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.

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