots,\, t^n \rangle, \end{align*} which has dimension $n + 1 = \deg D + 1$. So on $\mathbb{P}^1$, the formula $\ell(D) = \deg D + 1$ holds exactly. Does this formula persist on a general curve? It cannot, for the following reason: a general smooth projective curve $C$ carries no global regular functions except constants. The geometry of $C$ imposes constraints on sections that $\mathbb{P}^1$ does not feel. The Riemann–Roch theorem quantifies the discrepancy precisely: the failure of $\ell(D) = \deg D + 1$ is measured by a correction term $\ell(K_C - D)$, where $K_C$ is the canonical divisor introduced in Section 5.2. [/motivation] ### The Genus of a Curve We have defined the genus as $g(C) = \ell(K_C)$. But what does this number actually measure? Before stating Riemann–Roch, it is worth pausing to ask: given a specific curve, how do we compute $g$? For plane curves, the formula $g = (d-1)(d-2)/2$ answers this. For an abstract curve we must rely on the canonical divisor, whose degree $2g - 2$ the Riemann–Roch theorem will pin down. The genus is therefore simultaneously the dimension of the space of global regular differentials and the measure of how much the curve's function theory deviates from that of $\mathbb{P}^1$. [definition: Genus of a Curve] Let $C$ be a smooth projective curve over an algebraically closed field $k$. The **geometric genus** of $C$ is the integer \begin{align*} g = g(C) \coloneqq \ell(K_C). \end{align*} Equivalently, $g = \dim_k H^0(C, \Omega_C)$, the dimension of the space of global regular differentials on $C$. [/definition] The genus is the central intrinsic invariant of a curve. Over $\mathbb{C}$, a smooth projective curve of genus $g$ is homeomorphic (in the Euclidean topology) to a compact orientable surface of genus $g$, i.e., a sphere with $g$ handles. So $g = 0$ corresponds to $\mathbb{P}^1$ (the sphere), $g = 1$ to an elliptic curve (a torus), and $g \geq 2$ to surfaces of higher topological complexity. Before stating Riemann–Roch, we record a key constraint on $K_C$ that the theorem will imply. [remark: Degree of the Canonical Divisor] The Riemann–Roch theorem, applied with $D = K_C$, forces $\deg K_C = 2g - 2$. We prove this as Corollary 5.14 below. Note the sign: for $g = 0$, $\deg K_C = -2 < 0$; for $g = 1$, $\deg K_C = 0$ (so $K_C$ is linearly equivalent to the zero divisor); and for $g \geq 2$, $\deg K_C > 0$. This mirrors the Gauss–Bonnet theorem in differential geometry, where the curvature of a genus-$g$ surface is positive for $g = 0$, zero for $g = 1$, and negative for $g \geq 2$. [/remark] ### Statement of the Riemann–Roch Theorem The stage is set: we have the Riemann–Roch space $\mathcal{L}(D)$, the canonical divisor $K_C$, and the genus $g$. The Riemann–Roch theorem asks: what is the exact relationship between $\ell(D)$, $\deg D$, and $g$? On $\mathbb{P}^1$ we found $\ell(D) = \deg D + 1$ for effective $D$. The theorem says this formula holds for general $C$ up to a correction by $\ell(K_C - D)$, the dimension of the complementary space. The correction vanishes when $\deg D$ is large, recovering the $\mathbb{P}^1$ formula in the limit. [quotetheorem:2185] The term $\ell(K_C - D)$ is the correction to the naive formula. When $D$ has large degree, $K_C - D$ has negative degree, hence $\ell(K_C - D) = 0$, and the formula collapses to $\ell(D) = \deg D - g + 1$. The genus $g$ is thus the defect: it measures how much the arithmetic of the curve depresses the expected dimension. The proof of Riemann–Roch in its full generality uses either Serre duality or a direct linear-algebra argument involving the Riemann–Roch space and differentials. The source notes for this course do not give the proof; the theorem is taken as a black box whose consequences we develop extensively below. The duality perspective is worth sketching: Serre duality asserts an isomorphism $H^1(C, \mathcal{O}(D)) \cong H^0(C, \Omega_C(-D))^*$, so $\ell(K_C - D) = \dim H^1(C, \mathcal{O}(D))$. The Riemann–Roch formula then becomes the Euler characteristic formula $\chi(\mathcal{O}(D)) = h^0 - h^1 = \deg D + 1 - g$, a statement about the alternating sum of cohomology groups that can be established via cohomological methods. ### First Consequences We derive the most important structural corollaries. [quotetheorem:2186] [citeproof:2186] [quotetheorem:2187] [citeproof:2187] [remark: Genus in Low Degrees] For plane curves: degree $1$ (line) and degree $2$ (conic) both give $g = 0$; degree $3$ (cubic) gives $g = 1$; degree $4$ (quartic) gives $g = 3$; degree $5$ gives $g = 6$. Not every genus arises from a plane curve: the genus formula $g = \binom{d-1}{2}$ produces only the values $0, 1, 3, 6, 10, \ldots$, missing $g = 2, 4, 5, \ldots$. Other families of curves — e.g., a bidegree $(d_1, d_2)$ curve in $\mathbb{P}^1_k \times \mathbb{P}^1_k$, which has genus $(d_1 - 1)(d_2 - 1)$ — fill in these gaps. In particular, smooth curves of every genus $g \geq 0$ exist. [/remark] [quotetheorem:2188] [citeproof:2188] This corollary shows that for sufficiently large $\deg D$, the correction term vanishes and $\ell(D)$ is determined by $\deg D$ alone. The range $0 \leq \deg D \leq 2g - 2$ is where the geometry of the specific curve $C$ (encoded in the canonical divisor) genuinely affects the count. ### Examples by Genus [example: Genus Zero — the Projective Line] Let $C = \mathbb{P}^1_k$, so $g = 0$. Then $\deg K_C = -2$, consistent with $K_C \sim -2[P]$ for any point $P$ (since $\mathbb{P}^1$ has no nonzero global differentials). For any divisor $D$ with $\deg D \geq 0$, we have $\deg D \geq 0 \geq 2(0) - 1 = -1$, so the correction term $\ell(K_C - D)$ vanishes for all $D$ with $\deg D \geq 0$. Taking $D = n[P]$ for $n \geq 0$: $\ell(n[P]) = n - 0 + 1 = n + 1$. This matches the direct computation $\mathcal{L}(n[P]) = \langle 1, t, t^2, \ldots, t^n \rangle$ from the motivation. For $D$ with $\deg D < 0$: any effective divisor has nonnegative degree, so $\deg D < 0$ forces $\mathcal{L}(D) = 0$, i.e. $\ell(D) = 0$. [/example] [example: Genus One — Elliptic Curves] Let $C$ be a smooth curve of genus $g = 1$, so $\deg K_C = 0$ and $\ell(K_C) = 1$. Since $K_C$ has degree $0$ and $\ell(K_C) = 1$, the space $\mathcal{L}(K_C)$ contains a nonzero global regular differential, meaning $K_C$ is linearly equivalent to the zero divisor: $K_C \sim 0$. Fix a point $p_0 \in C$. For the divisor $p + q - p_0$ (where $p, q \in C$), we have $\deg(p + q - p_0) = 1 \geq 2(1) - 1 = 1$, so by the large-degree corollary, $\ell(p + q - p_0) = 1 - 1 + 1 = 1$. More generally, for any divisor $D$ of degree $d \geq 1$ on $C$: $\ell(D) = d$. For $d = 1$: $\ell(D) = 1$, meaning $D$ is linearly equivalent to a unique effective divisor of degree $1$, i.e., a unique point. This is the key input for the group law on elliptic curves (see Section 5.3 of the source notes). [/example] [example: Genus Two] Let $C$ be a smooth curve of genus $g = 2$, so $\deg K_C = 2$ and $\ell(K_C) = 2$. The Riemann–Roch theorem gives: - For $\deg D = 0$: $\ell(D) - \ell(K_C - D) = -1$. Since $\ell(D) \geq 0$ and $\ell(K_C - D) \geq 0$, the only way is $\ell(D) = 0$ and $\ell(K_C - D) = 1$ (unless $D \sim 0$, in which case $\ell(D) = 1$ and $\ell(K_C - D) = 2 = \ell(K_C)$). - For $\deg D = 1$: $\ell(D) - \ell(K_C - D) = 0$. Generically $\ell(D) = 0$, but for special $D$ (e.g., if $D$ is a Weierstrass point), $\ell(D) \geq 1$. - For $\deg D = 2$: $\ell(D) - \ell(K_C - D) = 1$. If $\ell(D) \geq 2$, then $D$ defines a degree-$2$ map $C \to \mathbb{P}^1$, making $C$ a hyperelliptic curve. - For $\deg D = 3 \geq 2(2) - 1 = 3$: $\ell(D) = 3 - 2 + 1 = 2$ with no correction. [/example] ### The Elliptic Curve Group Law via Riemann–Roch The computation $\ell(p + q - p_0) = 1$ for a genus-$1$ curve has a remarkable consequence. Recall from Section 5.1 (Lemma 5.17 in the source): [quotetheorem:2189] [citeproof:2189] This lemma, combined with $\ell(p + q - p_0) = 1$, defines the group law on a genus-$1$ curve: given $p, q \in C$, there is a unique point $r \in C$ such that $p + q - p_0$ is linearly equivalent to $[r] - [p_0]$, equivalently $p + q \sim r + p_0$ in $\mathrm{Div}(C)$. We set $p \oplus_{p_0} q = r$. [definition: Elliptic Curve] An **elliptic curve** is a pair $(C, p_0)$ where $C$ is a smooth projective curve of genus $1$ and $p_0 \in C$ is a distinguished base point, equipped with the group law $\oplus_{p_0}$ defined by the condition $p \oplus_{p_0} q = r$ iff $[p] + [q] \sim [r] + [p_0]$ in $\mathrm{Div}(C)$. [/definition] [quotetheorem:2190] [citeproof:2190] This theorem identifies the elliptic curve $C$ with its own Jacobian. The key point is that every degree-zero divisor class on a genus-$1$ curve is represented by a unique point. This is special to genus $1$: for $g = 0$ the Jacobian is trivial, and for $g \geq 2$ the Jacobian $\mathrm{Jac}(C)$ is a $g$-dimensional abelian variety strictly larger than $C$. The identification $\beta: C \xrightarrow{\sim} \mathrm{Jac}(C)$ is the reason the choice of base point $p_0$ matters: a different base point $p_0'$ gives a different identification $\beta': p \mapsto [p - p_0']$, which differs from $\beta$ by the group translation $[D] \mapsto [D + p_0' - p_0]$. Changing the base point thus corresponds to a translation automorphism of the Jacobian, and the elliptic curve has no canonical choice of identity — the theory is symmetric under all such translations. ### Plane Cubics and the Collinearity Criterion A concrete model for elliptic curves comes