Rational singularities of nested Hilbert schemes

The Hilbert scheme of points 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 Hilb^n(S) is a smooth variety of dimension 2n. In recent years there has been growing interest in a natural generalization of Hilb^n(S), the nested Hilbert scheme Hilb^(n1,n2)(S), which parametrizes nested pairs of zero-dimensional subschemes Z1 ⊇ Z2 of S with deg Zi = ni. In contrast to Fogarty’s theorem, Hilb^(n1,n2)(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 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 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 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.