On the contravariant of homogeneous forms arising from isolated hypersurface singularities

Abstract

Let [Formula: see text] be the vector space of homogeneous forms of degree [Formula: see text] on [Formula: see text], with [Formula: see text]. The object of our study is the map [Formula: see text], introduced in earlier papers by J. Alper, M. Eastwood and the author, that assigns to every form for which the discriminant [Formula: see text] does not vanish the so-called associated form lying in the space [Formula: see text]. This map is a morphism from the affine variety [Formula: see text] to the affine space [Formula: see text]. Letting [Formula: see text] be the smallest integer such that the product [Formula: see text] extends to a morphism from [Formula: see text] to [Formula: see text], one observes that the extended map defines a contravariant of forms in [Formula: see text]. In this paper, we obtain upper bounds for [Formula: see text] thus providing estimates for the contravariant’s degree.