Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system

Abstract

In this paper we study the isomorphism classes of Artinian Gorenstein local rings with socle degree three by means of Macaulay’s inverse system. We prove that their classification is equivalent to the projective classification of the hypersurfaces of P n of degree three. This is an unexpected result because it reduces the study of this class of local rings to the homogeneous case. The result has applications in problems concerning the punctual Hilbert scheme Hilbd(P n ) and