On the image of the associated form morphism

TL;DR

This paper investigates the image of a specific morphism related to forms and associated forms, showing it is an open subset of an irreducible component of certain algebraic varieties, and describes the complement in special cases.

Contribution

It establishes that the image of the morphism ${f A}$ is an open subset of an irreducible component of catalecticant varieties and describes the complement, extending known results.

Findings

The image of ${f A}$ is an open subset of an irreducible component of catalecticant varieties.

The complement of the image is characterized by specific invariants, such as the Aronhold invariant for $n=3$, $d=2$.

The results extend understanding of the structure of associated form morphisms and their images.

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 $d\ge 2$.