Syzygy Bundles on P² and the Weak Lefschetz Property

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.