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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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