` Hilbert series of certain jet schemes of determinantal varieties

# Hilbert series of certain jet schemes of determinantal varieties

### B. A. Sethuraman

#### Department of Mathematics California State University Northridge Northridge, CA 91330, USA E-Mail: al.sethuraman@csun.edu

Abstract: We consider the affine variety $Y= {\mathcal{Z}_{2,2}^{m,n}}$ of first order jets over $X$, where $X={\mathcal{Z}_{2}^{m,n}}$ 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 $Y$ 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 ${\mathbb{P}}^{2mn-1}$. 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 $m=n$.

 1 Introduction 1 2 Binomials and Lattice Paths 3 3 Multiplicity 7 4 Hilbert series 9 Acknowledgments 20 References 22

This paper is published in the Pacific Journal of Mathematics, Vol. 272, No. 1 (2014), pp. 147-175.