The weak Lefschetz property and mixed multiplicities of monomial ideals

TL;DR

This paper establishes a connection between the weak Lefschetz property of simplicial complexes' Stanley-Reisner ideals and mixed multiplicities of monomial ideals, providing criteria and bounds in various characteristics.

Contribution

It introduces a necessary and sufficient condition for the WLP in terms of mixed multiplicities and extends existing criteria to positive odd characteristics.

Findings

Provides a criterion for WLP in degree i in characteristic zero.

Offers an upper bound for WLP failures in positive characteristics.

Extends Dao and Nair's criterion to arbitrary monomial ideals in odd characteristics.

Abstract

Recently, H. Dao and R. Nair gave a combinatorial description of simplicial complexes $\Delta$ such that the squarefree reduction of the Stanley-Reisner ideal of $\Delta$ has the WLP in degree $1$ and characteristic zero. In this paper, we apply the connections between analytic spread of equigenerated monomial ideals, mixed multiplicities and birational monomial maps to give a sufficient and necessary condition for the squarefree reduction $A(\Delta)$ to satisfy the WLP in degree $i$ and characteristic zero in terms of mixed multiplicities of monomial ideals that contain combinatorial information of $\Delta$, we call them incidence ideals. As a consequence, we give an upper bound to the possible failures of the WLP of $A(\Delta)$ in degree $i$ in positive characteristics in terms of mixed multiplicities. Moreover, we extend Dao and Nair's criterion to arbitrary monomial ideals in positive odd characteristics.