Tag Archives: davesh maulik

What I learned from Zhiwei Yun about Hilbert schemes

One knows, of course, that Hilbert schemes of smooth curves and smooth surfaces are nice, and Hilbert schemes of varieties of dimension greater than two are terrifying.

Zhiwei Yun was here giving a talk about his work with Davesh Maulik on Hilbert schemes of curves with planar singularities, and he made a point I’d never appreciated; it’s not the dimension of the variety, but the dimension of its tangent space that really measures the terrifyingness of the  Hilbert space.  Singular curves C with planar singularities are not so bad — you still have a nice Hilbert scheme with an Abel-Jacobi map to the compactified Jacobian.  But let C be the union of the coordinate axes in A^3 and all bets are off.  Hideous extra high-dimensional components aplenty.  If I had time to write a longer blog post today I would think about what the punctual Hilbert scheme at the origin looks like.  But maybe one of you guys will just tell me.

Update:  Jesse Kass explains that I am wrong about C; its Hilbert scheme has a non-smoothable component, but it doesn’t have any components whose dimension is too large.

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Some visitors, and countable unions

A busy few days: we had a run of interesting visitors this week in Madison, including Thomas Lam, who gave a beautiful talk about total positivity (a subfield of algebraic combinatorics, not a self-help philosophy); Melanie Matchett Wood, who explained how to parametrize binary forms of degree n in the Bhargava style, not only over Z but over an arbitrary base scheme (which is to say, not really in the Bhargava style!); and Davesh Maulik, who showed us how one can rather miraculously count rational curves on a single K3 by counting rational curves on a suitably chosen one-parameter family of K3s, and then “dividing” by the Noether-Lefschetz theory attached to the family. Very agreeably for number theorists, a key point is the product formulae of Borcherds, which provide modular forms on the moduli space of K3s whose zeroes and poles are supported on the countable union of subvarieties where the Picard number jumps upwards from its generic value.

This led to an amusing conversation at lunch about countable unions of subvarieties. Here’s a remark: if A is an abelian variety over the complex numbers, it’s completely obvious that A(C) contains some non-torsion points; the torsion locus is a countable union of varieties of strictly lower dimension (in this case 0) and thus can’t cover A(C). On the other hand, if A is over Fpbar, every point of A(Fpbar) is defined over some finite field, and thus all these points are torsion. The case of Qbar is intermediate in difficulty; indeed, there are nontorsion points on every abelian variety over Qbar, but this is not, in some sense, by “pure thought” — one might, for instance, use the argument that torsion points have height 0 but that there are plainly points of arbitrarily large height on A(Qbar). This uses some actual theorems, not just a comparison of cardinalities. Similarly, one can ask: are there elliptic curves over an algebraically closed field k with End(E/k) = Z? When k = C, the answer is obviously yes. When k = Fpbar, the answer is no, thanks to Frobenius. And when k is Qbar, the answer is again no, but maybe one has to use a bit more — for instance, that a CM elliptic curve over a number field has potentially good reduction everywhere.

In general, it’s pretty hard to see whether a countable union of subvarieties of X/Qbar covers all the Qbar-points! Here are two well-known open questions in this vein.

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