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