[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]