Rational Singularities of Nested Hilbert Schemes

Abstract

The Hilbert scheme of points $\textrm {Hilb}^{n}(S)$ of a smooth surface $S$ is a well-studied parameter space, lying at the interface of algebraic geometry, commutative algebra, representation theory, combinatorics, and mathematical physics. The foundational result is a classical theorem of Fogarty, stating that $\textrm {Hilb}^{n}(S)$ is a smooth variety of dimension $2n$. In recent years there has been growing interest in a natural generalization of $\textrm {Hilb}^{n}(S)$, the nested Hilbert scheme$\textrm {Hilb}^{(n_{1}, n_{2})}(S)$, which parametrizes nested pairs of zero-dimensional subschemes $Z_{1} \supseteq Z_{2}$ of $S$ with $\deg Z_{i}=n_{i}$. In contrast to Fogarty’s theorem, $\textrm {Hilb}^{(n_{1}, n_{2})}(S)$ is almost always singular, and very little is known about its singularities. In this paper, we aim to advance the knowledge of the geometry of these nested Hilbert schemes. Work by Fogarty in the 70’s shows that $\textrm {Hilb}^{(n,1)}(S)$ is a normal Cohen–Macaulay variety, and Song more recently proved that it has rational singularities. In our main result, we prove that the nested Hilbert scheme $\textrm {Hilb}^{(n,2)}(S)$ has rational singularities. We employ an array of tools from commutative algebra to prove this theorem. Using Gröbner bases, we establish a connection between $\textrm {Hilb}^{(n,2)}(S)$ and a certain variety of matrices with an action of the general linear group. This variety of matrices plays a central role in our work, and we analyze it by various algebraic techniques, including the Kempf–Lascoux–Weyman technique of calculating syzygies, square-free Gröbner degenerations, and the Stanley–Reisner correspondence. Along the way, we also obtain results on classes of irreducible and reducible nested Hilbert schemes, dimension of singular loci, and $F$-singularities in positive characteristic.

On the period of Li, Pertusi, and Zhao’s symplectic variety

“…Indeed, it is known that the Hilbert scheme of 21 points on a smooth fourfold admits generically non-reduced components [26] but the question about reducedness of the Hilbert schemes of points on smooth threefold is still open. Analogously, in the nested case, the question about reducedness is already open in the case dim X = 2 [37]. Our main result in this direction is that, when dim X 4 nested Hilbert schemes become non-reduced as soon as possible.…”

“…For instance, nested Hilbert schemes are already reducible when dim X 2, while double nested Hilbert schemes of points on smooth curves are in general reducible and have many smoothable components, i.e. components whose generic point corresponds to a nesting involving only smooth subschemes, [37,14]. Another typical question about Hilbert schemes concerns their schematic structure.…”

“…For instance, when dim X = 2, the nested Hilbert scheme is irreducible for r 2 and reducible in general for r 5. The cases r = 3, 4 are not yet understood, [37,12].…”

On the Hilbert function of Artinian local complete intersections of codimension three

“…Can it be nonreduced? For d 1 ≤ 2 the singularities are rational, see [125,113]. Finally, for the Gorenstein locus we have also an additional structure: Iarrobino's symmetric decomposition [71].…”