Lefschetz properties of squarefree monomial ideals via Rees algebras

The theory of Rees algebras of monomial ideals has been extensively studied, and as a consequence, many (sometimes partial) equivalences between algebraic properties of monomial ideals, and combinatorial properties of simplicial complexes and hypergraphs are known. In this paper we show how this theory can be used to find interesting examples in the theory of Lefschetz properties. We explore the consequences of known results from Lefschetz properties to the Rees algebras of squarefree monomial ideals, for example in the calculation of analytic spread. In particular, we show a connection between symbolic powers and $f$-vectors of simplicial complexes. This perspective leads us to a generalization of Postnikov's "mixed Eulerian numbers". We prove the positivity of such numbers in our setting.