This course introduces algebraic geometry through the classical language of polynomial equations, affine and projective varieties, and the geometric meaning of their algebraic invariants. It asks how solution sets of polynomial systems are organized, how topology and algebra interact through the Zariski topology and coordinate rings, and how geometric objects can be studied using regular functions, rational maps, and local descriptions. The emphasis is on building a working dictionary between geometry and commutative algebra in a setting where the fundamental objects are concrete and explicit.

The chapters develop this dictionary step by step. The early chapters establish affine algebraic sets, the Nullstellensatz, and regular maps, then move to localization and rational functions as the local tools needed to analyze geometry on smaller open pieces. From there the course passes to projective space, homogeneous equations, projective closure, and elimination, which provide the natural compact setting for classical geometry and its embeddings. Later chapters study dimension and degree, tangent spaces and singularities, and hypersurface sections mostly in classical language: a fiber first means an inverse image of a point, a projective variety first means a zero set of homogeneous polynomials in projective space, and families of linear subspaces are introduced through explicit coordinates. Some quoted theorem cards in the later chapters are included as optional modern reference points rather than as part of the classical working level of the course. When those cards mention schemes, sheaves, Grassmannians, Cartier divisors, regular local rings, or scheme-theoretic length, they are recording the standard contemporary form of a result whose surrounding discussion is classical. A reader may use the card as a reference label and continue with the point-set, coordinate-ring, and explicit-equation interpretation developed in the main text.

# Introduction

This course studies geometric objects described by polynomial equations over an [algebraically closed field](/page/Algebraically%20Closed%20Field) $k$. Its guiding question is how much geometry can be recovered from commutative algebra, and how algebraic operations on polynomial rings appear as geometric operations on solution sets. The first half of the course builds the affine dictionary; the second half moves to projective space, where compactness-like phenomena and intersections at infinity enter naturally.

A scope convention will be used throughout. Unless a passage explicitly says otherwise, all varieties in this note are classical affine or projective varieties over $k$, meaning reduced algebraic sets cut out by polynomial or [homogeneous polynomial](/page/Homogeneous%20Polynomial) equations. If a map $f:X\to Y$ is discussed, the fiber over a point $y\in Y$ means the ordinary set-theoretic inverse image $f^{-1}(y)$, with its equations inherited from the defining equations of $X$ and the coordinate equations of $y$. Quoted theorem cards that use modern notation should be read as optional reference cards at a higher level of generality. In that notation, $\operatorname{Spec} A$ denotes the affine scheme attached to a ring $A$, $\operatorname{Proj} S$ denotes the projective scheme attached to a graded ring $S$, $\mathcal O_X(d)$ denotes the standard twisting sheaf on a projective variety, and $H^0(X,\mathcal O_X(d))$ denotes its space of global degree-$d$ sections. A scheme-theoretic fiber records the equations of a fiber with possible multiplicity or nilpotent structure; a reduced induced scheme forgets that extra nilpotent information; a Grassmannian parametrizes linear subspaces of a fixed dimension; a Cartier divisor is the modern way to package hypersurface-like codimension-one data; and the length of a zero-dimensional intersection is the algebraic multiplicity counted by its local coordinate ring. These explanations are a reading aid for the cited statements, not a prerequisite list for the course. The surrounding arguments continue to use the classical point-set and coordinate-ring language developed in the course, and any modern card can be skipped without losing the main line of exposition.

## The Central Problem

The starting point is a familiar observation from linear algebra and calculus: equations cut out shapes. Algebraic geometry asks what happens when the equations are polynomial, the ambient space is $k^n$, and we insist that the resulting theory remember both geometry and algebra.

[explanation: Polynomial Equations as Geometry] Let $k$ be an algebraically closed field. A finite family of polynomials $f_1,\dots,f_r \in k[x_1,\dots,x_n]$ determines a subset of affine space by imposing the simultaneous equations

\begin{align*} f_1(a)=\cdots=f_r(a)=0. \end{align*}

The geometric object is the common zero set, while the algebraic object is the ideal $(f_1,\dots,f_r)$ generated by the equations. The course is organised around making this passage reversible in the correct sense: geometry determines functions on the space, and functions determine the equations that vanish on it. [/explanation]

This reversal is not literal unless the algebra is adjusted. Different sets of equations may define the same subset, and nilpotent information in a [quotient ring](/page/Quotient%20Ring) is invisible to classical point sets. The Nullstellensatz will identify radical ideals as the correct algebraic objects for classical affine varieties.

