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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The punctual Hilbert scheme of a threefold is bad: Iarrobino. It has components of the wrong dimension. But I want to make another point. At least “locally” what is important is that the codimension is no greater than 2. So although the HIlbert scheme of points in threefolds are nasty, the Hilbert schemes of curves in threefolds are not as bad. They can be singular, but for “global” reasons rather than “local” reasons. This follows from the Hilbert-Burch-(Schaps) theorem.

Hmm … instead of “codimension no greater than 2”, I should have said “Cohen-Macaulay and codimension no greater than 2”. This is what is necessary for Hilbert-Burch-(Schaps).

Here is a description of the Hilbert scheme of points on the axes in 3-space. If $X$ is an integral curve with a unique singularity analytically equivalent to the axes in 3-space, then the compactified Jacobian has 2 irreducible components: the general element of one component is a line bundle and the general element of the other is the product of a line bundle and the dualizing module. Both components are of dimension equal to the genus (so there are no “hideous extra high-dimensional components”).

The Hilbert scheme of points has a similar description (which can be derived from the above description using an Abel map). For $d \ge 2$, the Hilbert scheme of $d$ points has 2 irreducible components: the general element is $d$ distinct points, and the general element of the second is the union of $d-2$ distinct points and a degree $2$ closed subscheme supported at the origin.

A few other examples of compactified Jaocbians of non-planar curves are given in my paper “An Explicit Non-smoothable Component of the Compactified Jacobian” (http://arxiv.org/abs/1108.2706).

Jesse’s computation seems to be correct. Maybe I am remembering the case of four lines through the origin in 3-space. There is also work of Voisin on the locus of “curvilinear schemes”. It seems that one of Jesse’s components is Voisin’s curvilinear locus.

@Jason: I am not familiar with that work of Voisin. Which paper(s?) are you referring to?

Dear Jesse,

Voisin looked at the locus of curvilinear schemes in her work on Green’s conjecture.

MR1941089 (2003i:14040)

Voisin, Claire(F-CNRS-MJ)

Green’s generic syzygy conjecture for curves of even genus lying on a surface.

J. Eur. Math. Soc. (JEMS) 4 (2002), no. 4, 363–404.

14H51 (14J28)

Thanks