Hilbert Schemes IV: A Definition

At this point in the project, we begin to understand why so many books and papers are written in dry, terse, often difficult mathematical prose. Discursiveness which is beneficial is risky, time consuming and hard to pull off! As we continue to hurtle preliminary posts, maybe a bit of motivation is in order.

At the highest level, far-out, hardcore algebraic geometry is finding application in strange places these days. We will just cite the three instances which are close to the heart of mathlab: Hodge Theory in Combinatorics, Geometric Complexity Theory, and the Macdonald Positivity Conjecture where Hilbert schemes played a starring role. Understanding this last one is what we are heading for in these posts. Moving on…

So let’s just do this the way where you put a loaded definition out front and then try to tease it apart, following Nakajima‘s exposition, repeated in seminar.

Definition Let {X}&fg=000000 be a projective scheme over an algebraically closed field {k}&fg=000000. The contravariant Hilbert functor {\mathcal{H}ilb_X:\mathop\mathrm{Sch}\rightarrow\mathop\mathrm{Sets}}&fg=000000 is the functor that associates to a scheme {Y}&fg=000000 the set of closed subschemes {Z\subset X\times Y}&fg=000000 such that the natural morphism from {Z}&fg=000000 to {Y}&fg=000000 is flat.

From the top, we are choosing an algebraically closed field {k}&fg=000000 so the “projective” here is with respect to {\mathop{\mathbb P}^n_k= \mathop\mathrm{Proj} k[x_0,\dots,x_n] = \mathop\mathrm{Spec} k \times \mathop{\mathbb P}^n_{\mathbb Z}}&fg=000000, as described in the last post. We would like to take this opportunity to express our regret for not really describing {\mathop\mathrm{Proj}}&fg=000000 or the fiber product in these posts even though we earlier gestured towards thoroughness. Hopefully the references serve.

Moving on, an arbitrary scheme is said to be projective (over {k}&fg=000000) if it is of the form {\mathop\mathrm{Proj} S}&fg=000000 for {S}&fg=000000 a finitely generated graded {k}&fg=000000-algebra. A subset {W}&fg=000000 of a scheme {V}&fg=000000 is said to be a closed subscheme iff:

  • The inclusion map {i: W\hookrightarrow V}&fg=000000 is an affine morphism, meaning that for each open affine subset {\mathop\mathrm{Spec} B \subset V}&fg=000000, the inverse image {i^{-1}(\mathop\mathrm{Spec} B)=\mathop\mathrm{Spec} A}&fg=000000 for some ring {A}&fg=000000. “The inverse image of affines is affine.”
  • Additionally, for each of these pairs, the map {B\rightarrow A}&fg=000000 is surjective.

\underline{Note}: In case {W}&fg=000000 is not a subset of {V}&fg=000000 but we still have a map {W\rightarrow V}&fg=000000 which satisfies the above properties, that map is said to be a closed embedding (or, less tastefully, a closed immersion) of schemes.

Back in the definition, the map from {Z}&fg=000000 to {Y}&fg=000000 is the composition of the inclusion {i:Z\hookrightarrow X\times Y}&fg=000000 and the projection map {\mathop\mathrm{pr_Y}:X \times Y \rightarrow Y}&fg=000000. Remember, this isn’t set-wise projection, our product is in the category of schemes, where we can also think of it as the fibered product over the terminal object, {\mathop\mathrm{Spec} {\mathbb Z}}&fg=000000. It is guaranteed to exist as in section 9.1 of The Rising Sea.

Finally we come to flatness, which is sort of exciting because this is the first time I’ve come across it in the wild since studying for my algebra qual, but still sort of mysterious. Apparently Mumford agrees, even as he reassures us that it is “technically the answer to many prayers,” in The Red Book (III.10). Remember that an {A}&fg=000000 module {M}&fg=000000 is said to be flat if tensoring with {M}&fg=000000 preserves short exact sequences. More loosely, flat modules are pretty close to (i.e. are direct limits of) free modules.

