` Hilbert series of certain jet schemes of determinantal varieties

Hilbert series of certain jet schemes of determinantal varieties


Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India

E-Mail: srg@math.iitb.ac.in

Boyan Jonov

Department of Mathematics
University of California Santa Barbara
Santa Barbara, CA 93106, USA

E-Mail: boyan@math.ucsb.edu

and

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.

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