from plane cubics. Suppose $k$ does not have characteristic $2$, and let $E = V(F) \subset \mathbb{P}^2_k$ where \begin{align*} F(X_0, X_1, X_2) = X_0 X_2^2 - \prod_{i=0}^{2}(X_1 - \lambda_i X_0) \end{align*} for distinct $\lambda_0, \lambda_1, \lambda_2 \in k$. Take $p_0 = [0:0:1] \in E$. [quotetheorem:2191] [citeproof:2191] The collinearity criterion is what makes plane cubics so special among all genus-$1$ curves. The key ingredient in the proof is the Riemann–Roch computation $\ell(3p_0) = 3$, which forces a basis of $\mathcal{L}(3p_0)$ to consist of exactly the affine coordinate functions — a three-dimensional space that can be identified with linear forms on $\mathbb{P}^2_k$. This identification is specific to genus $1$: for $g = 0$ the analogous space would not contain the coordinate functions, and for $g \geq 2$ the Riemann–Roch space of a degree-$3$ divisor could be smaller or would not have a clean basis of this form. In practice, the collinearity criterion gives an explicit algorithm for the group law on a plane cubic: to compute $p \oplus_E q$, draw the line through $p$ and $q$ (or the tangent line if $p = q$), find its third intersection point $r$ with $E$ (guaranteed by Bézout), then draw the line through $r$ and $p_0$ and take the third intersection point as $p \oplus_E q$. The entire group law is thus computable by intersecting lines with the cubic — no coordinate charts or local uniformisers needed. ### Existence of Curves of Every Genus The genus formula for plane curves — $g = \binom{d-1}{2}$ — shows that plane curves can only realize genera $0, 1, 3, 6, 10, \ldots$. To see that every genus $g \geq 0$ is realized by some smooth curve, one uses the family of bidegree $(d_1, d_2)$ curves in $\mathbb{P}^1_k \times \mathbb{P}^1_k$. Such a curve has genus $(d_1 - 1)(d_2 - 1)$, and by varying $d_1$ and $d_2$ independently one obtains every nonneg integer as a genus. ### The Canonical Embedding For curves of genus $g \geq 2$, the canonical divisor $K_C$ defines a natural embedding into projective space. [explanation: Canonical Map] Let $C$ be a smooth projective curve of genus $g \geq 2$. Since $\ell(K_C) = g$, the linear system $|K_C|$ defines a rational map \begin{align*} \phi_{K_C}: C &\dashrightarrow \mathbb{P}^{g-1}_k. \end{align*} Concretely, choose a basis $\omega_0, \ldots, \omega_{g-1}$ for $H^0(C, \Omega_C)$. In a local coordinate $t$ near a point $p \in C$, write $\omega_i = f_i(t)\, dt$. The map sends $p$ to $[f_0(p) : f_1(p) : \cdots : f_{g-1}(p)]$. For $C$ not hyperelliptic (i.e., admitting no degree-$2$ map to $\mathbb{P}^1$), this map is a closed embedding: $C$ embeds as a smooth curve of degree $2g - 2$ in $\mathbb{P}^{g-1}_k$. This is the **canonical embedding**, and the image is called a **canonically embedded curve**. For $g = 2$: all genus-$2$ curves are hyperelliptic, so the canonical map is a $2:1$ cover of $\mathbb{P}^1$. For $g = 3$: non-hyperelliptic genus-$3$ curves embed as smooth quartics in $\mathbb{P}^2$. For $g = 4$: non-hyperelliptic genus-$4$ curves embed as the intersection of a quadric and a cubic in $\mathbb{P}^3$. The Riemann–Roch theorem is what makes this work: for $D = K_C + p$ (where $p \in C$), we have $\deg(K_C + p) = 2g - 1 \geq 2g - 1$, so $\ell(K_C + p) = g$, which means adding one base point to $K_C$ does not increase the dimension. This forces the map to be base-point-free, and a separate argument using $\ell(2K_C) = 3g - 3$ shows it is an embedding when $C$ is not hyperelliptic. [/explanation] [illustration:ag-canonical-embedding-genus-3] ### The Riemann–Hurwitz Formula We close with a structural result about morphisms between curves that depends on the genus and generalizes the degree-of-canonical-divisor computation. [quotetheorem:2192] [citeproof:2192] [quotetheorem:2193] [citeproof:2193] This corollary has a pleasant geometric interpretation: you cannot map a curve onto a more complicated curve. Morphisms of curves only go "downhill" in genus. ## Hyperelliptic Curves [motivation] ### The question Riemann–Roch raises Riemann–Roch tells us the dimension of $\mathcal{L}(D)$ for any divisor $D$ on a smooth projective curve $C$ of genus $g$: once $\deg D > 2g-2$, we have $\ell(D) = \deg D - g + 1$ with no correction term. This is a powerful formula, but it prompts a more structural question. We know that the canonical divisor $K_C$ has degree $2g-2$ and $\ell(K_C) = g$. For $g \geq 2$ these are both positive, so $K_C$ is an effective divisor and the associated linear system $|K_C|$ is non-trivial. Can we use $K_C$ to embed $C$ into projective space? The answer turns out to depend on a single geometric property of $C$: whether or not $C$ admits a degree-2 map to $\mathbb{P}^1_k$. Genus-2 curves all share this property — as we shall see, it is forced by Riemann–Roch — but for $g \geq 3$ it becomes a special condition. This dichotomy, hyperelliptic versus non-hyperelliptic, governs the canonical map and is the organising theme of this section. ### What goes wrong for genus 2 For $g = 2$ the canonical class has degree $2$ and $\ell(K_C) = 2$, so the canonical map $\phi_{K_C}$ is a morphism $C \to \mathbb{P}^1_k$. A degree-2 map to $\mathbb{P}^1_k$ cannot be an embedding — the target has dimension 1 and $C$ is irreducible of the same dimension, so the map must be finite and of degree exactly 2. Every genus-2 curve is therefore a double cover of the line, and the canonical map provides the cover. The interesting question is how far this phenomenon extends to higher genera. [/motivation] ### Hyperelliptic curves A smooth curve $C$ of genus $g \geq 2$ carries $g$ independent global differentials, and these differentials span the canonical map $\phi_{K_C}: C \to \mathbb{P}^{g-1}_k$. For most curves this map is an embedding, but for a special class — the hyperelliptic curves — the map collapses pairs of points, being a $2:1$ cover rather than an injection. The fundamental question of this section is: which curves behave this way, and why? [definition: Hyperelliptic Curve] A smooth projective curve $C$ of genus $g(C) \geq 2$ is **hyperelliptic** if it admits a degree-2 morphism \begin{align*} \pi : C \longrightarrow \mathbb{P}^1_k. \end{align*} [/definition] The degree-2 map $\pi$ is called a **degree-2 cover** or a **hyperelliptic structure** on $C$. We will see that for $g \geq 3$ this structure is not automatic, but for $g = 2$ it always exists. [quotetheorem:2194] [citeproof:2194] [remark: Hyperelliptic curves in all genera] Hyperelliptic curves of every genus $g \geq 2$ exist and can be constructed explicitly. In an affine chart, a hyperelliptic curve of genus $g$ is birational to the smooth projective completion of the affine curve \begin{align*} y^2 = f(x), \end{align*} where $f(x) \in k[x]$ is a squarefree polynomial of degree $2g+1$ or $2g+2$. The two cases differ by whether there is a Weierstrass point at infinity. Smoothness of the affine model is equivalent to $f$ being squarefree. [/remark] ### Affine models and Weierstrass points The affine model $y^2 = f(x)$ makes the degree-2 structure transparent: the map $(x,y) \mapsto x$ is the composition of the inclusion into the smooth projective model with the hyperelliptic cover $\pi : C \to \mathbb{P}^1_k$. The generic fibre over $x = a$ consists of the two points $(a, \pm\sqrt{f(a)})$. The fibre degenerates to a single point exactly when $f(a) = 0$. [definition: Weierstrass Point] A point $p \in C$ is a **Weierstrass point** of the hyperelliptic curve $C$ if $\pi(p)$ is a branch point of the cover $\pi : C \to \mathbb{P}^1_k$, i.e. if $\pi^{-1}(\pi(p)) = \{p\}$. In the affine model $y^2 = f(x)$, the Weierstrass points in the affine chart correspond to the roots of $f$. [/definition] For $f$ of degree $2g+1$ the roots of $f$ in $k$ give $2g+1$ affine Weierstrass points, and the point at infinity is an additional Weierstrass point, for $2g+2$ total. For $f$ of degree $2g+2$ all $2g+2$ Weierstrass points are affine. In either case, by the Riemann–Hurwitz formula applied to the degree-2 map $\pi$, the total number of branch points is exactly $2g+2$. [example: Genus-2 Weierstrass points] Let $k = \mathbb{C}$ and $C$ be the smooth projective completion of $y^2 = f(x)$ where \begin{align*} f(x) = (x-a_1)(x-a_2)(x-a_3)(x-a_4)(x-a_5), \quad a_i \in k \text{ distinct}. \end{align*} This is a genus-2 curve: $\deg f = 5 = 2(2)+1$, so by the formula $g = \lfloor (\deg f - 1)/2 \rfloor = 2$. The five roots $a_1, \ldots, a_5$ of $f$ give five affine Weierstrass points (where the two sheets of the cover meet), and the point at infinity accounts for the sixth. The Riemann–Hurwitz formula confirms: $2g(C) - 2 = 2(2g(\mathbb{P}^1)-2) + 6 = 2(-2)+6 = 2$, so $g(C) = 2$. [/example] ### The hyperelliptic involution Every hyperelliptic curve carries a canonical involution arising from the degree-2 cover. The presence of this involution is what obstructs the canonical map from being an embedding: points in the same fibre of $\pi$ are identified by $\phi_{K_C}$, so the map cannot distinguish them. [definition: Hyperelliptic Involution] Let $\pi : C \to \mathbb{P}^1_k$ be the hyperelliptic cover. The **hyperelliptic involution** is the unique non-trivial automorphism $\iota : C \to C$ satisfying $\pi \circ \iota = \pi$. In the affine model $y^2 = f(x)$, it is given by $\iota(x, y) = (x, -y)$. [/definition] The involution $\iota$ swaps the two sheets of the cover $\pi$. Its fixed points are precisely the Weierstrass points: $\iota(p) = p$ if and only if $p$ is the unique point in its fibre, i.e. a branch point. [illustration:ag-hyperelliptic-double-cover] The involution $\iota$ is central to the arithmetic and geometry of hyperelliptic curves. The quotient $C/\langle \iota \rangle$ recovers $\mathbb{P}^1_k$, and the field extension $k(C)/k(\mathbb{P}^1)$ is a degree-2 extension ramified exactly at the $2g+2$ Weierstrass points. The involution also constrains the Jacobian: the $(-1)$-eigenspace of $\iota^*$ acting on $\mathrm{Jac}(C)$ gives a $(g-1)$-dimensional abelian subvariety, reflecting how much of the Jacobian is "new" compared with the base $\mathbb{P}^1$. ### Embeddings via linear systems and condition (*) To understand when a linear system gives an embedding, we introduce a key separability condition. Let $C \subset \mathbb{P}^N_k$ be a curve not contained in any hyperplane, and let $D = \operatorname{div}(X_0)$ for homogeneous coordinates $X_0, \ldots, X_N$ on $\mathbb{P}^N_k$. Smoothness of $C$ implies two geometric facts: distinct points can be separated by hyperplanes, and at any point $p$ there is a hyperplane containing $p$ but not tangent to $C$ at $p$. [definition: Condition (*)] A divisor $D$ on a curve $C$ satisfies **condition $(*)$** if for every $p, q \in C$ (not necessarily distinct), \begin{align*} \ell(D - p - q) = \ell(D) - 2. \end{align*} [/definition] The two geometric facts above translate precisely to condition $(*)$: the first (separating distinct points) gives $\ell(D-p-q) \leq \ell(D)-2$ for $p \neq q$, and the second (separating tangent directions) gives $\ell(D-2p) \leq \ell(D)-2$. The equality holds in both cases for curves in projective space. [quotetheorem:2195] [citeproof:2195] Condition $(*)$ is thus a clean algebraic reformulation of two separate geometric conditions — point-separation and tangent-direction-separation — combined into a single uniform vanishing statement $\ell(D - p - q) = \ell(D) - 2$. It can be checked purely from the Riemann–Roch space without constructing $\phi_D$ explicitly. In practice, the large-degree theorem below shows that condition $(*)$ is automatic once $\deg D$ exceeds a threshold depending on the genus, giving an unconditional embedding into projective space for any smooth curve. [quotetheorem:2196] [citeproof:2196] ### The canonical map and the hyperelliptic characterisation For $g \geq 2$ the canonical divisor $K_C$ has degree $2g-2 \geq 2$ and $\ell(K_C) = g$. The canonical map is \begin{align*} \phi_{K_C} : C \longrightarrow \mathbb{P}^{g-1}_k. \end{align*} Whether this map is an embedding is the key dichotomy. [quotetheorem:2197] [citeproof:2197] [remark: Canonical image of hyperelliptic curves] When $C$ is hyperelliptic, $\phi_{K_C}$ is not an embedding: it is a degree-2 map onto a rational normal curve in $\mathbb{P}^{g-1}_k$, with the hyperelliptic involution $\iota$ as the deck transformation. The image $\phi_{K_C}(C)$ is isomorphic to $\mathbb{P}^1_k$, not to $C$. [/remark] The canonical map thus gives a clean geometric test: to determine whether an abstract curve $C$ is hyperelliptic, one examines whether $\phi_{K_C}$ has degree 1 or degree 2 onto its image. ### Embedding hyperelliptic curves via $3K_C$ For hyperelliptic curves $\phi_{K_C}$ fails, but we can use a larger multiple of the canonical class. [quotetheorem:2198] [citeproof:2198] This shows that every smooth projective curve of genus $g \geq 2$ — whether hyperelliptic or not — embeds into some projective space via a power of the canonical class. The discussion above completes the picture: every curve of every prescribed genus can be placed inside a projective space, and the moduli problem of classifying curves of genus $g$ up to isomorphism becomes the problem of classifying their images in $\mathbb{P}^{g-1}_k$ (for non-hyperelliptic curves) or their branch divisors on $\mathbb{P}^1_k$ (for hyperelliptic ones). This is the starting point for the moduli space $\mathcal{M}_g$ of curves, a direction Riemann first explored in 1857. [explanation: Hyperellipticity is rare for large genus] Counting parameters reveals why hyperelliptic curves are special for $g \geq 3$. A hyperelliptic curve of genus $g$ is determined by the $2g+2$ branch points of $\pi : C \to \mathbb{P}^1_k$ up to the $3$-dimensional automorphism group $\operatorname{Aut}(\mathbb{P}^1_k) = \mathrm{PGL}_2(k)$. This gives a $(2g+2-3)$-dimensional family, i.e. a $(2g-1)$-dimensional family of hyperelliptic curves of genus $g$. By contrast, the full moduli space $\mathcal{M}_g$ of all genus-$g$ curves has dimension $3g-3$ (a result that follows from deformation theory). For $g \geq 3$ we have $2g-1 < 3g-3$, so the hyperelliptic curves form a proper subvariety of $\mathcal{M}_g$ of strictly smaller dimension. The "generic" curve of genus $g \geq 3$ is not hyperelliptic, and one can construct explicit non-hyperelliptic examples as smooth plane curves or as $(d_1,d_2)$-bihomogeneous curves in $\mathbb{P}^1_k \times \mathbb{P}^1_k$ with genus $(d_1-1)(d_2-1)$. [/explanation] By the end of Chapter 6, we have classified all smooth projective curves of each genus and solved the moduli problem of identifying when two curves are isomorphic. Chapter 7 asks a completely different question: what happens when we reduce a variety modulo a prime and count solutions in finite fields? # 7. Bonus: The Weil Conjectures ## The Weil Conjectures The previous chapters built algebraic geometry from the ground up: affine and projective varieties, tangent spaces and smoothness, curves via Bézout, divisors, and the Riemann–Roch theorem. All of this happened over an algebraically closed field $k$, typically $\mathbb{C}$. This bonus lecture asks a completely different question: what happens when you take a variety defined over $\mathbb{Z}$ and count its solutions in finite fields? The answer, first conjectured by André Weil in 1949 and fully proved over the following three decades, is staggering. The number of points over finite fields is controlled by the topology of the complex variety. Arithmetic and geometry are not merely analogous — they are unified by a single generating function whose structure mirrors the cohomology of a topological space. This lecture is an invitation, not a complete treatment. Proofs are deferred or sketched; the goal is to feel why the conjectures are natural, surprising, and deep. ## Two Ways to Study a Variety [motivation] ### The complex-analytic viewpoint Suppose $X$ is a projective variety defined by homogeneous polynomials $F_1, \ldots, F_r$ with integer coefficients. We can form the complex variety \begin{align*} X_{\mathbb{C}} = V(F_1, \ldots, F_r) \subset \mathbb{P}^N_{\mathbb{C}}. \end{align*} Because $X_\mathbb{C}$ sits inside complex projective space, it inherits the Euclidean topology and becomes a compact topological manifold (when smooth). We can then apply all the machinery of algebraic topology: singular homology, Euler characteristic, Betti numbers. These are genuine topological invariants of the space over $\mathbb{C}$. ### The arithmetic viewpoint Instead of working over $\mathbb{C}$, we can reduce the coefficients of $F_1, \ldots, F_r$ modulo a prime $p$ and count solutions in the finite fields $\mathbb{F}_{p^m}$ for $m = 1, 2, 3, \ldots$. This gives a sequence of non-negative integers \begin{align*} N_m = |X(\mathbb{F}_{p^m})|, \end{align*} the number of $\mathbb{F}_{p^m}$-points of $X$. The sequence $(N_m)_{m \geq 1}$ encodes arithmetic information about the variety. ### The surprising bridge The Weil conjectures assert that these two viewpoints — one topological, one arithmetic — are not independent. The sequence $(N_m)$ is controlled by the Betti numbers of $X_\mathbb{C}$. The bridge is a generating function called the Hasse–Weil zeta function. [/motivation] [example: Point Counts on Standard Varieties] Before we define the zeta function, it is worth computing $N_m$ for the simplest varieties, to develop intuition for the kind of patterns that arise. Fix a prime power $q = p^r$ and work over $\mathbb{F}_q$. Write $N_m = |X(\mathbb{F}_{q^m})|$. **Affine line.** For $X = \mathbb{A}^1$, we have $X(\mathbb{F}_{q^m}) = \mathbb{F}_{q^m}$, so $N_m = q^m$. **Projective line.** For $X = \mathbb{P}^1$, we add one point at infinity to $\mathbb{A}^1_{\mathbb{F}_{q^m}}$, so $N_m = q^m + 1$. **Projective plane.** For $X = \mathbb{P}^2$, we stratify: $\mathbb{P}^2 = \mathbb{A}^2 \cup \mathbb{P}^1$, giving $N_m = q^{2m} + q^m + 1$. **The general pattern for $\mathbb{P}^n$.** Stratifying $\mathbb{P}^n = \mathbb{A}^n \sqcup \mathbb{A}^{n-1} \sqcup \cdots \sqcup \mathbb{A}^0$ gives \begin{align*} N_m = 1 + q^m + q^{2m} + \cdots + q^{nm}. \end{align*} Notice: these counts are polynomial in $q^m$, with non-negative integer coefficients, and the exponents $0, 1, 2, \ldots, n$ match the even Betti numbers of $\mathbb{P}^n_\mathbb{C}$, which is $h^{2k} = 1$ for $0 \leq k \leq n$ and $h^{\text{odd}} = 0$. **Elliptic curves.