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.

In 2022, Jinha Kim proved a conjecture by Engström that states the independence complex of a graph with no induced cycle of length divisible by 3 is either contractible or homotopy equivalent to a sphere. We give criteria for when the independence complex of a ternary graph is contractible, and describe the dimension of the sphere when it is not. We then apply our results to describe the multigraded betti numbers of the edge ideal of a ternary graph. In particular, we show that the regularity and depth of edge ideals of a ternary graph G are equal if and only if the independence complex of G is not contractible. Finally, we apply our results to partially recover and generalize recent results on the depth of edge ideals of some unicyclic graphs.

We consider Artinian level algebras arising from the whiskering of a graph. Employing a result by Dao-Nair we show that multiplication by a general linear form has maximal rank in degrees 1 and n−1 when the characteristic is not two, where n is the number of vertices in the graph. Moreover, the multiplication is injective in degrees < n/2 when the characteristic is zero, following a proof by Hausel. Our result in the characteristic zero case is optimal in the sense that there are whiskered graphs for which the multiplication maps in all intermediate degrees n/2,…,n−2 of the associated Artinian algebras fail to have maximal rank, and consequently, the weak Lefschetz property.

Recently, H. Dao and R. Nair gave a combinatorial description of simplicial complexes Δ such that the squarefree reduction of the Stanley-Reisner ideal of Δ 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(Δ) to satisfy the WLP in degree i and characteristic zero in terms of mixed multiplicities of monomial ideals that contain combinatorial information of Δ, we call them incidence ideals. As a consequence, we give an upper bound to the possible failures of the WLP of A(Δ) 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

This work is based on the article Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, by J. Huh, published in Journal of the American Mathematical Society (2012). The dissertation is devoted to the study of the methods used by Huh to prove a particular case of the Rota-Heron-Welsh conjecture. J. Huh used results from algebraic topology, combinatorics, algebraic geometry and commutative algebra to prove the log-concavity of the coefficients of the characteristic polynomial of a matroid representable over a field of characteristic zero.