Associated forms: current progress and open problems

TL;DR

This paper surveys the current understanding of the associated form morphism in algebraic geometry, its connection to singularity reconstruction, and highlights open problems linking complex singularity theory and invariant theory.

Contribution

It compiles known results on the associated form morphism and its contravariant extension, and emphasizes open problems and the connection to the Mather-Yau theorem.

Findings

The associated form morphism is ${ m SL}_n$-equivariant.

The extended map defines a contravariant related to the discriminant.

Open problems remain in understanding the full scope of the associated form concept.

Abstract

Let $d\ge 3$, $n\ge 2$. The object of our study is the morphism $\Phi$, introduced in earlier articles by J. Alper, M. Eastwood and the author, that assigns to every homogeneous form of degree $d$ on ${\mathbb C}^n$ for which the discriminant $\Delta$ does not vanish the so-called associated form, which is a form of degree $n(d-2)$ on the dual space. This morphism is ${\mathrm{SL}}_n$-equivariant and is of interest in connection with the well-known Mather-Yau theorem, specifically, with the problem of explicit reconstruction of an isolated hypersurface singularity from its Tjurina algebra. Letting $p$ be the smallest integer such that the product $\Delta^p\Phi$ extends to the entire affine space of degree $d$ forms, one observes that the extended map defines a contravariant. In the present paper we survey known results on the morphism $\Phi$ as well as the contravariant $\Delta^p\Phi$, and state several open problems. Our goal is to draw the attention of complex analysts and geometers to the concept of the associated form and the intriguing connection between complex singularity theory and invariant theory revealed through it.