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

Authors: J. Elias and M. E. Rossi
Journal: Trans. Amer. Math. Soc. 364 (2012), 4589-4604
MSC (2010): Primary 13H10; Secondary 13H15, 14C05
Published electronically: April 11, 2012
MathSciNet review: 2922602
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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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Additional Information

J. Elias
Affiliation: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

M. E. Rossi
Affiliation: Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy

Keywords: Artinian Gorenstein local rings, inverse system, Hilbert functions, isomorphism classes.
Received by editor(s): November 18, 2009
Received by editor(s) in revised form: March 22, 2010, June 29, 2010, and July 16, 2010
Published electronically: April 11, 2012
Additional Notes: The first author was partially supported by MTM2010-20279-C02, Acción Integrada España-Italia 07-09
The second author was partially supported by M.I.U.R.: PRIN 07-09, Azione Integrata Italia-Spagna 07-09
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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