That the map {f:Z\rightarrow Y}&fg=000000 is flat means that for all {z\in Z}&fg=000000, the induced map on the stalks {f^\sharp:\mathcal{O}_{Y,f(z)}\rightarrow \mathcal{O}_{Z,z}}&fg=000000 makes {\mathcal{O}_{Z,z}}&fg=000000 into a flat {\mathcal{O}_{Y,f(z)}}&fg=000000-module. For motivation on why flatness is the “right” condition in these sorts of circumstances, see [EH] II.3.4.

An important theorem of Grothendieck says that {\mathcal{H}ilb_X}&fg=000000 with {X}&fg=000000 as above is representable by a scheme which is “locally of finite type” called a *gasp* HILBERT SCHEME! A version of the proof with additional discussion and background is here. It is also explained in section 9 there that we can weaken the requirements so that {X}&fg=000000 need only be quasi-projective. In fact, one of the most interesting cases for us will be where {X}&fg=000000 is {{\mathbb A}^2}&fg=000000 ({{\mathbb C}^2}&fg=000000 in particular). Further on, we will refine this by introducing Hilbert polynomials. Yay!

Hilbert Schemes III

Having established that to each scheme {X}&fg=000000 we can associate its functor of points {h_X}&fg=000000, we continue our introduction by mentioning some eminent examples.

First the projective space {\mathop{\mathbb P}^n}&fg=000000. This is sort of the basic example of a moduli space, which is also what were after with Hilbert Schemes. While {\mathop{\mathbb P}^n}&fg=000000 (over your ring of choice) has a few different constructions, it’s points correspond to other geometric objects: lines in the affine space {{\mathbb A}^{n+1}}&fg=000000. This is the essence of moduli, and their value for “counting” objects of a given type.

We also have a description of the functor of points for projective space, {h_{\mathop{\mathbb P}^n}}&fg=000000, recalling that {h_{\mathop{\mathbb P}^n}(X)=\mathop\mathrm{Mor}(X, \mathop{\mathbb P}^n)}&fg=000000 for a given scheme {X}&fg=000000. See Vakil, Theorem 16.4.1, or also Eisenbud and Harris [EH], Theorem III-37

Theorem Morphisms {X\rightarrow \mathop{\mathbb P}^n}&fg=000000 are in bijection with the data (up to isomorphism) of {(\mathcal{L},s_0,\dots,s_n)}&fg=000000 where {\mathcal{L}}&fg=000000 is a line bundle (invertible sheaf) on {X}&fg=000000 and the {s_i}&fg=000000 are global sections which do not vanish simultaneously.

By “up to isomorphism” here we mean that the value of the section {s_i}&fg=000000 at a given point on {X}&fg=000000 is depends on the trivialization used, but it will differ according to multiplication by a scalar which multiplies all {s_0,\dots,s_n}&fg=000000 simultaneously. Thus, the ensemble of all sections together effectively gives a point in {\mathop{\mathbb P}^n}&fg=000000 (or a fiber of the tautological bundle, if you prefer), provided they are not all zero. Ranging over all of {X}&fg=000000 we get a morphism of schemes.

You might wonder which projective space we are working with here, i.e. over which field/ring. As will usually be the case, it doesn’t much matter because maps behave nicely under base change, but if we want to be as general as possible we should probably use {{\mathbb Z}}&fg=000000. This is because {{\mathbb Z}}&fg=000000 is the initial object in the category {\mathop\mathrm{CRing}}&fg=000000, which makes it the final object in the category of affine schemes. Initialness comes from the fact that there is a unique morphism from {{\mathbb Z}}&fg=000000 to any other commutative ring with identity which just takes 1 to itself.

