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.