Syzygy bundles on P 2 and the weak Lefschetz property

Let K be an algebraically closed field of characteristic zero and let I=(f₁ fₙ) be a homogeneous R₊-primary ideal in $R:=K[X,Y,Z]$. If the corresponding syzygy bundle (f₁ fₙ) 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 Miró-Roig. We prove that an almost complete intersection has the Weak Lefschetz property if the corresponding syzygy bundle is not semistable.