Hilbert series of certain jet schemes of determinantal varieties

TL;DR

This paper studies the jet schemes of determinantal varieties, showing that the Hilbert series of the principal component is the square of the original, and characterizing when it is Gorenstein.

Contribution

It establishes the Hilbert series relationship for jet schemes of determinantal varieties and characterizes the Gorenstein property of the principal component.

Findings

Hilbert series of the principal component is the square of that of the original variety.

Degree of the principal component equals the square of the degree of the determinantal variety.

Principal component is Gorenstein if and only if the matrix dimensions are equal.

Abstract

We consider the affine variety ${\mathcal{Z}_{2,2}^{m,n}}$ (or just "$Y$") of first order jets over ${\mathcal{Z}_{2}^{m,n}}$ (or just "$X$"), where $X$ 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 $Y$ 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 $X$. This second component is referred to as the principal component of $Y$; it is, in fact, a cone and can also be regarded as a projective subvariety of $\mathbf{P}^{2mn-1}$. We prove that the degree of the principal component of $Y$ is the square of the degree of $X$ and more generally, the Hilbert series of the principal component of $Y$ is the square of the Hilbert series of $X$. As an application, we compute the $a$-invariant of the principal component of $Y$ and show that the principal component of $Y$ is Gorenstein if and only if $m=n$.