## Illinois Journal of Mathematics

### Syzygy bundles on $\Bbb P\sp 2$ and the weak Lefschetz property

#### Abstract

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.

#### Article information

Source
Illinois J. Math., Volume 51, Number 4 (2007), 1299-1308.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138545

Digital Object Identifier
doi:10.1215/ijm/1258138545

Mathematical Reviews number (MathSciNet)
MR2417428

Zentralblatt MATH identifier
1148.13007

#### Citation

Brenner, Holger; Kaid, Almar. Syzygy bundles on $\Bbb P\sp 2$ and the weak Lefschetz property. Illinois J. Math. 51 (2007), no. 4, 1299--1308. doi:10.1215/ijm/1258138545. https://projecteuclid.org/euclid.ijm/1258138545