PIMS Emergent Research Seminar: Emanuela Marangone

In this talk, we will introduce the Weak and the Strong Lefschetz Properties (WLP and SLP) focusing on Artinian complete intersections. A famous result of Stanley shows that every Artinian monomial complete intersection over a field of characteristic zero has the SLP and the WLP.

However, this result does not hold in positive characteristics. In the first part of the talk, we will present results on monomial complete intersections with 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 whetherevery complete intersection has WLP. Harima, Migliore, Nagel, and Watanabe prove the conjecture for complete intersections 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 complete intersection of height 3 has the Strong Lefschetz Property in degree 2 and study the non-Lefschetz locus of conics.