The Non-Lefschetz Locus of Lines and Conics

A graded Artinian algebra A has the Weak Lefschetz Property if there exists a linear form l such that the multiplication map by l has maximum rank in every degree. The linear forms satisfying this property form a Zariski-open set; its complement is called the non-Lefschetz locus of A. The study of the non-Lefschetz locus started with Boij--Migliore--Miró-Roig--Nagel, who proved that the non-Lefschetz locus of a general complete intersection of height 3 has the expected codimension. This thesis aims to generalize this result in two main directions. The first concerns the non-Lefschetz locus of the first-cohomology module of rank 2 vector bundles E over P^2. This is motivated by Failla--Flores--Peterson who proved the WLP for such modules. The natural question is whether the non-Lefschetz locus has the expected codimension. We first prove that the non-Lefschetz locus in this context is the set of jumping lines of E. Using this classification, we show that, assuming generality, the non-Lefschetz locus has the expected codimension. It is a hypersurface if E is unstable or its first Chern class is even, and it is a finite set otherwise. In the second direction, we investigate analogous questions for degree-two forms rather than lines. The main result shows that the non-Lefschetz locus of conics for a general complete intersection of height 3 has the expected codimension in P^5. The hypothesis of generality is necessary: we include examples in which the non-Lefschetz locus has different codimension. To extend a similar result to the first cohomology modules of rank 2 vector bundles over P^2, we explore the connection between non-Lefschetz conics and jumping conics, defined first by Vitter. The non-Lefschetz locus of conics is a subset of the jumping conics, that can be proper when E is semistable with first Chern class even.