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

2009/11/18 by Joan Elias, Elias, Joan, Maria Evelina Rossi +1
Mathematics · #13H10 #13H15 #14C05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #math.AC #math.AG #msc:13H10 #msc:13H15 #msc:14C05

paper · pdf · doi:10.48550/arxiv.0911.3565

20 pages

arxiv created 2009/11/18 · openalex publication_date 2009/11/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbb 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 Hilb_d (\mathbb Pⁿ) and in relation to the problem of the rationality of the Poincaré series of local rings.

Related

• Inverse Systems of Zero-dimensional Schemes in Pⁿ
• A Criterion for Isomorphism of Artinian Gorenstein Algebras
• A structure theorem for 2-stretched Gorenstein algebras
• The explicit minimal resolution constructed from a Macaulay inverse system
• Artinian Gorenstein algebras of embedding dimension four and socle degree three over an arbitrary field