** If $C$ is an elliptic curve over $\mathbb{F}_q$, a deep theorem (proved by Hasse) gives \begin{align*} N_m = q^m + 1 - \alpha^m - \bar{\alpha}^m \end{align*} for some algebraic integer $\alpha$ with $|\alpha| = q^{1/2}$. So the deviation from $q^m + 1$ is oscillatory and bounded in size by $2q^{m/2}$ — this is the Hasse bound, proved by Hasse in 1936. [/example] [remark: Why the Hasse Bound Is Surprising] For a single prime $p$, the number of points on an elliptic curve over $\mathbb{F}_p$ could in principle be anything from $1$ to $p^2$. The Hasse bound says it lies in the interval $[p + 1 - 2\sqrt{p},\, p + 1 + 2\sqrt{p}]$, a window of width only $4\sqrt{p}$ around $p + 1$. This tight constraint is arithmetic, but its explanation is cohomological. [/remark] ## The Hasse–Weil Zeta Function Given a projective variety $X$ over $\mathbb{F}_q$, the point counts $N_m = |X(\mathbb{F}_{q^m})|$ encode arithmetic information, but the raw sequence grows without bound and its structure is not immediately transparent. The natural question is whether this data assembles into a simple algebraic object — a single generating function with a closed-form expression, ideally a rational function. This question motivates packaging the sequence into an exponential generating function. [definition: Hasse–Weil Zeta Function] Let $X$ be a projective variety over $\mathbb{F}_q$, and let $N_m = |X(\mathbb{F}_{q^m})|$. The **Hasse–Weil zeta function** of $X$ is the formal power series \begin{align*} Z_X(t) = \exp\!\left(\sum_{m=1}^{\infty} N_m \frac{t^m}{m}\right) \in \mathbb{Z}[[t]]. \end{align*} [/definition] The exponential packaging is deliberate: it converts a sequence of counts into a product formula, mimicking the Euler product of the Riemann zeta function. To see why, note that if $X = \{*\}$ is a single point, then $N_m = 1$ for all $m$ and \begin{align*} Z_{\{*\}}(t) = \exp\!\left(\sum_{m=1}^\infty \frac{t^m}{m}\right) = \exp(-\log(1-t)) = \frac{1}{1-t}. \end{align*} More generally, a disjoint union of $n$ points gives $Z(t) = (1-t)^{-n}$. [example: Zeta Function of Projective Space] For $X = \mathbb{P}^n$ over $\mathbb{F}_q$, we have $N_m = 1 + q^m + \cdots + q^{nm}$, so \begin{align*} Z_{\mathbb{P}^n}(t) &= \exp\!\left(\sum_{m=1}^\infty (1 + q^m + \cdots + q^{nm})\frac{t^m}{m}\right) \\ &= \exp\!\left(\sum_{k=0}^n \sum_{m=1}^\infty \frac{(q^k t)^m}{m}\right) \\ &= \prod_{k=0}^n \frac{1}{1 - q^k t}. \end{align*} This is a rational function in $t$ — in fact, a product of geometric series. The poles occur at $t = q^{-k}$ for $k = 0, 1, \ldots, n$, one for each cohomological degree of $\mathbb{P}^n$. [/example] ## The Weil Conjectures For $\mathbb{P}^n$, the zeta function has a striking closed form: it is a product of factors $1/(1 - q^k t)$, one for each even cohomological degree. In particular, the exponents $k = 0, 1, \ldots, n$ match the non-zero Betti numbers of $\mathbb{P}^n(\mathbb{C})$, and the factors of the denominator encode the eigenvalues of Frobenius on the cohomology of the complex variety. Is this a coincidence special to projective space, or does every smooth projective variety $X$ over $\mathbb{F}_q$ have a zeta function whose structure is governed by the topology of $X_\mathbb{C}$? Weil's 1949 conjectures give a precise and affirmative answer. [quotetheorem:2199] The proof of (W1), (W2), and (W4) is due to Grothendieck, working in the 1960s via the theory of étale cohomology. The Riemann hypothesis (W3) — the hardest part — was proved by Deligne in 1974. The statement of the theorem requires two remarks about its hypotheses before we move on. **Smoothness is essential.** The Weil conjectures as stated break down for singular varieties. On a smooth variety, the eigenvalues $\alpha_{ij}$ of Frobenius on $H^i_\mathrm{et}$ all have the single absolute value $q^{i/2}$ — this is the "purity" statement at the heart of the Riemann hypothesis. On a singular variety, purity fails: a single cohomology group can carry Frobenius eigenvalues of several different absolute values (a phenomenon Deligne later systematised as the weight filtration of mixed Hodge structures). The zeta function is still rational, but the clean factorisation and the connection to Betti numbers no longer hold in the same form. **Projectivity is essential.** For varieties that are not proper (not compact in the algebraic sense), the rationality of $Z_X(t)$ still holds by Grothendieck's trace formula, but the functional equation symmetry $t \leftrightarrow 1/(q^n t)$ relies on Poincaré duality, which requires compactness. Without properness one obtains a zeta function with a rational structure but without the symmetry between numerator and denominator that characterises the smooth projective case. **Open problems.** Grothendieck's étale cohomology resolved conjectures (W1)–(W4), but the deeper questions it raised remain open. The Standard Conjectures — Grothendieck's 1968 programme to establish algebraic cycle class maps with properties analogous to the Lefschetz decomposition in Hodge theory — would provide a purely algebraic proof of the Riemann hypothesis (W3) and unify the Weil conjectures with classical Hodge theory. They remain largely unproved. Grothendieck's theory of motives, intended as a universal cohomology theory explaining why étale cohomology, de Rham cohomology, and crystalline cohomology all compute the same invariants, is still under construction. Deligne's proof of (W3) uses the weight filtration on $\ell$-adic sheaves, a structure that has since grown into the theory of perverse sheaves and has driven much of modern arithmetic geometry. [explanation: Unpacking the Riemann Hypothesis Conjecture] The name "Riemann hypothesis" is chosen deliberately. The classical Riemann hypothesis says that the non-trivial zeros of the Riemann zeta function $\zeta(s)$ all have real part $1/2$. Here, the analogy works as follows. If we substitute $t = q^{-s}$ into $Z_X(t)$, the zeros and poles of $Z_X(q^{-s})$ occur at values where $\alpha_{ij} q^{-s} = 1$, i.e., $q^{-s} = \alpha_{ij}^{-1}$, so $q^s = \alpha_{ij}$. Since $|\alpha_{ij}| = q^{i/2}$, we have $|q^s| = q^{\operatorname{Re}(s)} = q^{i/2}$, which means $\operatorname{Re}(s) = i/2$. In other words: the zeros of $P_i(q^{-s})$ all lie on the vertical line $\operatorname{Re}(s) = i/2$. The structure is exactly like the Riemann hypothesis, but with a different half-plane for each cohomological degree $i$. The Hasse bound for elliptic curves — $|N_m - q^m - 1| \leq 2q^{m/2}$ — is precisely the Riemann hypothesis for curves: the single polynomial $P_1(t)$ has roots of absolute value $q^{-1/2}$, equivalently, Frobenius eigenvalues on $H^1_\mathrm{et}$ have absolute value $q^{1/2}$. [/explanation] [remark: The Betti Numbers Conjecture as a Structural Miracle] Conjecture (W4) says the numerator and denominator polynomials of $Z_X(t)$ have degrees equal to the Betti numbers of the complex manifold $X_\mathbb{C}$. This is genuinely remarkable: $Z_X(t)$ is defined entirely in terms of $\mathbb{F}_{q^m}$-point counts, yet its degrees of freedom are topological invariants of a space in characteristic zero. The finite-field arithmetic and the complex topology are counting the same underlying structure. For a curve of genus $g$, the Betti numbers are $h^0 = h^2 = 1$ and $h^1 = 2g$, so \begin{align*} Z_C(t) = \frac{P_1(t)}{(1-t)(1-qt)}, \quad \deg P_1 = 2g. \end{align*} The $2g$ roots of $P_1$ encode all the oscillatory behaviour of the point count sequence $(N_m)$. [/remark] ## When Smoothness Fails: A Singular Variety The pattern established for $\mathbb{P}^n$ and smooth curves — that $N_m$ is a polynomial in $q^m$ with coefficients encoding Betti numbers — depends crucially on the variety being smooth. To see what can go wrong, consider the simplest example of a singular curve. [example: Nodal Cubic] Let $C \subset \mathbb{P}^2_{\mathbb{F}_q}$ be the nodal cubic $V(Y^2 Z - X^2(X + Z))$, which has a single node at the point $[0:0:1]$. As a topological space over $\mathbb{C}$, the nodal cubic is a torus with one handle pinched to a point — topologically a sphere $S^2$, so its Betti numbers are $h^0 = h^2 = 1$, $h^1 = 0$, giving $\chi = 2$. If the Weil conjectures applied naively, we would expect $Z_C(t) = 1 / ((1-t)(1-qt))$ (the formula for a genus-$0$ curve), and hence $N_m = q^m + 1$. But counting over $\mathbb{F}_q$: the normalisation of $C$ is $\mathbb{P}^1$, and the two preimages of the node over $\overline{\mathbb{F}}_q$ are the solutions to $X^2 = -Z^2$ in projective coordinates, i.e., the points $[1:\pm i: -1]$. Whether these two preimage points are defined over $\mathbb{F}_q$ or only over $\mathbb{F}_{q^2}$ depends on whether $-1$ is a square in $\mathbb{F}_q^*$, i.e., on whether $q \equiv 1 \pmod{4}$ or $q \equiv 3 \pmod{4}$. **Case 1: $q \equiv 1 \pmod{4}$.** Both preimages of the node are $\mathbb{F}_q$-rational. The $\mathbb{F}_{q^m}$-points of $C$ are the $\mathbb{F}_{q^m}$-points of $\mathbb{P}^1$ (which are $q^m + 1$) minus the two preimage points of the node (which get identified to one), giving $N_m = q^m + 1 - (2 - 1) = q^m$. Wait — more carefully: $\mathbb{P}^1(\mathbb{F}_{q^m})$ has $q^m + 1$ points; the normalisation map collapses the two preimage points $\{P_+, P_-\}$ to the single node, so $|C(\mathbb{F}_{q^m})| = q^m + 1 - 1 = q^m$ for all $m \geq 1$. **Case 2: $q \equiv 3 \pmod{4}$.