Then, {\mathop\mathrm{Spec}({\mathbb Z})}&fg=000000 is also final in {\mathop\mathrm{Sch}}&fg=000000 because we have this ring morphism (which is an {\mathop\mathrm{Aff}}&fg=000000 morphism in reverse) into all of the open affine covering neighborhoods of a given scheme, and these glue up nicely. So, whether we construct {\mathop{\mathbb P}^n_{\mathbb Z}}&fg=000000 from a graded ring as {\mathop\mathrm{Proj} Z[x_0,\dots,x_n]}&fg=000000 or as the gluing of a bunch of {{\mathbb A}^n_{\mathbb Z}=\mathop\mathrm{Spec}{\mathbb Z}[\frac{x_0}{x_i},\dots,\frac{x_n}{x_i}]}&fg=000000, we again have a canonical scheme morphism {\mathop{\mathbb P}^n_{\mathbb Z}\rightarrow\mathop\mathrm{Spec}{\mathbb Z}}&fg=000000. This together with {A\rightarrow\mathop\mathrm{Spec}{\mathbb Z}}&fg=000000 lets us construct the fiber product {A\times_{\mathop\mathrm{Spec} {\mathbb Z}}\mathop{\mathbb P}^n_{\mathbb Z} =: \mathop{\mathbb P}^n_A}&fg=000000; see [EH] III.2.5 for a more thorough treatment of all of this. I’m still essentially just recording the things I had to remind myself of in jumping into this so it’s maybe more garbled than pedagogical.

[EH] here also gives the description of {\mathop\mathrm{Mor}(X, \mathop{\mathbb P}^n)}&fg=000000 as subsheafs of {\mathcal{O}^{n+1}}&fg=000000 which are locally of rank {n}&fg=000000, which we think of as the kernels of the surjection from the direct sum sheaf {\mathcal{O}^{n+1} \twoheadrightarrow \mathcal{L}}&fg=000000, surjection being guaranteed by the non-vanishing of at least one of the sections above (at every point). This leads naturally into our other key example, the Grassmannian.

According to the Yoneda philosophy (which is often attributed to Grothendieck as well) we should actually think of the Grassmannian (and the special case of projective space) not just as the “space of {k}&fg=000000-planes in {n}&fg=000000-space,” but as a moduli space for certain bundles (locally free sheaves in AG-speak) and their sections. Really, it is a functor which can be applied to other schemes, and returns the collection of morphism which define the scheme that represents the functor. : S

The change with the Grassmann functor {G(k,n)}&fg=000000 is that now we look at surjections from {\phi:\mathcal{O}_X^{n}\twoheadrightarrow \mathcal{Q}}&fg=000000 where {\mathcal{Q}}&fg=000000 is locally free of rank {k}&fg=000000. At neighborhoods in a locally trivializing cover, we identify {\mathcal{Q}}&fg=000000 with {\mathcal{O}^k}&fg=000000 and we can treat {\phi}&fg=000000 as a {k\times n}&fg=000000 matrix. The Grassmannian will be covered by open sets which are the nonvanishings of the determinant of the {\binom{n}{k}}&fg=000000 maximal minors. This leaves the other {k(n-k)}&fg=000000 entries of the matrix free, so they can vary and together determine a map {X\rightarrow{\mathbb A}^{k(n-k)}}&fg=000000 in the way the sections {s_0,\dots,s_n}&fg=000000 gave a map to {\mathop{\mathbb P}^n}&fg=000000 before. Thus, we can cover {G(k,n)}&fg=000000 by a bunch of affine spaces which glue together. In the parlance, these are open subfunctors represented by the schemes {{\mathbb A}^{k(n-k)}}&fg=000000.

It’s worth noting that the full Grassman functor {G(k,n)}&fg=000000 is represented by a bona fide scheme of the form {\mathop\mathrm{Proj}({\mathbb Z}[\dots, x_I,\dots]/I_{k,n})}&fg=000000 where {I}&fg=000000 denotes a subset of {[n]=\{1,\dots,n\}}&fg=000000 of size {k}&fg=000000, which indexes the variables, an {I_{k,n}}&fg=000000 is a particular homogeneous ideal given by the Plücker equations. So we have a more traditional geometric object to think of as well. We may come back to this in more detail later if we need to. For now it suffices to say that applying the {k^{th}}&fg=000000 exterior power functor {\wedge^k}&fg=000000 to {[\phi:\mathcal{O}_X^{n}\twoheadrightarrow \mathcal{Q}]}&fg=000000 gives a surjection from a locally free sheaf of rank {\binom{n}{k}}&fg=000000 onto a line bundle. So by the prior discussion, we have a map from the Grassmannian into {\mathop{\mathbb P}^{\binom{n}{k}-1}}&fg=000000 which is the Plücker embedding.

