On the image of the associated form morphism

Abstract

Let ${\mathbb C}[x_1,\dots,x_n]_{d+1}$ be the vector space of homogeneous forms of degree $d+1$ on ${\mathbb C}^n$, with $n,d\ge 2$. 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 ${\mathbb C}[y_1,\dots,y_n]_{n(d-1)}$. 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 ${\mathrm {GL}}_n$-invariant functions on the affine open subvariety ${\mathbb C}[x_1,\dots,x_n]_{d+1,\Delta}$ of forms with nonvanishing discriminant can be obtained as the pull-backs by means of $A$ of the rational ${\mathrm {GL}}_n$-invariant functions on ${\mathbb C}[y_1,\dots,y_n]_{n(d-1)}$ defined on ${\mathrm {im}}(A)$. The morphism $A$ factors as $A={\mathbf A}\circ {\mathrm {grad}}$, 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 ${\mathbb C}[y_1,\dots,y_n]_{n(d-1)}$ 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 $d\ge 2$.