** The two preimages of the node are conjugate over $\mathbb{F}_{q^2}$ but not $\mathbb{F}_q$-rational. For $m$ odd, neither preimage is in $\mathbb{F}_{q^m}$, and the normalisation map is injective on $\mathbb{F}_{q^m}$-points, giving $N_m = q^m + 1$. For $m$ even, both preimages are $\mathbb{F}_{q^m}$-rational, and they collapse to one node, giving $N_m = q^m$. In Case 2, the counts $N_m$ alternate between $q^m + 1$ and $q^m$ — the oscillatory term is $(-1)^m$, corresponding to a Frobenius eigenvalue of $-1$ on $H^1$ of the singular curve. This is not of the form $q^{1/2}$ — the Riemann hypothesis fails. The zeta function in this case is \begin{align*} Z_C(t) = \frac{1}{(1-t)(1+t)(1-qt)} \quad (\text{Case 2}), \end{align*} which has a factor $(1+t)$ in the denominator corresponding to the Frobenius eigenvalue $-1$. This is not pure of weight one, reflecting the singularity. The lesson: for a singular variety, the zeta function is still rational (by Grothendieck's trace formula, which applies in full generality), but the strict weight purity of the Weil Riemann hypothesis fails. The eigenvalues of Frobenius on the cohomology can take values of different absolute values within the same cohomological degree — the cohomology is "mixed" rather than "pure." [/example] ## Rationality for Curves: A Proof Sketch Of the four Weil conjectures, rationality (W1) is often the most immediate to grasp. For curves of low genus one can verify it by direct computation — the $\mathbb{P}^1$ case above gives a clean product formula — but verifying rationality for a curve of arbitrary genus $g$ requires understanding where the finiteness comes from. The key insight is that the generating series for point counts can be reorganised into a sum over divisor classes, and Riemann–Roch then controls those classes completely for high degree, turning the infinite series into a finite rational expression. [explanation: Why Rationality Follows from the Jacobian] Fix a smooth projective curve $C$ over $\mathbb{F}_q$ and a base point $p_0 \in C(\mathbb{F}_q)$. The key identity is \begin{align*} Z_C(t) = \sum_{k=0}^\infty |\operatorname{Sym}^k(C)(\mathbb{F}_q)|\, t^k, \end{align*} where $\operatorname{Sym}^k(C) = C^k / S_k$ is the $k$-th symmetric product of $C$. This is the variety parametrising effective divisors of degree $k$ on $C$. Now recall the Abel–Jacobi map $\mathrm{AJ}_k : \operatorname{Sym}^k(C) \to \mathrm{Jac}(C)$ sending $\sum_i a_i p_i \mapsto \bigl[\sum_i a_i p_i - k p_0\bigr]$. For a divisor class $[D] \in \mathrm{Jac}(C)$, the fibre $\mathrm{AJ}_k^{-1}([D])$ is the projective space $\mathbb{P}(\mathcal{L}(D))$, where $\mathcal{L}(D)$ is the Riemann–Roch space of $D$ as defined in Chapter 6. By Riemann–Roch, for $k > 2g - 2$, every divisor class of degree $k$ has $\ell(D) = k - g + 1$, so every fibre is isomorphic to $\mathbb{P}^{k-g}$. This means the map $\mathrm{AJ}_k$ is a projective bundle over $\mathrm{Jac}(C)$ when $k$ is large. Counting $\mathbb{F}_q$-points on a projective bundle is multiplicative: $|\mathbb{P}^{k-g}(\mathbb{F}_q)| = 1 + q + \cdots + q^{k-g}$. So for $k > 2g - 2$, the count $|\operatorname{Sym}^k(C)(\mathbb{F}_q)|$ is a sum over the $|\mathrm{Jac}(C)(\mathbb{F}_q)|$ classes of $\mathbb{F}_q$-rational divisors, each contributing $1 + q + \cdots + q^{k-g}$. Plugging this back into the generating series and separating the finitely many terms $k \leq 2g - 2$ from the geometric tail $k > 2g - 2$, one obtains $Z_C(t)$ as a rational function in $t$. The verification for $\mathbb{P}^n$ reduces to the direct calculation \begin{align*} Z_{\mathbb{P}^n}(t) = \frac{1}{(1-t)(1-qt)\cdots(1-q^n t)}, \end{align*} which is manifestly rational. [/explanation] ## The Frobenius Map and Grothendieck's Approach Weil's proof for curves relied on the Abel–Jacobi map and the Jacobian — tools specific to dimension one. How does one prove rationality, the functional equation, and the Riemann hypothesis for a variety of arbitrary dimension $n$, where no Jacobian is available? The guiding idea is striking: the $\mathbb{F}_{q^m}$-points of $X$ are precisely the fixed points of the $m$-fold iterate of a canonical algebraic map, the Frobenius morphism. Exploiting this connection demands a new cohomology theory that works in characteristic $p$, and produces a Lefschetz-type fixed-point formula that encodes all four Weil conjectures simultaneously. [explanation: Fixed Points and Cohomology] The proof of the remaining conjectures required an entirely new cohomology theory — étale cohomology — built by Grothendieck and his collaborators in the Séminaire de Géométrie Algébrique (SGA). The central insight is a beautiful parallel with the Lefschetz fixed point theorem in topology. **The Frobenius morphism.** Over $\mathbb{F}_q$ with $q = p^r$, the Frobenius morphism is the map \begin{align*} \mathrm{Frob}_X : X &\longrightarrow X \\ (z_1, \ldots, z_r) &\longmapsto (z_1^q, \ldots, z_r^q). \end{align*} The crucial observation: a point $x \in X$ is an $\mathbb{F}_{q^m}$-rational point if and only if $\mathrm{Frob}_X^m(x) = x$, i.e., $x$ is a fixed point of the $m$-fold iterate of Frobenius. So \begin{align*} N_m = |\mathrm{Fix}(\mathrm{Frob}_X^m)|. \end{align*} **The Lefschetz analogy.** In ordinary topology, the Lefschetz fixed point theorem says that for a continuous map $f: M \to M$ on a compact manifold, \begin{align*} |\mathrm{Fix}(f)| = \sum_i (-1)^i \mathrm{tr}(f^* \mid H^i(M; \mathbb{Q})). \end{align*} Grothendieck's étale cohomology groups $H^i_\mathrm{et}(X, \mathbb{Q}_\ell)$ play the role of $H^i(M; \mathbb{Q})$ in characteristic $p$. The **Grothendieck–Lefschetz trace formula** then gives \begin{align*} N_m = \sum_{i=0}^{2n} (-1)^i \mathrm{tr}(\mathrm{Frob}^m \mid H^i_\mathrm{et}(X, \mathbb{Q}_\ell)). \end{align*} From this formula, the rationality of $Z_X(t)$ follows from standard linear algebra: the generating function $\sum_m N_m t^m / m$ is the logarithm of a ratio of characteristic polynomials of the Frobenius acting on the étale cohomology groups, which is rational. The Betti number conjecture (W4) is immediate: $\deg P_i = \dim H^i_\mathrm{et}(X, \mathbb{Q}_\ell) = h^i(X_\mathbb{C}; \mathbb{Q})$ (the latter comparison being a theorem of Artin). **Deligne's contribution.** What remained after Grothendieck was the Riemann hypothesis: the eigenvalues of Frobenius on $H^i_\mathrm{et}$ have absolute value $q^{i/2}$. Grothendieck had hoped to prove this via a "standard conjecture" about algebraic cycles, but this remains open. Deligne found a different route in 1974, using $L$-functions of families of varieties and an inductive strategy reminiscent of Rankin's method in analytic number theory. [/explanation] [remark: What Étale Cohomology Does] The difficulty in running this programme is that ordinary singular cohomology does not work in characteristic $p$: it is insensitive to the field. Étale cohomology replaces open sets in the Zariski topology with "étale covers" — a characteristic-$p$ substitute for the local homeomorphisms of the complex-analytic setting. The resulting cohomology groups live over $\mathbb{Q}_\ell$ (for a prime $\ell \neq p$) and carry an action of the Frobenius. The choice of $\ell$ is auxiliary: the trace formula and the absolute values of eigenvalues are independent of $\ell$. [/remark] ## The Case of Curves: Emil Artin's Conjecture [explanation: Historical Priority] The special case of the Weil conjectures for curves — that $Z_C(t) = P_1(t) / (1-t)(1-qt)$ with $P_1$ a polynomial of degree $2g$ whose roots have absolute value $q^{-1/2}$ (equivalently, whose reciprocal roots, i.e. the Frobenius eigenvalues on $H^1_\mathrm{et}$, have absolute value $q^{1/2}$) — was actually conjectured by Emil Artin in his 1921 thesis on quadratic function fields, decades before Weil's general formulation. Artin computed examples over small finite fields and noticed the pattern. Weil proved the curve case in 1948 using his theory of algebraic functions over finite fields, as a warmup for the general conjectures. The proof used the Riemann–Roch theorem, the Jacobian, and an explicit calculation with the intersection pairing on $C \times C$. This historical sequence — Artin conjectures for curves, Weil proves curves and conjectures generally, Grothendieck proves all but RH, Deligne proves RH — spans over fifty years and required building much of modern algebraic geometry along the way. [/explanation] ## A Structural Example: The Fermat Hypersurface The Weil conjectures are most revealing when applied to a concrete family where the zeta function can be computed explicitly. The Fermat hypersurfaces $V(X_0^d + \cdots + X_n^d) \subset \mathbb{P}^n$ form exactly such a family: Weil carried out this computation in 1949 and the pattern he observed in the character sum expressions for $N_m$ is what suggested the general structure of the conjectures. [example: Fermat Hypersurface] To see the conjectures in action beyond projective spaces, consider the Fermat hypersurface \begin{align*} X = V(X_0^d + X_1^d + \cdots + X_n^d) \subset \mathbb{P}^n_{\mathbb{F}_q}, \end{align*} where $\gcd(d, q) = 1$. This is a smooth hypersurface of dimension $n-1$. The point count $N_m = |X(\mathbb{F}_{q^m})|$ can be computed explicitly using multiplicative characters of $\mathbb{F}_{q^m}$. If $\chi$ ranges over the $d$-th power residue characters of $\mathbb{F}_{q^m}^*$, then \begin{align*} N_m = \frac{q^{mn} - 1}{q^m - 1} + \sum_{\substack{\chi_0, \ldots, \chi_n \\ \chi_i^d = 1,\, \chi_i \neq 1 \\ \chi_0 \cdots \chi_n = 1}} J(\chi_0, \ldots, \chi_n), \end{align*} where $J(\chi_0, \ldots, \chi_n)$ are Jacobi sums — explicit algebraic integers whose absolute values equal $q^{m(n-1)/2}$ by Weil's theorem on character sums. The Jacobi sums are exactly the $\alpha_{ij}$ in the Weil conjectures: they are the eigenvalues of Frobenius on the middle cohomology $H^{n-1}_\mathrm{et}(X, \mathbb{Q}_\ell)$, and they satisfy $|\alpha_{ij}| = q^{(n-1)/2}$, consistent with $i = n-1$ in (W3). The roots of the corresponding polynomial $P_{n-1}(t)$ are $\alpha_{ij}^{-1}$ with absolute value $q^{-(n-1)/2}$. The