Phew, okay I think in the next post we might actually define a Hilbert Scheme.

Hilbert Schemes II

Next, we introduce the Hilbert scheme functorially following notes by Maclagan. See also Bertram’s course notes, who points out that we (as in Grothendieck) are trying to develop a category-theoretic framework for counting objects with a given property. First we should say a little bit about morphisms of schemes.

To do this, one should indicate that schemes (like manifolds) actually carry the additional structure of a locally ringed space. This means that when you take a kind of (filtered co-)limit on the structure sheaf {\mathcal{O}_X}&fg=000000 over all the open neighborhoods containing some point {p\in X}&fg=000000, then we get a local ring {\mathcal{O}_{X,p}}&fg=000000. In this case of an affine scheme {\mathop\mathrm{Spec}{R}}&fg=000000 it works out that {\mathcal{O}_{\mathop\mathrm{Spec} R,p}}&fg=000000 is actually the localization {R_p}&fg=000000, as {p}&fg=000000 is a prime ideal of {R}&fg=000000, and localization satisfies exactly the universal property prescribed by the colimit.

First, remember that schemes are ringed spaces, and that a morphism of ringed spaces {f:X\rightarrow Y}&fg=000000 is a continuous map of topological spaces together with a morphism of sheaves from {\mathcal{O}_Y}&fg=000000 to the push-forward {f_*\mathcal{O}_X}&fg=000000. This induces a map on stalks

\displaystyle f^\sharp:\mathcal{O}_{Y,q} \rightarrow \mathcal{O}_{X,p}&fg=000000

where {f(p)=q}&fg=000000 and supposing {f^\sharp}&fg=000000 sends the maximal ideal of one to the other, then we have a morphism of locally ringed spaces, and it is THIS, my friends, which will be a scheme morphism when the locally ringed spaces happen to be schemes… oh hell, this can’t be useful for anyone as exposition; just seriously go skim the first six chapters of The Rising Sea to clear things up. It’s only 200 pages.

Now we have objects and morphisms so we can talk about the category {\mathop\mathrm{Sch}}&fg=000000 and functors therefrom. We are interested in the confusingly named functor of points for a given scheme {X}&fg=000000, which is really the one that comes up in many versions of the Yoneda Lemma. The idea is that if category is locally small, meaning the collection of morphisms (arrows) {\mathop\mathrm{Mor}(A,B)}&fg=000000 between any two objects forms a set, then we can understand an object by the morphisms going into (or out of) it. Bearing in mind that a contravariant functor between categories {F: \mathcal{C}\rightarrow \mathcal{D}}&fg=000000 is the same as a covariant functor {\mathcal{C}^{op}\rightarrow \mathcal{D}}&fg=000000, we have the following.

Definition Suppose {X}&fg=000000 is an object of {\mathop\mathrm{Sch}}&fg=000000, the category of schemes. Then there is a contravariant functor {h_X: \mathop\mathrm{Sch}\rightarrow \mathop\mathrm{Set}}&fg=000000 such that on objects we have {h_X(Y)=\mathop\mathrm{Mor}(Y,X)}&fg=000000, and for a given morphism {f:Y\rightarrow Z}&fg=000000 we have

\displaystyle \begin{array}{rcl} h_X(f):& \mathop\mathrm{Mor}(Z,X) \rightarrow \mathop\mathrm{Mor}(Y,X) \\ & g \mapsto g\circ f \end{array} &fg=000000

In general, functors that come from such {h_X}&fg=000000‘s are said to be representable. Elements of the set {\mathop\mathrm{Mor}(Y,X)}&fg=000000 are called the Y-valued points of X, which is weird, because on sight they are just maps. Further, by laziness, in case {Y=\mathop\mathrm{Spec A}}&fg=000000 for some ring (or field) A, then we will call theses maps the “{A}&fg=000000-valued points of X.”

