On the contravariant of homogeneous forms arising from isolated hypersurface singularities

TL;DR

This paper investigates the contravariant associated with homogeneous forms from isolated hypersurface singularities, providing bounds on its degree by extending a previously defined morphism to a larger domain.

Contribution

It introduces bounds for the minimal power of the discriminant needed to extend the associated form map as a contravariant, advancing understanding of its algebraic properties.

Findings

Derived upper bounds for the extension power p of the associated form map.

Provided estimates for the degree of the contravariant in terms of form parameters.

Extended the morphism to a contravariant for forms with non-vanishing discriminant.

Abstract

Let ${\mathcal Q}_n^d$ be the vector space of homogeneous forms of degree $d\ge 3$ on ${\mathbb C}^n$, with $n\ge 2$. The object of our study is the map $\Phi$, introduced in earlier articles by J. Alper, M. Eastwood and the author, that assigns to every form for which the discriminant $\Delta$ does not vanish the so-called associated form lying in the space ${\mathcal Q}_n^{n(d-2)*}$. This map is a morphism from the affine variety $X_n^d:=\{f\in{\mathcal Q}_n^d:\Delta(f)\ne 0\}$ to the affine space ${\mathcal Q}_n^{n(d-2)*}$. Letting $p$ be the smallest integer for which the product $\Delta^p\Phi$ extends to a morphism from ${\mathcal Q}_n^d$ to ${\mathcal Q}_n^{n(d-2)*}$, one observes that the extended map defines a contravariant of forms in ${\mathcal Q}_n^d$. In the present paper we obtain upper bounds for $p$ thus providing estimates for the contravariant's degree.