fact that Jacobi sums have the right absolute value $q^{m(n-1)/2}$ is precisely the Riemann hypothesis for the Fermat hypersurface, verifiable without the full machinery of Deligne's proof. This example is historically important: it provided the template for how Weil expected the general result to look, and Weil's own proof of the curve case used character sum methods of exactly this type. [/example] ## Philosophical Aftermath The Weil conjectures reveal a profound analogy between number theory and topology that has driven mathematics ever since. Motives — introduced by Grothendieck as a conjectural "universal cohomology theory" — are meant to explain why the same topological data appears on both sides. The Standard Conjectures, still largely open, would give the remaining structural properties of this theory. More immediately, Deligne's proof of the Riemann hypothesis became one of the central tools of modern arithmetic geometry: it underlies bounds in exponential sums (including those used in cryptography and coding theory), the proof of the Ramanujan conjecture on modular forms, and large parts of the Langlands programme. The zeta function $Z_X(t)$, born from counting points over finite fields, turned out to be a window into the deepest structure of arithmetic and geometry simultaneously. [illustration:ag-weil-conjectures-bridge] The Weil conjectures establish a stunning bridge between arithmetic and topology: the number of solutions over finite fields is controlled by the cohomology of the associated complex variety. Chapter 8 steps back to ask a structural question—why is the world of classical varieties insufficient—and introduces schemes as the natural language for deformations, limits, and arithmetic. # 8. Bonus: Why Schemes? The course so far has taken place entirely within the world of classical varieties: affine varieties defined by radical ideals over an algebraically closed field, their projective counterparts, morphisms, divisors, and the Riemann–Roch theorem. This final bonus lecture, delivered in Lent 2022, steps back and asks a structural question — why is this world not enough? What are the genuine mathematical pressures that force us to move to the broader category of schemes? The answer involves three distinct failure modes of the classical theory, each illuminating a different facet of what schemes are designed to fix. ## What Varieties Cannot See: Nilpotents The cleanest way to understand the inadequacy of classical varieties is through limits of families. Consider a smooth elliptic curve, that is, a smooth projective curve $C \subset \mathbb{P}^2$ defined by a homogeneous cubic $f_3 \in \mathbb{C}[X_0, X_1, X_2]$. For each $t \in \mathbb{C}$, define \begin{align*} V_t = \mathcal{V}(t f_3 + X_0^3) \subset \mathbb{P}^2. \end{align*} For $t \neq 0$, $V_t$ is a smooth cubic curve. As $t \to 0$, one expects the family to degenerate to the vanishing locus of $X_0^3$. But $X_0^3$ is not a radical polynomial — the ideal $(X_0^3)$ strictly contains its own radical $(X_0)$. The classical variety associated to $X_0^3$ is just the line $\mathcal{V}(X_0) \cong \mathbb{P}^1$, which is a linear object with a completely different Hilbert polynomial from a cubic curve. So if we insist on taking limits only within the world of varieties (equivalently, only within the world of radical ideals), the limit of a sequence of smooth cubic curves becomes a line. Something has gone badly wrong: we have discarded the multiplicity information encoded in $X_0^3$, and in doing so we have lost control of the Hilbert polynomial, the degree, and every other numerical invariant we care about. The fix is to take the limit as the scheme $\mathcal{V}(X_0^3)$, whose coordinate ring is $\mathbb{C}[X_0, X_1, X_2]/(X_0^3)$. This ring contains the nilpotent element $X_0$ (since $X_0^3 = 0$ in the ring but $X_0 \neq 0$). As a function on the underlying topological space $\mathcal{V}(X_0)$, the element $X_0$ vanishes everywhere — it is identically zero as a function — yet it is not the zero element of the ring. This is the key distinction: schemes track nilpotent data in their structure sheaves, data that is invisible to the classical variety associated to the same zero locus. [remark: Nilpotents as Infinitesimal Data] The nilpotent $X_0 \in \mathbb{C}[X_0]/(X_0^2)$ has a concrete geometric meaning: it is an infinitesimal normal direction to the subscheme $\{X_0 = 0\}$. The ring $\mathbb{C}[X_0]/(X_0^2)$ should be thought of as functions on a "fat point" — a point together with a first-order infinitesimal neighbourhood in the $X_0$ direction. More precisely, $\operatorname{Spec}(\mathbb{C}[X_0]/(X_0^2))$ is the "dual numbers" scheme, which appears naturally as the scheme-theoretic tangent vector. This is why scheme theory is the natural language for deformation theory. [/remark] ## The Hilbert Scheme: Classifying Subvarieties Forces Schemes The second failure mode is more subtle and arises from the moduli problem of classifying subvarieties of projective space. From the outset of the course, we have been implicitly working toward classification: we classified linear subvarieties via the Grassmannian, and the Weil conjectures relate point counts over finite fields to the topology of complex varieties. The natural next step is to classify all subvarieties of $\mathbb{P}^n$. ### Grassmannians and the Plücker Embedding Before discussing the Hilbert scheme, we record the classical construction of the Grassmannian, since it is the local model for the more general theory. Throughout this chapter the base field is $\mathbb{C}$ (we reserve $k$ for another use below). [definition: Grassmannian] Let $0 \leq k \leq n$. The **Grassmannian** $\operatorname{Gr}(k, n)$ is the set of all $k$-dimensional linear subspaces of $\mathbb{C}^n$: \begin{align*} \operatorname{Gr}(k, n) = \{ V \leq \mathbb{C}^n : \dim V = k \}. \end{align*} [/definition] [example: Projective Space as Grassmannian] The simplest case is $\operatorname{Gr}(1, n) = \mathbb{P}^{n-1}$, since a one-dimensional subspace of $\mathbb{C}^n$ is exactly a point of projective space. This example already shows that $\operatorname{Gr}(k, n)$ carries a natural topology inherited from $\mathbb{P}^{n-1}$ in the case $k = 1$. [/example] Every element $V \in \operatorname{Gr}(k, n)$ can be represented by an $n \times k$ matrix of full rank, well-defined up to right multiplication by elements of $\mathrm{GL}_k$. The **Plücker embedding** sends such a matrix to its $\binom{n}{k}$ maximal minors, assembled into a point of $\mathbb{P}^{\binom{n}{k} - 1}$: \begin{align*} \iota : \operatorname{Gr}(k, n) \to \mathbb{P}^{\binom{n}{k} - 1}. \end{align*} [quotetheorem:2200] [citeproof:2200] [illustration:ag-plucker-embedding-gr24] ### From Grassmannians to the Hilbert Scheme The Grassmannian classifies linear subspaces. To classify all subvarieties of $\mathbb{P}^n$, one needs a richer construction. The key observation is that a subvariety $V \subset \mathbb{P}^n$ is determined by its homogeneous ideal $I(V) \subset \mathbb{C}[X_0, \ldots, X_n]$. The ideal $I(V)$, truncated to the degree-$d$ piece, gives a vector subspace $I(V)_d \subset \mathbb{C}[X_0, \ldots, X_n]_d$. As $d$ varies, the dimensions $\dim I(V)_d$ assemble into a polynomial in $d$ for large $d$ — the **Hilbert polynomial** of $V$. The Hilbert polynomial is a basic invariant: it encodes the degree and dimension of $V$. Two subvarieties of $\mathbb{P}^n$ are "of the same type" if and only if they have the same Hilbert polynomial. One then asks: can the set of all subvarieties with a fixed Hilbert polynomial $P$ be given the structure of a variety? The answer requires fixing a large truncation degree $M$ and embedding the set of degree-$M$ pieces of ideals with Hilbert polynomial $P$ into a Grassmannian $\operatorname{Gr}(K, N)$ for appropriate $K$ and $N$. Grothendieck's theorem says that this construction works: [quotetheorem:2201] The theorem is not proved in this course; the key inputs are flatness and Castelnuovo–Mumford regularity. The crucial point is what happens when we take the closure. The set of honest subvarieties of $\mathbb{P}^n$ (those corresponding to radical ideals) is a Zariski open subset of $\operatorname{Hilb}^P(\mathbb{P}^n)$. But the Hilbert scheme is a projective variety, hence compact, so it must include limit points that do not correspond to any subvariety. What are these limit points? [explanation: Limits in the Hilbert Scheme Are Not Varieties] Return to the family $V_t = \mathcal{V}(tf_3 + X_0^3)$. For each $t \neq 0$, the Hilbert polynomial of $V_t$ is that of a smooth cubic curve. The limit $t \to 0$ must land somewhere in $\operatorname{Hilb}^P(\mathbb{P}^2)$, because the Hilbert scheme is compact. But the limit point corresponds to the ideal $(X_0^3)$, which is not radical. Its radical is $(X_0)$, the ideal of the line $\mathcal{V}(X_0)$, which is a curve with a completely different Hilbert polynomial. So the Hilbert scheme "knows" that the limit is $\mathcal{V}(X_0^3)$, not the line $\mathcal{V}(X_0)$. It keeps track of the non-radical ideal. But a classical variety cannot represent a non-radical ideal — the variety associated to $(X_0^3)$ and the variety associated to $(X_0)$ are the same set $\mathcal{V}(X_0)$. The additional information encoded in $X_0^3$ versus $X_0$ is invisible to the category of classical varieties. This is the precise sense in which the Hilbert scheme forces schemes upon us: the compactification