On the contravariant of homogeneous forms arising from isolated hypersurface singularities

Abstract: Let ${\mathcal Q}_n<sup>d$ be the vector space of homogeneous forms of degree $d\ge 3$ on ${\mathbb C}<sup>n$, 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 ${\mathcal Q}_n<sup>{n(d-2)*}$. 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 ${\mathcal Q}_n<sup>{n(d-2)*}$. Letting $p$ be the smallest integer for which the product $\Delta<sup>p\Phi$ extends to a morphism from ${\mathcal Q}_n<sup>d$ to ${\mathcal Q}_n<sup>{n(d-2)*}$, one observes that the extended map defines a contravariant of forms in ${\mathcal Q}_n<sup>d$. In the present paper we obtain upper bounds for $p$ thus providing estimates for the contravariant's degree.