A famous result of Stanley shows that every Artinian monomial complete intersection (CI) over a field of characteristic zero has the Strong Lefschetz Property (SLP). However, this result does not hold in positive characteristics. In the first part of the talk, we will focus on characteristics p. While Lundqvist-Nicklasson provides a complete characterization for the monomial CI for having the SLP, the Weak Lefschetz Property (WLP) is still open. We will present results on monomial CI having the WLP in characteristic p, and give a complete classification for p=2.

In the second part of the talk, we will work over characteristic zero. It remains an open conjecture whether every CI has WLP. Harima, Migliore, Nagel, and Watanabe prove the conjecture for CIs of height 3, and Boij, Migliore, Miró-Roig, and Nagel introduce the study of the non-Lefschetz locus. We will address a similar question for forms of degree 2 instead of lines. Specifically, we will show that every CI of height 3 has the Strong Lefschetz Property in degree 2 and study the non-Lefschetz locus of conics.