of the space of varieties naturally includes objects whose coordinate rings contain nilpotent elements, and there is no way to describe these objects within the classical framework. [/explanation] ## Schemes: Keeping Track of Nilpotents The preceding discussion identifies the core problem. Classical algebraic geometry studies varieties, which correspond to radical ideals. Passing to the radical discards nilpotent information. Scheme theory retains this information by replacing the variety $\mathcal{V}(I)$ with the scheme $\operatorname{Spec}(R/I)$, where $R/I$ may contain nilpotent elements. [definition: Scheme (Informal)] A **scheme** is a locally ringed space $(X, \mathcal{O}_X)$ that is locally isomorphic to $\operatorname{Spec}(A)$ for some commutative ring $A$, where $\operatorname{Spec}(A)$ carries the Zariski topology and the structure sheaf whose sections over a basic open $D(f) = \{P : f \notin P\}$ are the localisation $A_f$. [/definition] The key difference from classical varieties: the ring $A$ need not be reduced (i.e., need not be equal to its own nilradical). The nilpotent elements of $A$ are genuinely part of the data of the scheme, even though they vanish as functions at every point of the underlying topological space. [example: The Scheme of a Double Point] Let $A = \mathbb{C}[X]/(X^2)$. As a topological space, $\operatorname{Spec}(A)$ has a single point (the maximal ideal $(X)$). But the ring $A$ is not reduced: $X \in A$ is a nonzero nilpotent. As a function on $\operatorname{Spec}(A)$, $X$ evaluates to zero at the single point — but $X \neq 0$ in $A$. This scheme $\operatorname{Spec}(\mathbb{C}[X]/(X^2))$ is the "dual numbers" scheme: it represents a point together with a tangent direction. It appears naturally as the kernel of the map $\mathbb{C}[\varepsilon]/(\varepsilon^2) \to \mathbb{C}$ and encodes first-order deformation data. Two schemes over $\operatorname{Spec}(\mathbb{C}[X]/(X^2))$ can disagree even when their underlying topological spaces are identical — their structure sheaves carry the distinction. [/example] ## The Generic/Closed Point Dichotomy and Non-Closed Fields A second major advantage of schemes — distinct from nilpotents — is that they provide the right framework for working over non-algebraically-closed fields and for relating varieties over different fields. Consider a polynomial $f \in \mathbb{Z}[X_0, \ldots, X_n]$. There are two natural ways to study its vanishing locus: - **Over $\mathbb{C}$:** reduce $f$ modulo no prime and consider $V(\mathbb{C}) = \mathcal{V}(f) \subset \mathbb{P}^n_\mathbb{C}$ as a complex variety. This gives access to Betti numbers, Euler characteristic, and the full apparatus of algebraic topology. - **Over $\mathbb{F}_{p^r}$:** reduce $f$ modulo a prime $p$ to get $\bar{f} \in \mathbb{F}_{p^r}[X_0, \ldots, X_n]$, and count the finite set $V(\mathbb{F}_{p^r}) = \mathcal{V}(\bar{f}) \subset \mathbb{P}^n_{\mathbb{F}_{p^r}}$. These two perspectives are related by the geometry of $\operatorname{Spec}(\mathbb{Z})$. In the scheme-theoretic picture, $f$ defines a scheme $X \to \operatorname{Spec}(\mathbb{Z})$. Each prime $p$ corresponds to a closed point $(p) \in \operatorname{Spec}(\mathbb{Z})$, and the fibre $X_p$ is the variety over $\mathbb{F}_p$. The generic point $(0) \in \operatorname{Spec}(\mathbb{Z})$ corresponds to the generic fibre $X_\mathbb{Q}$, which is the variety over $\mathbb{Q}$. This generic/closed point dichotomy is one of the most powerful features of scheme theory: a single geometric object — the scheme $X \to \operatorname{Spec}(\mathbb{Z})$ — simultaneously encodes arithmetic information (point counts over finite fields) and topological information (the complex variety), and the passage between them is a change of base along a map of schemes. ### The Weil Conjectures The Weil conjectures, proved by Grothendieck and Deligne, make the relationship between arithmetic and topology precise. Their full statement — rationality, functional equation, Riemann hypothesis, and Betti number correspondence — is developed in Chapter 7. Here we note only what the scheme-theoretic framework buys in this context. [remark: Why the Weil Conjectures Need Schemes] The Weil conjectures (see Chapter 7) assert that the Hasse–Weil zeta function $Z_X(t)$ of a smooth projective variety $X/\mathbb{F}_q$ is a rational function whose degree equals the Euler characteristic $\chi(X(\mathbb{C}))$ of the associated complex variety. This is a comparison between two a priori unrelated objects: a finite-field point count and a topological invariant of a complex manifold. Making this comparison rigorous requires a cohomology theory that works uniformly over all fields — not just $\mathbb{C}$. Such a theory must be defined for schemes, not just for classical varieties over algebraically closed fields. This is the technical content of étale cohomology, and it was the primary motivation for Grothendieck's development of the theory of schemes. [/remark] ## Why the Classical Theory Runs Out To summarise, classical algebraic geometry — varieties over algebraically closed fields with coordinate rings that are finitely generated reduced $k$-algebras — runs into at least three fundamental walls: **Limits of families leave the category.** The limit of a flat family of varieties need not be a variety: it may be a scheme with nilpotents in its structure sheaf. The Hilbert scheme is projective (hence compact), but the open subset parameterising honest varieties is not closed, so limits of varieties are schemes. **Arithmetic geometry is invisible.** A polynomial over $\mathbb{Z}$ lives simultaneously over $\mathbb{Q}$, over every $\mathbb{F}_p$, and over $\mathbb{C}$. Classical variety theory over $\mathbb{C}$ cannot see the finite-field fibres, and variety theory over $\mathbb{F}_p$ cannot see the complex fibre. Scheme theory over $\operatorname{Spec}(\mathbb{Z})$ sees all fibres at once. **Infinitesimal data is lost.** Tangent vectors, deformations, and intersection multiplicities are naturally described by rings with nilpotent elements. The scheme $\operatorname{Spec}(k[X]/(X^2))$ is the universal first-order deformation, and without it the moduli theory of varieties becomes pathological. Grothendieck's response to all three problems was a single conceptual move: replace "radical ideal in a polynomial ring over an algebraically closed field" with "arbitrary commutative ring." The resulting category of schemes is vastly larger than the category of classical varieties, but it contains the classical theory as a full subcategory, and it is stable under all the operations — limits, base change, deformation — that the classical theory cannot handle. [explanation: The Coordinate Ring Perspective] Here is a way to see the transition concretely. In the classical theory, the coordinate ring of an affine variety $V \subset \mathbb{A}^n_k$ is $k[X_1, \ldots, X_n]/I(V)$, where $I(V)$ is radical. The radical condition is forced by the requirement that functions on $V$ are determined by their values at points: if $f^2(p) = 0$ for all $p \in V$ then $f(p) = 0$ for all $p \in V$, so $f \in I(V)$. In scheme theory, we drop the radical condition and allow $A = k[X_1, \ldots, X_n]/I$ for any ideal $I$. The ring $A$ may contain nonzero nilpotents $a \in A$ with $a^n = 0$. Such an element vanishes at every point of $\operatorname{Spec}(A)$ as a function (since $a \in \mathfrak{m}$ for every prime $\mathfrak{m}$), but is retained as a section of the structure sheaf. The scheme "remembers" $a$ even though the underlying topological space cannot distinguish $\operatorname{Spec}(A)$ from $\operatorname{Spec}(A/\sqrt{0})$. This is precisely what happens for $\mathcal{V}(X_0^3)$: as a topological space it is the line $\mathcal{V}(X_0)$, but as a scheme its structure sheaf has the ring $\mathbb{C}[X_0, X_1, X_2]/(X_0^3)$, which remembers that $X_0$ vanished to order $3$. [/explanation] ## The Hilbert Scheme in Practice To close the chapter, let us make the Hilbert scheme discussion more concrete by tracing through what "being a limit point" means combinatorially. Fix $\mathbb{P}^n$ and a Hilbert polynomial $P$. For a subscheme $Z \subset \mathbb{P}^n$ with Hilbert polynomial $P$, the degree-$d$ piece of the saturated ideal of $Z$ is a vector subspace $I(Z)_d \subset H^0(\mathbb{P}^n, \mathcal{O}(d))$. For $d$ large enough (controlled by the Castelnuovo–Mumford regularity of $Z$), the dimension of $I(Z)_d$ is $\binom{n+d}{n} - P(d)$, and $Z$ is completely determined by $I(Z)_d$ for any single such $d$. This means the map $Z \mapsto I(Z)_d$ gives an injection from $\operatorname{Hilb}^P(\mathbb{P}^n)$ into the Grassmannian $\operatorname{Gr}(\binom{n+d}{n} - P(d),\, \binom{n+d}{n})$. The image is cut out by algebraic conditions (the "multiplication" conditions that say $I(Z)_d$ is an ideal piece), and these conditions define a Zariski closed subset — the Hilbert scheme. [example: Three Collinear Points in the Plane] Let $P = 3$ (constant polynomial, so $Z$ is a scheme of length $3$) in $\mathbb{P}^2$. The Hilbert scheme $\operatorname{Hilb}^3(\mathbb{P}^2)$ parameterises all zero-dimensional subschemes of $\mathbb{P}^2$ of length $3$. This includes: - Three distinct points $\{p_1, p_2, p_3\}$: a reduced scheme, three closed points. - A reduced point $p_1$ and a "double point" $Z_2$ at $p_2 \neq p_1$: here $Z_2 = \operatorname{Spec}(\mathcal{O}_{\mathbb{P}^2, p_2}/\mathfrak{m}_{p_2}^2)$ encoded by a tangent