Title: The Weak Lefschetz Property, almost complete intersections and monomial ideals

Abstract: We describe joint work with Rosa Miró-Roig and Uwe Nagel. Let R be a polynomial ring with the usual grading. An Artinian graded algebra, A = R/I, has the Weak Lefschetz Property (WLP) if multiplication by a general linear form, L, from any component to the next, has maximal rank. In characteristic zero it's known that monomial complete intersections in any number of variables have WLP, as does *every* complete intersection in three variables. On the other hand, Gorenstein algebras in codimension 4 or more do not necessarily have WLP. (References to these results will be given in the talk.)

Many natural questions arise. In particular, Migliore and Miró-Roig asked whether every almost complete intersection (ACI) necessarily has WLP. A counterexample was given recently by Brenner and Kaid: the ideal (x^{3}, y^{3}, z^{3}, xyz) in k[x,y,z]. Notice that this is an almost complete intersection, i.e. its codimension (3) is one less than the number of generators (4).
A useful technique for addressing WLP is via restriction modulo the linear form L. Geometrically, this turns out to involve studying the syzygy bundle, *E* on **P**^{2}, associated to I, and the restriction of *E* to a general line. This approach was introduced by Harima, Migliore, Nagel and Watanabe, and carried farther by Brenner and Kaid. We discuss this approach and its limitations in the context of the Brenner-Kaid example.
We then give our first main result, namely that in r variables the ideal (x_{1}^{r}, ... , x_{r}^{r}, x_{1}...x_{r}) fails WLP. This is surprisingly difficult to prove, and we describe how this was done. We also describe families of level monomial quotients of k[x,y,z], where the characteristic of the field enters into the question of whether WLP holds or not.