Hilbert series of certain jet schemes of determinantal varieties

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 $m=n$.

References in corpus (1)

• Arcs on Determinantal Varieties

Cited by in corpus (3)

• Computing with jets
• On jet schemes of pfaffian ideals
• Jet Graphs