Gorenstein algebras presented by quadrics

Abstract

We establish restrictions on the Hilbert function of standard graded Gorenstein algebras with only quadratic relations. Furthermore, we pose some intriguing conjectures and provide evidence for them by proving them in some cases using a number of different techniques, including liaison theory and generic initial ideals.

Complete intersections of quadrics and the Weak Lefschetz Property

“…An immediate consequence of the proposition above significantly extends the Injectivity Lemma of Proposition 4.3 of [5], proved in that paper in the case d = 2.…”

Lefschetz properties for Artinian Gorenstein algebras presented by quadrics

“…In this way we construct a concrete family of algebras presented by quadrics and whose Hilbert vector is non unimodal. These algebras provide counterexamples for two conjectures posed in [MN1,MN2].…”

“…In [MN2], the authors studied Artinian Gorenstein algebras presented by quadrics, they provided some constructions of such algebras and described their possible Hilbert vectors in low codimension. In [MN1] and [MN2] the authors proposed two conjectures.…”

On mixed Hessians and the Lefschetz properties

“…In recent times the first author, in [Go], constructed families of algebras failing WLP. We recall the following result: On the other hand, for algebras presented by quadrics there was a conjecture posed in [MN1,MN2]:…”

“…We also construct new examples of Artinian Gorenstein algebras presented by quadrics that does not satisfy the Weak Lefschetz Property (WLP); we construct minimal examples of such algebras and we give bounds, depending on the degree, for their existence. Artinian Gorenstein algebras presented by quadrics were conjectured to satisfy WLP in [MN1,MN2], and in a previous paper we construct the first counter-examples (see [GZ]). *Partially supported by the CAPES postdoctoral fellowship, Proc.…”