Syzygy bundles on P-2 and the weak Lefschetz property

Abstract

Let K be an algebraically closed field of characteristic zero and let I = (f(1),..., f(n)) be a homogeneous R+-primary ideal in R := K[X, Y, Z]. if the corresponding syzygy bundle Syz(f(1),....,f(n)) on the projective plane is semistable, we show that the Artinian algebra R/I has the Weak Lefschetz property if and only if the syzygy bundle has a special generic splitting type. As a corollary we get the result of Harima et alt., that every Artinian complete intersection (n = 3) has the Weak Lefschetz property. Furthermore, we show that an almost complete intersection (n = 4) does not necessarily have the Weak Lefschetz property, answering negatively a question of Migliore and Miro-Roig. We prove that an almost complete intersection has the Weak Lefschetz property if the corresponding syzygy bundle is not semistable.