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Syzygy bundles on \(\mathbb P^2\) and the weak Lefschetz property. (English) Zbl 1148.13007
Let \(0\to S\to F\to \mathcal{O}_{\mathbb{P}^n}\to 0\) be an exact sequence on \(\mathbb{P}^2\) where \(F\) is a direct sum of line bundles. Let \(A_m=H^1(\mathbb{P}^2,F(m))\) and let \(A=\bigoplus A_m\). Then \(A\) is a graded Artinian algebra which is a quotient of the polynomial ring \(R=\bigoplus_m H^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^n}(m))\), by an ideal genrated by some \(f_i, 1\leq i\leq k\). If the above exact sequence is minimal, then clearly we may assume \(k\) is the rank of \(F\) and we say that \(S\) is the syzygy bundle of the ideal generated by \(f_i\)’s. One says that \(A\) satisfies the Weak Lefscetz Property (WLP) if for a generic linear form \(l\in R\), the multiplication map \(l:A_m\to A_{m+1}\) is either injective or surjective for all \(m\).
The authors relate WLP with some stability properties of \(S\) in characteristic zero. For example, they show that if \(S\) is semi-stable, then \(A\) has WLP if and only if the splitting type of \(S\) restricted to a general line is of the form \(\oplus \mathcal{O}_{\mathbb{P}^1}(a)\oplus \mathcal{O}_{\mathbb{P}^1}(a+1)\) for some \(a\). This easily shows that a complete intersection has WLP, a result proved by T. Harima et al. [J. Algebra 262, No. 1, 99–126 (2003; Zbl 1018.13001)]. They also show that if \(A\) is an almost complete intersection and the syzygy bundle \(S\) is not semistable, then \(A\) has WLP.

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
13C13 Other special types of modules and ideals in commutative rings
13C40 Linkage, complete intersections and determinantal ideals
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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