Let $K$ be an algebraically closed field of characteristic zero and let $I=(f_1 \komdots f_n)$ be a homogeneous $R_+$-primary ideal in $R:=K[X,Y,Z]$. If the corresponding syzygy bundle $\Syz(f_1 \komdots f_n)$ 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 Miró-Roig. We prove that an almost complete intersection has the Weak Lefschetz property if the corresponding syzygy bundle is not semistable.
"Syzygy bundles on $\Bbb P\sp 2$ and the weak Lefschetz property." Illinois J. Math. 51 (4) 1299 - 1308, Winter 2007. https://doi.org/10.1215/ijm/1258138545