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Hilbert series of certain jet schemes of determinantal varieties
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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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