HILBERT SERIES OF CERTAIN JET SCHEMES OF DETERMINANTAL VARIETIES

We consider the affine variety Z(2,2)(m,n) of first-order jets over Z(2)(m,n), where Z(2)(m,n) is the classical determinantal variety given by the vanishing of all 2 x 2 minors of a generic m x n matrix. When 2 < m <= n, this jet scheme Z(2,2)(m,n) 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 Z(2)(m,n). This second component is referred to as the principal component of Z(2,2)(m,n); 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 Z(2,2)(m,n) is the square of the degree of Z(2)(m,n) and, more generally, the Hilbert series of the principal component of Z(2,2)(m,n) is the square of the Hilbert series of Z(2)(m,n). As an application, we compute the alpha-invariant of the principal component of Z(2,2)(m,n) and show that the principal component of Z(2,2)(m,n) is Gorenstein if and only if m = n.