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.