Endnote: for an interesting discussion on whether scheme theory can be taught using just {\mathop\mathrm{MaxSpec}}&fg=000000, the maximal ideals of a ring, instead of the primes, that then digresses into thinking of schemes as their functors of points instead of the whole locally ringed space shebang, as well as other AG/Berkeley cultural inside jokes by a bunch of dudes, see the post and extensive comments here. Other interesting background can be found on nLab.

Hilbert Schemes I

In this series of posts, we will report and editorialize the goings-on of the algebraic geometry (AG) and combinatorics learning seminar. The goal is to learn something about Hilbert Schemes and the relevant combinatorics.

First, since we are working with the category of schemes, we should indicate what those are and where to find out more about them. This feels like joining the chorus or preaching to the choir or {<}&fg=000000insert perfunctory sentimental metaphor{>}&fg=000000, but it’s hard to overstate the value of Vakil’s The Rising Sea/FOAG, for the background category and scheme theory: an innovation in mathematical text writing.

Okay so a scheme is like a variety, which is to say it’s like a manifold, which is to say it’s locally like other spaces we know and love and go under the general banner affine. The central metaphor here is that it comes from gluing simpler things together along the places they match up, and that the information about the spaces themselves really comes from information about the functions on the space… Ahem, made precise by the following gesture at a definition which really takes scores of pages to build up properly, and relies on a multitude of other definitions:

Definition 1 A Scheme is a ringed space {(X, \mathcal{O}_X)}&fg=000000 such that every point in {X}&fg=000000 has a neighborhood {U}&fg=000000 such that {(U, \mathcal{O}_X |_U)}&fg=000000 is an affine scheme.

{\mathcal{O}_X}&fg=000000 is a sheaf (of rings). That is, it’s a (contravariant) functor from the category of open subsets of {X}&fg=000000, sometimes written {\mathop\mathrm{Op}(X)}&fg=000000, to {\mathop\mathrm{CRing}}&fg=000000, the category of commutative rings (with identity) and their homomorphisms. As elsewhere in life, if you find the category theoretic phrasing more obnoxious than illuminating, please ignore and seek the alternative. Or spend some time discovering the joy of cats, and join us. : D

So what’s an affine scheme? Turns out the affine scheme category is nothing but {\mathop\mathrm{CRing}^{op}}&fg=000000! You know, the category with the same objects but with the arrows reversed? OK, I shut up. Just take your favorite ring {R}&fg=000000. The collection of prime ideals of {R}&fg=000000 is (hopefully) a set, denoted {\mathop\mathrm{Spec}(R)}&fg=000000, which can be given a topology by taking sets of the form

\displaystyle D(f)=\{\mathfrak{p}\in\mathop\mathrm{Spec}(R) \mid f\not\in \mathfrak{p}\}&fg=000000

as basis for the open sets, where {f}&fg=000000 is some element of {R}&fg=000000. We get the sheaf (also called a structure sheaf) for an affine scheme by taking

\displaystyle \mathcal{O}_{\mathop\mathrm{Spec}{R}}(D(f))=R_f,&fg=000000

where the right hand side denotes localization at the multiplicative subset generated by {f}&fg=000000.

Why use the prime ideals? I guess because localization works nicely for prime ideals. But also if we have an algebraically closed field {K}&fg=000000, then Hilbert’s Nullstellensatz gives an identification of the points in {K^n}&fg=000000 with maximal ideals of the polynomial ring {K[x_1,\dots,x_n]}&fg=000000.

I always groan when I see people spend a page introducing like a semester’s worth of algebraic geometry, but, well, now I’ve done it too. Maybe this is another one of those rites of passage. Next up: some representable functors.