direction at $p_2$. - A "triple point" $Z_3$ at a single point $p$: the scheme $\operatorname{Spec}(\mathcal{O}_{\mathbb{P}^2, p}/I)$ for an ideal $I$ of colength $3$. The latter two types involve nilpotents and cannot be described as classical varieties. Yet they appear naturally as limits of families of three distinct points — for example, by letting two or all three points collide. The Hilbert scheme is the correct compactification of the configuration space of $3$ distinct points in $\mathbb{P}^2$. [/example] The lesson of the Hilbert scheme is exactly the lesson of the bonus lecture as a whole: the classical category of varieties is not closed under natural limiting operations. Scheme theory is the minimal enlargement of this category that is stable under limits, base change, and deformation — and this is why every serious treatment of modern algebraic geometry begins with schemes. Having developed classical algebraic geometry from varieties to divisors to the Weil conjectures, this final chapter provides two indispensable supplements: worked examples showing how rational maps can extend to morphisms in surprising ways, and the full proof of Hilbert's Nullstellensatz, which closes the logical foundation on which all preceding chapters rest. # A. Appendix The main body of the course develops the general theory of varieties, morphisms, and their geometric invariants. This appendix collects two pieces of supplementary material that deepen specific results from earlier chapters: first, a worked example illustrating how rational maps on projective curves can extend to genuine morphisms in non-obvious ways, and second, the full proof of Hilbert's Nullstellensatz — the foundational theorem whose statement was used throughout Chapters 1–2 but whose proof the course deferred. Together these sections close the logical gaps in the exposition and reward the reader who has absorbed the main theory. ## A Conic and Its Two Projection Morphisms Projection maps from a point in projective space to a lower-dimensional projective space are among the simplest examples of rational maps between varieties. When the centre of projection lies outside a given curve, the restriction to the curve is automatically a morphism. When it lies on the curve, however, the situation is more delicate: the rational map may appear undefined at that point, yet still extend to a genuine morphism by finding an alternative regular local description. This example from the conic $C = V(X_0 X_2 - X_1^2) \subset \mathbb{P}^2_k$ makes the distinction concrete. [example: Two Projections of the Conic] Let $C = V(X_0 X_2 - X_1^2) \subset \mathbb{P}^2_k$, where $k$ is an algebraically closed field. The conic $C$ is a smooth rational curve; it will turn out to be isomorphic to $\mathbb{P}^1_k$. **The projection $\pi_1$ from $[0:1:0]$.** Define $\pi_1 : \mathbb{P}^2_k \dashrightarrow \mathbb{P}^1_k$ to be the projection from the point $[0:1:0]$, so on points not equal to $[0:1:0]$ it acts by $[X_0 : X_1 : X_2] \mapsto [X_0 : X_2]$. First, we check that $[0:1:0] \notin C$: substituting gives $0 \cdot 0 - 1^2 = -1 \neq 0$. Since the centre of projection is not on $C$, the rational map $\pi_1$ restricts to a well-defined morphism $\pi_1|_C : C \to \mathbb{P}^1_k$. The fibre over a point $[a_0 : a_2] \in \mathbb{P}^1_k$ consists of all $[a_0 : s : a_2] \in C$, i.e., points satisfying $a_0 a_2 - s^2 = 0$, so $s^2 = a_0 a_2$. The number of solutions $s \in k$ is either $1$ (when $a_0 a_2 = 0$, giving $s = 0$) or $2$ (when $a_0 a_2 \neq 0$, giving $s = \pm\sqrt{a_0 a_2}$). Geometrically, one pictures a pencil of lines emanating from the centre $[0:1:0]$ sweeping across the conic and projecting it onto a line; lines through the centre generically meet the conic in two points, tangentially at the two points where the conic meets the line $X_0 X_2 = 0$ at infinity. **The projection $\pi_2$ from $[0:0:1]$.** Define $\pi_2 : \mathbb{P}^2_k \dashrightarrow \mathbb{P}^1_k$ to be the projection from $[0:0:1]$, sending $[X_0 : X_1 : X_2] \mapsto [X_0 : X_1]$ wherever defined. Now check whether $[0:0:1] \in C$: substituting gives $0 \cdot 1 - 0^2 = 0$, so $[0:0:1]$ lies on $C$. The map $\pi_2$ is therefore a rational map $C \dashrightarrow \mathbb{P}^1_k$ that is only manifestly a morphism on $C \setminus \{[0:0:1]\}$. On the open locus $C \setminus \{[0:0:1]\}$ we have $X_0 \neq 0$ (since $X_0 = 0$ on $C$ forces $X_1 = 0$, giving the single point $[0:0:1]$). So for $[a_0 : a_1 : a_2] \in C$ with $a_0 \neq 0$, the defining relation gives $a_2 = a_1^2/a_0$, and the fibre of $\pi_2$ over $[a_0 : a_1]$ is the unique point \begin{align*} \left[a_0 : a_1 : \frac{a_1^2}{a_0}\right]. \end{align*} At the missing point $[0:0:1]$, the formula $[X_0 : X_1 : X_2] \mapsto [X_0 : X_1]$ is undefined since both $X_0$ and $X_1$ vanish there. However, consider the alternative formula $[X_0 : X_1 : X_2] \mapsto [X_1 : X_2]$. At $[0:0:1]$ this gives $[0:1]$, which is a well-defined point of $\mathbb{P}^1_k$. On the overlap $C \setminus \{[0:0:1]\}$, using $a_2 = a_1^2/a_0$: \begin{align*} [a_0 : a_1] = [a_0 a_1 : a_1^2] = [a_0 a_1 : a_0 a_2] = [a_1 : a_2], \end{align*} so the two descriptions agree wherever both are defined. Patching them together gives a globally defined morphism $\pi_2|_C : C \to \mathbb{P}^1_k$. **The resulting morphism is an isomorphism.** A further analysis shows that both descriptions of $\pi_2|_C$ are injective with single-point fibres everywhere: for any $[b_0 : b_1] \in \mathbb{P}^1_k$, there is a unique point of $C$ mapping to it. Since $C$ and $\mathbb{P}^1_k$ are smooth projective curves and the morphism is bijective, it is in fact an isomorphism $C \cong \mathbb{P}^1_k$. [/example] [remark: Regularity at Apparent Indeterminacy] The key lesson of the $\pi_2$ analysis is that a rational map can be regular at a point even when one particular local formula breaks down. Regularity is intrinsic: a rational map $\phi : X \dashrightarrow Y$ is regular at $p$ if there exists any representative defined and non-vanishing at $p$. Finding the alternative formula $[X_1 : X_2]$ on $C$ near $[0:0:1]$ is exactly the geometric content of removing the indeterminacy. [/remark] [illustration:ag-conic-two-projections] ## Hilbert's Nullstellensatz: Full Proof The Nullstellensatz is the bridge between algebra and geometry in the affine setting. Its weak form guarantees that a proper ideal has a geometric realisation — a non-empty vanishing set — and its strong form gives the precise algebraic description of the ideal of functions vanishing on a given variety. Both statements were stated in Chapter 2 and used throughout Chapters 2–6 to build the Zariski correspondence between ideals and varieties; this section provides the complete proofs, supplying the argument deferred there. ### The Weak Nullstellensatz via a Dimension Count The weak Nullstellensatz is an existence result: every proper ideal in $k[X_1, \ldots, X_n]$ has at least one common zero. The proof we give works for uncountable algebraically closed fields and proceeds by showing that the residue field of any maximal ideal must equal $k$ itself. [quotetheorem:2202] [citeproof:2202] This immediately yields the geometric content: [quotetheorem:2123] [citeproof:2123] ### The Strong Nullstellensatz via Rabinowitsch's Trick The weak Nullstellensatz tells us that proper ideals have zeros. The strong form, due to Hilbert, identifies the ideal of all polynomial functions vanishing on $V(I)$ with the radical $\sqrt{I}$. The inclusion $\sqrt{I} \subset \mathcal{I}(V(I))$ is immediate from definitions: if $f^m \in I$ then $f^m$ vanishes on $V(I)$, hence so does $f$. The hard direction is the reverse. The elegant proof of the reverse inclusion uses a technique introduced by Rabinowitsch: introduce a new variable $T$ to turn the vanishing condition on $f$ into membership in an ideal, then apply the weak Nullstellensatz in $k[X_1, \ldots, X_n, T]$. [quotetheorem:2124] [citeproof:2124] [remark: Projective Analogues] Projective versions of the Nullstellensatz — identifying the homogeneous radical ideal of $V(I) \subset \mathbb{P}^n_k$ — follow from the affine statement by passing to the cone over a projective variety and applying the affine result to the corresponding homogeneous ideal. The only subtlety is the irrelevant ideal $(X_0, \ldots, X_n)$, which must be handled separately: one needs $I \neq (X_0, \ldots, X_n)$ to ensure $V(I) \neq \varnothing$ in projective space. [/remark] [explanation: Why Rabinowitsch's Trick Works] The substitution $T = 1/f$ is the conceptual heart of the argument. At the algebraic level, it converts the geometric statement "$f$ vanishes on $V(I)$" — which is a condition on infinitely many points — into a single algebraic identity in a ring. The auxiliary variable $T$ forces $fT - 1 = 0$, i.e., $T = 1/f$, so any element of the form $(1 - fT)g$ vanishes under the substitution. What remains, after multiplying through to clear denominators, is an algebraic certificate that $f^m \in I$. This technique recurs throughout algebra: trading a geometric vanishing condition for algebraic membership via a localisation or a new variable. It is conceptually similar to the trick of inverting an element to detect non-membership in an ideal. [/explanation] ## References Based on Zhiyuan Bai's notes for the Cambridge Mathematical Tripos Part II course *Algebraic Geometry*, taught by Dr. D. Ranganathan in Lent 2021 (with bonus material from 2022).