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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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