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Elements of Classical Algebraic Geometry

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Affine sets, the Zariski topology, and Hilbert’s Nullstellensatz

Affine varieties are the fundamental building blocks of the central objects we study in classical algebraic geometry, quasi-projective varieties. In essence, an affine variety is the solution set of a system of polynomial equations $$f_1(x_1, …, x_n) = f_2(x_1, …, x_n) = … =fm(x_1, …, x_n) = 0$$ with coefficients in some field $\k$. This solution set is made into a geometric object after we introduce the Zariski topology on it. Furthermore, two such solutions sets are allowed to communicate with each other only through a restricted class of functions, called regular maps. Setting things up this way allows us to define what it means for two solution sets to be isomorphic, even though they may arise in totally different contexts. (Much like in group theory, the notion of homomorphism and isomorphism allows us to extract an abstract group from its various guises.) An affine variety $X$ is then basically an equivalence class of solution sets under the notion of regular isomorphism. It turns out that $X$ is uniquely recoverable from the ring of regular functions on it. This ring is called its coordinate ring, denoted $\k[X]$. The basic philosophy in affine algebraic geometry is then to translate “geometric notions” about $X$ into ring-theoretic notions about $\k[X]$, thereby supporting our geometric intuitions with rigorous algebra.

The relationship between the originating system of equations and its solution set is cleanest and makes most intuitive sense when the ground field $\k$ is algebraically closed, and the main theorem(s) describing this relationship is Hilbert’s Nullstellensatz in its various incarnations. This theorem is the most important piece of commutative algebra allowing us to set up the theory of affine varieties. Because algebraic geometry is easiest to understand when $\k$ is algebraically closed, we will assume this throughout. (Later, we will see that the characteristic $0$ situation is especially nice, but for now algebraic closed-ness is all we assume.)

Affine sets

We begin by defining the central geometric objects of this chapter.

It is immediate that if $S ⊂ \k[x_1, …, x_n]$ is any subset, and if $I$ is the ideal generated by $S$, then $\V(S) = \V(I)$. So, our focus quickly moves to ideals in $\k[x_1, …, x_n]$. The following exercise justifies the use of the words “closed/open.”

In light of this exercise, in order to create an algebraic set, we need only consider $\V(I)$’s for ideals $I ⊂ \k[x_1, …, x_n]$. Whenever we say “open” or “closed”, we refer to the Zariski topology.

The beauty in classical algebraic geometry is only seen through its countless, interesting examples. Certain examples are more educational than others and have special names. Here are a few to start:

Pictures

In the list of examples above, what do those pictures mean? Clearly they are real images, meaning they exist in $\A^2\R$ or $\A3\R$. But this is not our true context: we’ve assumed $\k$ is algebraically closed! Unfortunately, it is not possible to draw a true representation in this case, so we refer to these sorts of images as “real cartoons.”

Operations $\V$ and $\I$

The operation $\V$ is a surjective function: \[\V(-): \big\{\,\textrm{ideals in} \,\, \k[x_1, …, x_n] \big\} → \big\{\,\textrm{affine subsets of} \,\, \An\k \big\}.\]

But we can also go backwards, from sets to ideals:

And so, $\I$ defines a function: \[\I(-): \big\{\,\textrm{affine subsets of} \,\, \An\k \big\} → \big\{\,\textrm{ideals in} ⊂ \k[x_1, …, x_n] \big\}.\]

$\I(-)$ satisfies some obvious properties, for example if $Z_1 ⊂ Z_2$ is a proper containment of affine subsets, then we get a strict, opposite containment of ideals $\I(Z_2) ⊃ \I(Z_1)$.

Since $\V$ and $\I$ point in opposite directions, it’s natural to wonder that their compositions (both ways) are. One composition, found in the next exercise, is easy to understand. The other composition leads naturally to Hilbert’s Nullstellansatz, and so we postpone it.

Prime ideals and irreducible affine sets

Affine sets have “main, atomic pieces,” called irreducible components. This requires that we define the notion of reducibility, which we elaborate on here.

For affine sets, irreducibility has a nice algebraic interpretation.

Algebra: Hilbert’s Nullstellensatz

Our first fundamental theorem, which appears in several different forms in several different levels of generality, explains the relationship between the operations $\V(-)$ and $\I(-)$. The exact message of Hilbert’s Nullstellensatz can be confusing for newcomers, so I’ll try to give many different viewpoints here.

