Title: The weak Lefschetz property and powers of linear forms (joint work with A. Seceleanu and B. Harbourne/A. Seceleanu)

Abstract: Let I be an ideal of S=K[x_{1}, ... , x_{r}]
such that A = S/I is Artinian. Then A has the ** weak Lefschetz
property** (WLP) if there is an l in S_{1} such that for all
m, the map

given by multiplying by l is either injective or surjective. Motivation to study WLP comes both from algebraic geometry (via the relation to the Lefschetz hyperplane theorem) and the fact that WLP imposes strong restrictions on Hilbert functions. Important work in analyzing WLP has been done by Harima-Migliore-Nagel-Watanabe, Anick, Stanley, Brenner-Kaid, Migliore-Miró-Roig-Nagel, and others. I will describe joint work with Seceleanu (PAMS, 2010) and Harbourne-Seceleanu arXiv:1008.2377 on WLP for ideals generated by uniform powers of linear forms. I'll describe two useful general tools: the syzygy bundle of an ideal, and inverse systems. In the case r = 3, WLP always holds for ideals generated by powers of linear forms, while for r = 4 and up to eight general forms, if the power is large, WLP fails. We conjecture that for r > 3 and I not a complete intersection, WLP always fails asymptotically.