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

Abstract

Let $ K$ be an algebraically closed field of characteristc zero. In this paper we study the isomorphism classes of Artinian Gorenstein local $ K$-algebras with socle degree three by means of Macaulay's inverse system. We prove that their classification is equivalent to the projective classification of cubic hypersurfaces in $ \mathbb{P}_K ^{n }$. This is an unexpected result because it reduces the study of this class of local rings to the graded case. The result has applications in problems concerning the punctual Hilbert scheme $ \operatorname {Hilb}_d (\mathbb{P}_K^n) $ and in relation to the problem of the rationality of the Poincaré series of local rings.