2016/09/05 by Alexander Isaev, Isaev, Alexander · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.1609.01027
Let \mathbb C[x1,…,xn]d+1 be the vector space of homogeneous forms of degree d+1 on \mathbb Cn, with n,d≥ 2. In earlier articles by J. Alper, M. Eastwood and the author, we introduced a morphism, called A, that assigns to every nondegenerate form the so-called associated form lying in the space \mathbb C[y1,…,yn]n(d-1). One of the reasons for our interest in A is the conjecture---motivated by the well-known Mather-Yau theorem on complex isolated hypersurface singularities---asserting that all regular \mathrm GLn-invariant functions on the affine open subvariety \mathbb C[x1,…,xn]d+1,Δ of forms with nonvanishing discriminant can be obtained as the pull-backs by means of A of the rational \mathrm GLn-invariant functions on \mathbb C[y1,…,yn]n(d-1) defined on \mathrm im(A). The morphism A factors as A=\mathbf A∘ \mathrm grad, where \mathrm grad is the gradient morphism and \mathbf A assigns to every n-tuple of forms of degree d with nonvanishing resultant a form in \mathbb C[y1,…,yn]n(d-1) defined analogously to A(f) for a nondegenerate f. In order to establish the conjecture, it is important to study the image of \mathbf A. In the present paper, we show that \mathrm im(\mathbf A) is an open subset of an irreducible component of each of the so-called catalecticant varieties V, \mathrm Gor(T) and describe the closed complement to \mathrm im(\mathbf A), at the same time clarifying and extending known results on these varieties. Furthermore, for n=3, d=2 we give a description of the complement to \mathrm im(\mathbf A) via the zero locus of the Aronhold invariant of degree 4, which is analogous to the case n=2 where this complement is known to be the vanishing locus of the catalecticant for any d≥ 2.