The weak Lefschetz property and mixed multiplicities of monomial ideals

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.