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

Elias, J.; Rossi, M.

doi:10.1090/S0002-9947-2012-05430-4

Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system
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by J. Elias and M. E. Rossi PDF

Trans. Amer. Math. Soc. 364 (2012), 4589-4604 Request permission

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.

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