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Hilbert Series Of Certain Jet Schemes Of Determinantal Varieties
We consider the affine variety Zm,n2,2 of first-order jets over Zm,n2, where Zm,n2 is the classical determinantal variety given by the vanishing of all 2×2 minors of a generic m×n matrix. When 2<m≤n, this jet scheme Zm,n2,2 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 Zm,n2. This second component is referred to as the principal component of Zm,n2,2; it is, in fact, a cone and can also be regarded as a projective subvariety of P2mn−1. We prove that the degree of the principal component of Zm,n2,2 is the square of the degree of Zm,n2 and, more generally, the Hilbert series of the principal component of Zm,n2,2 is the square of the Hilbert series of Zm,n2. As an application, we compute the a-invariant of the principal component of Zm,n2,2 and show that the principal component of Zm,n2,2 is Gorenstein if and only if m=n.