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!

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