A note on the weak Lefschetz property of monomial complete intersections in positive characteristic

TL;DR

This paper characterizes when monomial complete intersections in positive characteristic have the weak Lefschetz property, providing explicit conditions based on the degree and characteristic, and confirms a recent conjecture for characteristic 2.

Contribution

It offers an explicit description of the weak Lefschetz property for monomial Artinian complete intersections in positive characteristic, answering a previously open question.

Findings

For p=2, the algebra has the weak Lefschetz property if and only if d=(2^t+1)/3 or d=(2^t-1)/3.

Provides an explicit criterion based on d and p for the weak Lefschetz property.

Confirms a recent conjecture by Li and Zanello for characteristic 2.

Abstract

Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of C. Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersection A = K[X,Y,Z]/(X^d,Y^d,Z^d) in terms of d and p. This answers a question of J. Migliore, R. M. Miro-Roig and U. Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(X^d,Y^d,Z^d) on PP^2 satisfies the Grauert-Muelich theorem. As a corollary we obtain that for p=2 the algebra A has the weak Lefschetz property if and only if d=(2^t+1)/3 or d=(2^t-1)/3 for some positive integer t. This was recently conjectured by J. Li and F. Zanello.