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

From the top, we are choosing an algebraically closed field {k} 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}}, as described in the last post. We would like to take this opportunity to express our regret for not really describing {\mathop\mathrm{Proj}} 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}) if it is of the form {\mathop\mathrm{Proj} S} for {S} a finitely generated graded {k}-algebra. A subset {W} of a scheme {V} is said to be a closed subscheme iff:

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

\underline{Note}: In case {W} is not a subset of {V} but we still have a map {W\rightarrow V} 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} to {Y} is the composition of the inclusion {i:Z\hookrightarrow X\times Y} and the projection map {\mathop\mathrm{pr_Y}:X \times Y \rightarrow Y}. 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}}. 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} module {M} is said to be flat if tensoring with {M} 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} is flat means that for all {z\in Z}, the induced map on the stalks {f^\sharp:\mathcal{O}_{Y,f(z)}\rightarrow \mathcal{O}_{Z,z}} makes {\mathcal{O}_{Z,z}} into a flat {\mathcal{O}_{Y,f(z)}}-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} with {X} 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} need only be quasi-projective. In fact, one of the most interesting cases for us will be where {X} is {{\mathbb A}^2} ({{\mathbb C}^2} in particular). Further on, we will refine this by introducing Hilbert polynomials. Yay!

Hilbert Schemes III

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

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

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

By “up to isomorphism” here we mean that the value of the section {s_i} at a given point on {X} is depends on the trivialization used, but it will differ according to multiplication by a scalar which multiplies all {s_0,\dots,s_n} simultaneously. Thus, the ensemble of all sections together effectively gives a point in {\mathop{\mathbb P}^n} (or a fiber of the tautological bundle, if you prefer), provided they are not all zero. Ranging over all of {X} 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}}. This is because {{\mathbb Z}} is the initial object in the category {\mathop\mathrm{CRing}}, 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}} to any other commutative ring with identity which just takes 1 to itself.

Then, {\mathop\mathrm{Spec}({\mathbb Z})} is also final in {\mathop\mathrm{Sch}} because we have this ring morphism (which is an {\mathop\mathrm{Aff}} 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}} from a graded ring as {\mathop\mathrm{Proj} Z[x_0,\dots,x_n]} 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}]}, we again have a canonical scheme morphism {\mathop{\mathbb P}^n_{\mathbb Z}\rightarrow\mathop\mathrm{Spec}{\mathbb Z}}. This together with {A\rightarrow\mathop\mathrm{Spec}{\mathbb Z}} lets us construct the fiber product {A\times_{\mathop\mathrm{Spec} {\mathbb Z}}\mathop{\mathbb P}^n_{\mathbb Z} =: \mathop{\mathbb P}^n_A}; 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)} as subsheafs of {\mathcal{O}^{n+1}} which are locally of rank {n}, which we think of as the kernels of the surjection from the direct sum sheaf {\mathcal{O}^{n+1} \twoheadrightarrow \mathcal{L}}, 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}-planes in {n}-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)} is that now we look at surjections from {\phi:\mathcal{O}_X^{n}\twoheadrightarrow \mathcal{Q}} where {\mathcal{Q}} is locally free of rank {k}. At neighborhoods in a locally trivializing cover, we identify {\mathcal{Q}} with {\mathcal{O}^k} and we can treat {\phi} as a {k\times n} matrix. The Grassmannian will be covered by open sets which are the nonvanishings of the determinant of the {\binom{n}{k}} maximal minors. This leaves the other {k(n-k)} entries of the matrix free, so they can vary and together determine a map {X\rightarrow{\mathbb A}^{k(n-k)}} in the way the sections {s_0,\dots,s_n} gave a map to {\mathop{\mathbb P}^n} before. Thus, we can cover {G(k,n)} 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)}}.

It’s worth noting that the full Grassman functor {G(k,n)} is represented by a bona fide scheme of the form {\mathop\mathrm{Proj}({\mathbb Z}[\dots, x_I,\dots]/I_{k,n})} where {I} denotes a subset of {[n]=\{1,\dots,n\}} of size {k}, which indexes the variables, an {I_{k,n}} 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}} exterior power functor {\wedge^k} to {[\phi:\mathcal{O}_X^{n}\twoheadrightarrow \mathcal{Q}]} gives a surjection from a locally free sheaf of rank {\binom{n}{k}} onto a line bundle. So by the prior discussion, we have a map from the Grassmannian into {\mathop{\mathbb P}^{\binom{n}{k}-1}} 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} over all the open neighborhoods containing some point {p\in X}, then we get a local ring {\mathcal{O}_{X,p}}. In this case of an affine scheme {\mathop\mathrm{Spec}{R}} it works out that {\mathcal{O}_{\mathop\mathrm{Spec} R,p}} is actually the localization {R_p}, as {p} is a prime ideal of {R}, 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} is a continuous map of topological spaces together with a morphism of sheaves from {\mathcal{O}_Y} to the push-forward {f_*\mathcal{O}_X}. This induces a map on stalks

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

where {f(p)=q} and supposing {f^\sharp} 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}} and functors therefrom. We are interested in the confusingly named functor of points for a given scheme {X}, 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)} 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}} is the same as a covariant functor {\mathcal{C}^{op}\rightarrow \mathcal{D}}, we have the following.

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

In general, functors that come from such {h_X}‘s are said to be representable. Elements of the set {\mathop\mathrm{Mor}(Y,X)} 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}} for some ring (or field) A, then we will call theses maps the “{A}-valued points of X.”

Endnote: for an interesting discussion on whether scheme theory can be taught using just {\mathop\mathrm{MaxSpec}}, 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, 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)} such that every point in {X} has a neighborhood {U} such that {(U, \mathcal{O}_X |_U)} is an affine scheme.

{\mathcal{O}_X} is a sheaf (of rings). That is, it’s a (contravariant) functor from the category of open subsets of {X}, sometimes written {\mathop\mathrm{Op}(X)}, to {\mathop\mathrm{CRing}}, 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}}! You know, the category with the same objects but with the arrows reversed? OK, I shut up. Just take your favorite ring {R}. The collection of prime ideals of {R} is (hopefully) a set, denoted {\mathop\mathrm{Spec}(R)}, 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}\}

as basis for the open sets, where {f} is some element of {R}. 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,

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

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

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.