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Uniform bundles on quadric surfaces and some related varieties. (English) Zbl 0535.14009
A rank 2 vector bundle E on \({\mathbb{P}}_ 1\times {\mathbb{P}}_ 1\) is said to be weakly uniform of type (a,b) if there exists integers c,d such that \(E| {\mathbb{P}}_ 1\times \{x\}\cong {\mathcal O}_{{\mathbb{P}}_ 1\times \{x\}}(c)\oplus {\mathcal O}_{{\mathbb{P}}_ 1\times \{x\}}(c-a)\) and \(E| {\mathbb{P}}_ 1\times \{y\}\cong {\mathcal O}_{\{y\}\times {\mathbb{P}}_ 1}(d)\oplus {\mathcal O}_{\{y\}\times {\mathbb{P}}_ 1}(d-b)\) for every \(x,y\in {\mathbb{P}}_ 1\). This notion is equivalent to the notion of doubly reducible bundle introduced by R. L. E. Schwarzenberger [Proc. Camb. Philos. Soc. 58, 209-216 (1962; Zbl 0219.14010)] and studied by P. E. Newstead and R. L. E. Schwarzenberger [ibid. 60, 421-424 (1964; Zbl 0129.129)]. E is uniform if \(a=b\) and \(c=d\). In this paper we study weakly uniform rank 2 vector bundles on \({\mathbb{P}}_ 1\times {\mathbb{P}}_ 1\) giving necessary and sufficient conditions in terms of a,b,c,d or of \(c_ 1(E)\), \(c_ 2(E)\) for their existence. Indecomposable weakly uniform rank 2 bundles have fine moduli spaces. We give necessary and sufficient conditions for stability; in particular any indecomposable weakly uniform bundle of type (a,b) with \(a>0,b>0\) is stable in the sense of Mumford-Takemoto with respect to the line bundle \({\mathcal O}_{{\mathbb{P}}_ 1\times {\mathbb{P}}_ 1}(a,b)\). Then we consider the same notions on \(({\mathbb{P}}_ 1)^ n\). In the last section we improve the main results of E. Ballico in Ann. Univ. Ferrara, Nuova Ser., Sez. VII 27, 135-146 (1981; Zbl 0495.14008), proving that any uniform vector bundle of rank t on a smooth quadric Q, dim Q\(=2r\) or \(2r+1\), is decomposable if \(t\leq r+1\), \(r\geq 3\).

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J25 Special surfaces
14D22 Fine and coarse moduli spaces
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