  1. When $\k$ is algebraically closed, the Nullstellensatz tells us that the only way for $\V(I)$ to be empty is if $I = (1)$, the unit ideal. In other words, any system of polynomial equations which is not equivalent to $1=0$ /must have/ solutions in $\A^n\k$.
  2. When $\k$ is algebraically closed, the Nullstellensatz provides the precise list of maximal ideals in $\k[x_1, …, x_n]$.

Regular functions

Next, we must set the rules for how algebraic sets and quasi-affine sets are allowed to communicate with each other. This will also allow us to talk about a quasi-affine set without reference to an ambient $\An\k$. In short, we allow ourselves only to use functions which are locally describable using ratios of polynomial functions. These functions are called regular functions.

It would be nice if this restricted class of functions behaved well with respect to other choices we’ve made. For instance, it comes as a relief that:

After doing the previous exercise, one sees that the composition of regular mappings is again regular.

If $U ⊂ \An\k$ is a quasi-affine set, then there is a very quick way to produce regular functions on $U$: restrict a polynomial $f(x_1, …, x_n)$ to $U$. Therefore, we always have a ring homomorphism $\k[x_1, …, x_n] → \k[U]$. Secondly, clearly if $f ∈ \I(U)$, then $f$ restricts to the zero function $0 ∈ \k[U]$. Therefore, restriction of polynomials induces a homomorphism \[\k[x_1, …, x_n]/\I(U) → \k[U].\] One fundamental use of the Nullstellensatz provides great news in the case that $U ⊂ \An\k$ is affine.

This is why, when $Z ⊂ \An\k$ is affine, the ring of regular functions $\k[Z]$ is called the coordinate ring of $Z$ – it just turns out to be the algebra of functions obtained by playing around with only the coordinate functions $x_1, …, x_n$ restricted to $Z$.

The various rings attached to an affine set

The ring \(\O(U)\) for an open set \(U ⊂ X\)

The local ring \(\OX,p\) attached to point \(p ∈ X\)

Projective space $\P^n\k$ and projective varieties

For a multitude of reasons, it is useful to want to compactify affine space. Take, for instance, the concept of the slope of a line from high school algebra. Slopes were numbers, typically labelled $m$. Except, hold on, what about those pesky “vertical” lines? It is clear that, geometrically, vertical lines are no more special than any other kind of line. So we should extend the concept of slope $m$ to include an “infinity” slope $m =∞$. By adding $∞$ to the $\A^1$ of slopes, we arrive at the projective line $\P^1$.

Another way that projective space arises is in the following context: We have a vector space $V$, but we only really care about elements of $V$ up to scaling. For instance, if we only care about the location of the roots of a quadratic polynomial $ax^2+bx+c$, then for our purposes there is no difference between polynomials $ax^2+bx+c$ and $λ a x^2 + λ b x + λ c =0$, where $λ$ a nonzero number.

Projective varieties

For simplicity, we work throughout over an algebraically closed field \(\k\).

Projective morphisms

Morphisms between projective spaces

(Proof: For all representatives $(ξα, Uα)$ representing $\varphi$, define $U\textrm{max}$ to be the union of all $U_α$. Suppose $(ξα, Uα)$ and $(ξβ, Uβ)$ both represent $\varphi$. Then $Uα ∩ Uβ$ is an irreducible quasi-projective variety. The two regular maps $ξα|U_{α ∩ Uβ}$ and $ξβ|U_{α ∩ Uβ}$ agree on an open set, and therefore agree everywhere. Thus the $ξα$’s glue together to produce a regular map on $U\textrm{max}$, which is what we needed.)

Tangent spaces

To each point $p$ on a variety $X$ we attach a \(k\)-vector space $TX,p$, called the /Zariski tangent space/. This vector space unlocks the important concept of /smoothness/ of $X$ at $p$, which is the algebro-geometric formulation of the intuition that $X$ appears “locally linear” around $x$.

In what follows, we will provide several definitions of $TX,p$, with the first being the most concrete, and the last being the most intrinsic.

