Syzygy Bundles on P^2 and the Weak Lefschetz Property

By Holger Brenner, Almar Kaid. 2006

73 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c4d1955aa5b01ff7cf7d198564ac63c6923c9aff

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Citations in the corpus

Monomial ideals, almost complete intersections and the Weak Lefschetz property (2008, methodology): “In any case, in [ 4 ] it was shown for n = 3 and characteristic zero that A(3) = 3 (also using a result of [6]), as noted in the introduction.”
On the Weak Lefschetz Property for Powers of Linear Forms (2010, background): “Kaid [2] gave very simple examples to show that even level monomial ideals need not have this property, and the latter gave an example that was even an almost complete intersection (the ideal was in a ring with three variables and had four minimal generators).”