The weak Lefschetz property for artinian Gorenstein algebras of small Sperner number

TL;DR

This paper proves that all artinian Gorenstein algebras with small Sperner number relative to socle degree satisfy the Weak Lefschetz Property, establishing a sharp bound and advancing understanding in algebraic geometry.

Contribution

It establishes a sharp bound for Sperner number ensuring WLP in artinian Gorenstein algebras, filling a gap in codimension three cases.

Findings

All such algebras with Sperner number ≤ socle degree + 1 satisfy WLP.

Counterexamples exist for Sperner number = socle degree + 2.

The result is independent of codimension.

Abstract

For artinian Gorenstein algebras in codimension four and higher, it is well known that the Weak Lefschetz Property (WLP) does not need to hold. For Gorenstein algebras in codimension three, it is still open whether all artinian Gorenstein algebras satisfy the WLP when the socle degree and the Sperner number are both higher than six. We here show that all artinian Gorenstein algebras with socle degree $d$ and Sperner number at most $d+1$ satisfy the WLP, independent of the codimension. This is a sharp bound in general since there are examples of artinian Gorenstein algebras with socle degree $d$ and Sperner number $d+2$ that do not satisfy the WLP for all $d\ge 3$.