On the contravariant of homogeneous forms arising from isolated hypersurface singularities

Let Q^d _n be the vector space of homogeneous forms of degree d ≥ 3 on Cn, with n ≥ 2. The object of our study is the map φ, introduced in earlier papers by J. Alper, M. Eastwood and the author, that assigns to every form for which the discriminant δ does not vanish the so-called associated form lying in the space Qⁿ (d-2) n . This map is a morphism from the affine variety X^dn := {f Q^d _n : δ(f) = 0} to the affine space Q^n(d-2)∗ _n . Letting p be the smallest integer such that the product δφ extends to a morphism from Q^d _n to Qⁿ (^d-2)∗ _n , one observes that the extended map defines a contravariant of forms in Qd n. In this paper, we obtain upper bounds for p thus providing estimates for the contravariant's degree.