Hilbert series of certain jet schemes of determinantal varieties

Sudhir R. Ghorpade, Boyan Jonov and B. A. Sethuraman

Vol. 272 (2014), No. 1, 147–175

DOI: 10.2140/pjm.2014.272.147

Abstract

We consider the affine variety ${\mathcal{Z}}_{2,2}^{m,n}$ of first-order jets over ${\mathcal{Z}}_2^{m,n}$, where ${\mathcal{Z}}_2^{m,n}$ is the classical determinantal variety given by the vanishing of all $2\times 2$ minors of a generic $m\times n$ matrix. When $2<m\le n$, this jet scheme ${\mathcal{Z}}_{2,2}^{m,n}$ has two irreducible components: a trivial component, isomorphic to an affine space, and a nontrivial component that is the closure of the jets supported over the smooth locus of ${\mathcal{Z}}_2^{m,n}$. This second component is referred to as the principal component of ${\mathcal{Z}}_{2,2}^{m,n}$; it is, in fact, a cone and can also be regarded as a projective subvariety of ${\mathbb{P}}^{2mn-1}$. We prove that the degree of the principal component of ${\mathcal{Z}}_{2,2}^{m,n}$ is the square of the degree of ${\mathcal{Z}}_2^{m,n}$ and, more generally, the Hilbert series of the principal component of ${\mathcal{Z}}_{2,2}^{m,n}$ is the square of the Hilbert series of ${\mathcal{Z}}_2^{m,n}$. As an application, we compute the $a$-invariant of the principal component of ${\mathcal{Z}}_{2,2}^{m,n}$ and show that the principal component of ${\mathcal{Z}}_{2,2}^{m,n}$ is Gorenstein if and only if $m=n$.

Keywords

jet schemes, Hilbert series, determinantal varieties

Mathematical Subject Classification 2010

Primary: 14M12

Secondary: 05E40

Milestones

Received: 5 July 2013

Accepted: 4 November 2013

Published: 9 October 2014