2012/10/31 by Sudhir R. Ghorpade, Boyan Jonov, B. A. Sethuraman
Mathematics · #Advanced Combinatorial Mathematics #Affine transformation #Affine variety #Algebraic Geometry and Number Theory #Combinatorics #Commutative Algebra and Its Applications #Component (thermodynamics) #Hilbert scheme #Hilbert–Poincaré series #Invariant (physics) #Irreducible component #Locus (genetics) #Mathematical analysis #Mathematical physics #Mathematics #Physics #Principal component analysis #Pure mathematics #Series (stratigraphy) #Subvariety #Variety (cybernetics) #math.AC #math.AG #math.CO
paper · pdf · doi:10.2140/pjm.2014.272.147
published as Pacific J. Math. 272 (2014), No. 1, 147-175
arxiv created 2013/11/15 · openalex publication_date 2014/10/09 · arxiv updated 2015/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Abstract. 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. 1.