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arXiv · 1210.3841

Hilbert series of certain jet schemes of determinantal varieties

Sudhir R. Ghorpade·Boyan Jonov·B. A. Sethuraman
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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 $m=n$.

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BibTeXRIS

Sudhir R. Ghorpade, Boyan Jonov, B. A. Sethuraman. 2012-10-14. Hilbert series of certain jet schemes of determinantal varieties. https://doi.org/10.2140/pjm.2014.272.147

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