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Winter 2007 Syzygy bundles on $\Bbb P\sp 2$ and the weak Lefschetz property
Holger Brenner, Almar Kaid
Illinois J. Math. 51(4): 1299-1308 (Winter 2007). DOI: 10.1215/ijm/1258138545

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.

Citation

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Holger Brenner. Almar Kaid. "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

Information

Published: Winter 2007
First available in Project Euclid: 13 November 2009

zbMATH: 1148.13007
MathSciNet: MR2417428
Digital Object Identifier: 10.1215/ijm/1258138545

Subjects:
Primary: 14J60
Secondary: 13D02

Rights: Copyright © 2007 University of Illinois at Urbana-Champaign

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Vol.51 • No. 4 • Winter 2007
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