Now, an important point. Notice that this most concrete definition produces a linear space inside $\An\k$. But the Zariski tangent space $TX,p$ actually has a specific structure of $\k$-vector space. The vector space structure is as follows. To each point $q = (q_1, …, q_n) ∈ TX,p$ we associate the displacement vector $\langle q_1 - p_1, …, q_n - p_n \rangle$. Addition is by component-wise addition in these displacements, and scalar multiplication acts by scaling these displacements.

The ring $\k[ε]/(ε^2)$ provides a sometimes-useful alternate interpretation of the Zariski tangent space. This is encoded in the following critical exercise.

Singular points and smooth points

The nature of the local ring at a smooth point

Bertini’s theorem(s)

Constancy of number

Hypersurfaces

Let $X = \V(F) ⊂ \P^n\k$ be a hypersurface, defined by a degree $d$ homogeneous form $F(x_0, …, x_n)$. Let \(p = (p_0, \ldots, p_n) ∈ \left( \An+1 \right)×\) be a point such that $F(p) = 0$, and let $\hat{X} ⊂ \An+1$ denote the affine cone over $X$. Then $X$ is singular at $[p]$ if and only if $\hat{X}$ is singular at $p$.

Basics on vector bundles

The definition of a vector bundle

The concept of a /vector bundle/ is ubiquitous in mathematics, and captures the idea of a “family of vector spaces parametrized by a base space.” In algebraic geometry, the definition is as follows:

Line bundles

Line bundles on $\A^n\k$.

Curiously, for affine varieties, the capacity for carrying interesting lines bundles is related to unique factorization.

Line bundles on $\P^n\k$.

The basic geometry of Grassmannians

Algebraic Geometry in Linear Algebra

Rank-dropping varieties

Suppose $V$ and $W$ are \(\k\)-vector spaces of dimensions $n$, and $m$ respectively. In this document we will establish fundamental facts about the geometry of a particularly ubiquitous stratification of the vector space of linear homomorphisms \[\Hom(V,W)\]

On the extremes, \(\mathsf{R}_m = \Hom(V,W)\) and \(\mathsf{R}0 = \{0\}\). After choosing bases of $V$ and $W$, every element of $\Hom(V,W)$ is represented by a $m × n$ matrix. Let us write the /universal $m × n$ matrix/ as \[A = \big\{ xij \big\} \,\,\, 1 \leq i \leq m, 1 \leq j \leq n.\] Linear algebra says:

For each $r$, the subset $\mathsf{R}r$ is the common vanishing locus of all determinants of $(r+1) × (r+1)$ minors of the universal matrix $A$.

Thus, $\mathsf{R}r$ is a sub-variety of $\Hom(V,W)$. (In fact, it comes with a “usual” scheme structure given by the ideal in $\k\left[\{Xi,j\}\right]$ generated by \((r+1) × (r+1)\)-minors of $A$.)

In the language of schemes, this translates to: \(\mathsf{Rr-1}\) is the singular scheme of \(\mathsf{Rr}\).

The relevant incidence variety

For each $1 \leq r \leq m$ we define the subset \[\widetilde{\mathsf{R}}r ⊂ \Hom(V,W) × \Gr(n-r,V)\] as the set of pairs $(\varphi, Λ)$ satisfying \[ Λ ⊂ ker \varphi.\]

Let $π1: \widetilde{\mathsf{R}}r → \Hom(V,W)$ denote the projection onto the first factor, $π1(\varphi, Λ) := \varphi$. Then $π1$ is a projective morphism, and $π1(\widetilde{\mathsf{R}}r) = \mathsf{R}r$.

The conclusion from all exercises above is:

Symmetric linear algebra and the geometry of quadric hypersurfaces

Let $V$ be an \((n+1)\)-dimensional vector space /over a ground field $\k$ of characteristic not $2$./

Let $Q ∈ \Sym2V∨$ be a quadratic form on $V$. Choose a basis $(vi)i=0, \ldots, n$ of $V$, and the corresponding dual basis $(Xi)$ of $V∨$. Then, the quadratic form $Q$ is a homogeneous quadratic polynomial $Q(X_0, \ldots, X_n)$, and defines a corresponding quadric hypersurface $$\mathsf{Q} ⊂ \P V$$ defined as the vanishing (scheme) of $Q$.

The geometry of smooth cubic surfaces

The geometry of smooth cubic surfaces is extremely rich and interesting, and constitutes one of the earliest chapters in classical algebraic geometry. The first fundamental theorem, due to Cayley and Salmon, is: