Algebraic Geometry
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 14 (Algebraic geometry)

Prerequisites:

# Getting Oriented

 Rough Guides to Algebra
 General Algebra Special Topics

Algebraic geometry can be thought of as the study of generalizations of the Fundamental Theorem of Algebra. Precisely, it is the study of the zero sets of polynomials. So the FTA is just the restriction to the case of a single polynomial in one variable.

# Affine Varieties

Varieties are similar to manifolds, with the primary difference being their algebraic nature: they are defined by polynomials rather than maps and open sets. Even more algebraically, they can be defined as ideals of a certain ring. They are much more structured than manifolds, and of course their study uses a lot more algebra.

## The Basics

The primary object of study in algebraic geometry is the variety:

Affine Variety
the zero locus of a collection of polynomials (possibly in more than one variable) over some field $K$. If these polynomials are $F_1,\ldots,F_k$, then the variety is denoted $V(F_1,\ldots,F_k)=\{F_1,F_2,\ldots,F_k=0\}$.} Generally, the field will be assumed to be $\mathbb{C}$. A simple example in this case is $x^2+y^2=1$, whose real solutions form the unit circle. The word affine separates this variety from projective varieties, where the underlying space is $\mathbb{P}^n$ rather than $K^n$. We will discuss these more later.

A variety which cannot be broken up into two smaller varieties, or subvarieties, in a nontrivial way is called irreducible. Precisely, a variety $X$ is irreducible if one cannot write $X=X_1\cup X_2$ where $X_i\neq 0,X$. Irreducibility is often assumed, sometimes even in the definition of an affine variety. Every irreducible variety over $\mathbb{C}$ has a dense subvariety which is a manifold. The above example is an irreducible variety; the ‘cross’ $xy=0$ is an example of a reducible variety. All varieties are built from irreducible varieties: we can break down a variety $X$ by

(1)
\begin{align} X=X_n \supseteq X_{n-1} \supseteq \cdots \supseteq X_0, \end{align}

where $X_i$ is obtained from $X_{i-1}$ by adding an irreducible component.

Speaking of a zero set is a fairly geometric representation; there is also an equivalent algebraic representation: a variety $V(F_1,\ldots,F_k)$ corresponds to a unique ideal of the polynomial ring $K[x_1,x_2,\ldots,x_n]$, specifically the radical ideal $\sqrt{(F_1,\ldots,F_k)}$. A polynomial $f$ is in the radical ideal iff $f^k$ is in the ideal $(F_1,\ldots,F_k)$ for some $k$. In the example above, the ideal is just $\fgp{x^2+y^2-1}$. We'll denote the ideal corresponding to a variety $X$ by $I(X)$. We can then define the

Coordinate Ring $K[X]$
given a variety $X$, it is the ring $K[x_1,\ldots,x_n]/I(X)$, or intuitively the set of polynomials each having unique values on the variety. Functions in $K[X]$ are called regular functions.

Note that $X_1\subset X_2$ iff $I(X_2)\subset I(X_1)$: thus large ideals correspond to small subvarieties, and vice versa. In particular, we have correspondences $K(x_1,\ldots,x_n) \leftrightarrow \{\}$ (the empty variety), maximal ideals $\leftrightarrow$ single points, and prime ideals $\leftrightarrow$ irreducible subvarieties.

## The Zariski Topology and Dimension

There is a topology induced on affine varieties in which points are close only if they lie in many of the same varieties:

Zariski Topology
the topology whose closed sets are exactly the zero locus of polynomials $F_1,\ldots,F_k=0$. This is much coarser than the standard topology.

Often, an affine variety is defined as a subset of a variety $X$ which is Zariski open (in the induced subspace topology). This basically allows varieties to be “subtracted” from each other. It also allows bad points, such as intersections, to be removed from consideration.

Defining the dimension of a variety can be a little tricky, since there may be pieces of several different dimensions. Thus, we restrict ourselves to irreducible varieties, and for the remaining cases let the dimension be the largest dimension of the irreducible subvarieties. In this case, all Zariski open subsets are dense, and we can define the dimension of the variety to be the dimension of the subset.

There are alternate ways to define the dimension. One can define the tangent space to $X=V(F_1,\ldots,F_k)$ at a point $(p_1,\ldots,p_n)$ as the set of solutions, in the variables $\{x_j\}$, to

(2)
\begin{align} \sum_j \delfrac{F_i}{x_j}(p)(x_j-p_j)=0 \text{ for all } i. \end{align}

Points with degenerate tangent spaces are called singular points. The dimension of $X$ is also equal to the dimension of the tangent space to $X$ at a generic point.

A third definition is algebraic: the dimension of $X$ is the Krull Dimension of the ring $K[X]$, that is the largest $n$ such that $K[X]$ contains a chain of increasing prime ideals

(3)
\begin{align} p_0\subseteq p_1\subseteq \cdots \subseteq p_n. \end{align}

All three definitions described here give the same result.

## Maps on Affine Varieties

Recall that regular functions are those in the coordinate ring $K[X]$. Between varieties, one has:

**Regular Map*
a map $f:X\to Y$ between $X\subset K^n$ and $Y\subset K^m$ which can be written in the form $f=(f_1,\ldots,f_m)$ where the $f_i$ are all regular functions. Equivalently, the pullback $f^*:K[Y]\to K[X]$ is a ring homomorphism.

Note that the image $f(X)$ is dense in $Y$ iff the pullback $f^*$ has trivial kernel, meaning all functions which vanish on $f(X)$ vanish on $Y$. As we might expect, a regular isomorphism is just a bijective regular map, and $X\cong Y$ iff $K[X]\cong K[Y]$.

The coordinate ring $K[X]$ of an irreducible variety has field of fractions $K(X)$ which consists of rational functions of the form $f/g$ with $g \not\in I(X)$. A rational function $f/g\in F(X)$ is regular at a point $p\in X$ if $g(p)\neq 0$. If a rational function is regular at every point, then it is actually a regular function. Also, two rational functions which agree everywhere on an open set of $X$ are equal.

Between algebraic varieties, one has rational maps $\phi=(\phi_1,\ldots,\phi_m): X \to Y$ taking values in $Y$ wherever the $\phi_i$ are all defined. This is equivalent to having all $\phi_i$ defined on a Zariski open set. Bijective rational maps are called birational maps. Affine varieties $X$ and $Y$ are birationally equivalent iff the pullback $\phi^*$ gives an isomorphism $K(X)\cong K(Y)$ iff there exist isomorphic Zariski open subsets $U\subset X$ and $V\subset Y$.

## Induced Varieties

There are a number of ways in which a variety $X$ may induce new varieties. To start, we list a few special subvarieties:

• The intersection of varieties $X\cap Y$, which consists of points on which functions $f \in I(X)\cap I(Y)$ vanish.
• The singular subvariety consisting of points at which $\dim T_p(X) > \dim X$, that is points with degenerate tangent spaces.
• The complement of the singular subvariety, called the nonsingular subvariety.

Both intersections $X\cap Y$ and singular subvarieties may be reducible, even if $X$ and $Y$ are irreducible. Nonsingular subvarieties behave much better: they are dense, open, irreducible subvarieties.

There are other ways to combine two varieties $X$ and $Y$: one has a union variety [[$X\cup Y$, and a product variety $X\times Y \subset \mathbb{A}^{n+m}$.

## Blow-Ups

One way to resolve singular points on a curve is called the blow-up variety. Essentially, replace a singular point by the set of possible tangent lines to that point, which is just a copy of projective space $\mathbb{P}^{n-1}$. Formally, at the point $p$ one has the variety in $\mathbb{A}^n\times\mathbb{P}^{n-1}$ defined by the equations $F_1,\ldots,F_k=0$ as well as $\frac{x_i-p_i}{x_j-p_j}=\frac{y_i}{y_j}$ or

(4)
\begin{align} (x_i-p_i)(y_j)=(x_j-p_j)(y_i), \quad x_i\in\mathbb{A}^n, y_i\in\mathbb{P}^{n-1}. \end{align}

This looks exactly like the variety $X$ everywhere except the point $p$, which is replaced by the entirety of $\mathbb{P}^{n-1}$ since the $y_i$ may be chosen arbitrarily. The blow-up of $X$ is birational to $X$. Applying a series of blow-ups, one can resolve all singularities of $X$ and so obtain a nonsingular variety.

One can also define the blow-up along a subvariety $Y$ given by $G_1,\ldots,G_l=0$ as the closure of the set of points $(x,\phi(x)) \in \mathbb{A}^n\times \mathbb{P}^{l-1}$, where $\phi(x)$ is the “direction” of the variety $X$ along the subvariety $Y$. More precisely, it is the line through the point $[G_1(x),\ldots,G_l(x)]$. This coincides with the previous formulation, except that it does not include the extra copy of $\mathbb{P}^{n-1}$.

# Projective Varieties

Projective Variety
the zero locus of a collection of homogeneous polynomials in projective space.

Projective space $\mathbb{P}^n$ is the set of $(n+1)$-tuples $(x_0,\ldots,x_n)$, excluding the origin, with the equivalence relation $(x_0,\ldots,x_n)\simeq(\lambda x_0,\ldots, \lambda x_n)$ for $\lambda\neq 0$. This can also be thought of as the set of lines through the origin in $\K^{n+1}$. Homogeneous just means that all terms of the polynomial have the same degree. An analogous example to that above would be $X^2+Y^2=Z^2$.