4 research outputs found
Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras
In this paper, we exploit some geometric-differential techniques to prove the
strong Lefschetz property in degree for a complete intersection standard
Artinian Gorenstein algebra of codimension presented by quadrics. We prove
also some strong Lefschetz properties for the same kind of Artinian algebras in
higher codimensions. Moreover, we analyze some loci that come naturally into
the picture of "special" Artinian algebras: for them, we give some geometric
descriptions and show a connection between the non emptiness of the so-called
non-Lefschetz locus in degree and the "lifting" of a weak Lefschetz
property to an algebra from one of its quotients.Comment: 21 page
Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras
In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree 1 for a complete intersection standard Artinian Gorenstein algebra of codimension 6 presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of “special” Artinian algebras: for them we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree 1 and the “lifting” of a weak Lefschetz property to an algebra from one of its quotients
A theorem of Gordan and Noether via Gorenstein rings
Gordan and Noether proved in their fundamental theorem that an hypersurface
with is a cone if and only if has
vanishing hessian (i.e. the determinant of the Hessian matrix). They also
showed that the statement is false if , by giving some
counterexamples. Since their proof, several others have been proposed in the
literature. In this paper we give a new one by using a different perspective
which involves the study of standard Artinian Gorenstein -algebras
and the Lefschetz properties. As a further application of our setting, we prove
that a standard Artinian Gorenstein algebra
with generated by a regular sequence of quadrics has the strong Lefschetz
property. In particular, this holds for Jacobian rings associated to smooth
cubic threefolds.Comment: 21 page