The non-Lefschetz locus

TL;DR

This paper investigates the conditions under which artinian Gorenstein algebras and complete intersections exhibit the weak Lefschetz property, providing a complete description for monomial cases and partial results for higher codimensions.

Contribution

It characterizes the non-Lefschetz locus for monomial complete intersections and establishes expected codimension results for general complete intersections and Gorenstein algebras in codimension three and four.

Findings

Complete description of non-Lefschetz locus for monomial complete intersections.

Expected codimension of non-Lefschetz locus in general codimension three and four cases.

Full description of non-Lefschetz locus for codimension two algebras.

Abstract

We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra $A$ we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms $\ell$ give maximal rank for all the multiplication maps $\times \ell: [A]_i \longrightarrow [A]_{i+1}$. This is a Zariski open set and its complement is the \emph{non-Lefschetz locus}. For monomial complete intersections, we completely describe the non-Lefschetz locus. For general complete intersections of codimension three and four we prove that the non-Lefschetz locus has the expected codimension, which in particular means that it is empty in a large family of examples. For general Gorenstein algebras of codimension three with a given Hilbert function, we prove that the non-Lefschetz locus has the expected codimension if the first difference of the Hilbert function is of decreasing type. For completeness we also give a full description of the non-Lefschetz locus for artinian algebras of codimension two.