The module of derivations of certain hypersurfaces

Abstract

Let [Formula: see text] be an algebraically closed field of characteristic 0. Further, [Formula: see text], where [Formula: see text] is a polynomial in [Formula: see text] such that [Formula: see text]. We show that the module of derivations of [Formula: see text], namely [Formula: see text] is generated by [Formula: see text] elements. We also compute the generators explicitly. It is well known that, in such cases, [Formula: see text] is stably free of rank [Formula: see text]. As a special case, when [Formula: see text] and [Formula: see text] is quasi-homogeneous, we give an explicit minimal generating set for [Formula: see text], consisting of two derivations.