Abstract. Let K be an algebraically closed field of characteristc zero. In this paper we study the isomorphism classes of Artinian Gorenstein local Kalgebras 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 P n K . 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 Hilb d (P n K ) and in relation to the problem of the rationality of the Poincaré series of local rings.

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In this paper we attack the problem of the classification, up to analytic isomorphism, of Artinian Gorenstein local k-algebras with a given Hilbert Function. We solve the problem in the case the square of the maximal ideal is minimally generated by two elements and the socle degree is high enough.

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Let AK be an Artinian Gorenstein local ring with K an algebraically closed field of characteristic 0. In the present article, we prove a structure theorem describing the Artinian Gorenstein local K-algebras satisfying 4 = 0. We use this result in order to prove that such a K-algebra has rational Poincaré series and it is smoothable in any embedding dimension, provided dim K 2 / 3 ≤ 4. We also prove that the generic Artinian Gorenstein local K-algebra with 4 = 0 has rational Poincaré series.