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.