[example: Two Equations With the Same Classical Points] Over a field $k$, the two ideals $(x)$ and $(x^2)$ in $k[x]$ have the same vanishing set in $\mathbb A_k^1$. Indeed, for $a \in k$, \begin{align*} a \in V(x) \text{ if and only if } a=0. \end{align*} Also, \begin{align*} a \in V(x^2) \text{ if and only if } a^2=0. \end{align*} Since a field has no nonzero nilpotent elements, $a^2=0$ implies $a=0$, so \begin{align*} V(x)=V(x^2)=\{0\}. \end{align*} The quotient rings remember different algebra. In $k[x]/(x)$, the class of $x$ is zero because $x \in (x)$. In $k[x]/(x^2)$, the class $\overline{x}$ is not zero: if $\overline{x}=0$, then $x \in (x^2)$, so $x=x^2g(x)$ for some $g(x)\in k[x]$, which is impossible because the right side has degree at least $2$ unless it is zero. But $\overline{x}$ is nilpotent, since \begin{align*} \overline{x}^2=\overline{x^2}=0. \end{align*} Thus $(x)$ and $(x^2)$ define the same classical point set but give different quotient rings; the classical dictionary keeps the radical ideal $(x)$ and deliberately forgets the nilpotent thickening encoded by $k[x]/(x^2)$. Later scheme theory is designed to retain such nilpotent information, but that language is not part of the working framework of this course. [/example]

## Affine Space and Coordinate Rings

Once a subset of affine space has been cut out by equations, the next problem is to decide which functions on it should count as polynomial functions. Restricting polynomials from the ambient affine space gives the right class, but two ambient polynomials give the same function on the subset precisely when their difference vanishes there.

[definition: Affine Space] Let $k$ be a field and let $n \in \mathbb N$. Affine $n$-space over $k$ is the set \begin{align*} \mathbb A_k^n := k^n. \end{align*} Its elements are written $a=(a_1,\dots,a_n)$. [/definition]

The notation treats $k^n$ as a space of points rather than as a [vector space](/page/Vector%20Space) with a preferred origin. Having named the ambient space, the next task is to name the subsets produced by polynomial equations inside it.

[definition: Vanishing Set] Let \begin{align*} V:\mathcal P(k[x_1,\dots,x_n]) \to \mathcal P(\mathbb A_k^n) \end{align*} be the operation \begin{align*} S \mapsto V(S) \end{align*} that sends a subset $S \subset k[x_1,\dots,x_n]$ to its vanishing set. For such an $S$, define \begin{align*} V(S) := \{a \in \mathbb A_k^n : f(a)=0 \text{ for all } f \in S\}. \end{align*} [/definition]

The operation $S \mapsto V(S)$ is a map from the power set of $k[x_1,\dots,x_n]$ to the power set of $\mathbb A_k^n$. The set $S$ may be infinite, but the [Hilbert Basis Theorem](/theorems/860) will imply that over a Noetherian [polynomial ring](/page/Polynomial%20Ring) every such vanishing set is already defined by finitely many equations. Subsets that arise in this way receive a name, because they are the basic geometric objects of the affine part of the course.

[definition: Affine Algebraic Set] A subset $X \subset \mathbb A_k^n$ is an affine algebraic set if there exists a subset $S \subset k[x_1,\dots,x_n]$ such that \begin{align*} X=V(S). \end{align*} [/definition]

To compare two different defining families for the same affine algebraic set, we need to move in the reverse direction. Starting from a subset $X$, the relevant algebraic data is not a chosen list of equations but the complete collection of polynomial equations that vanish at every point of $X$. This construction records exactly when two ambient polynomials define the same function on $X$, and it will become the kernel used to form the coordinate ring.