On the contravariant of homogeneous forms arising from isolated hypersurface singularities

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 ${\mathcal{Q}}_n^{n(d-2)\ast }$. 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 ${\mathcal{Q}}_n^{n(d-2)\ast }$. Letting $p$ be the smallest integer for which the product ${\Delta }^p\Phi$ extends to a morphism from ${\mathcal{Q}}_n^d$ to ${\mathcal{Q}}_n^{n(d-2)\ast }$, 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.

arXiv commentFinal version, to appear in the International Journal of Mathematics