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Rank two globally generated vector bundles with \(c_1\leq 5\). (English) Zbl 1271.14020
The classification of globally generated vector bundles on the projective spaces has been given by J. C. Sierra and L. Ugaglia in the case \(c_1=2\) [J. Pure Appl. Algebra 213, No. 11, 2141–2146 (2009; Zbl 1166.14011)]. Huh considered the case \(c_1=3\), obtaining the list of the triple embeddings of projective spaces in a suitable Grassmannian of projective lines [S. Huh, Math. Nachr. 284, No. 11–12, 1453–1461 (2011; Zbl 1279.14014)]. Here the authors classify the globally generated rank two vector bundles with \(c_1\leq 5\), on the projective spaces of dimension \(n\geq 3\).
The main theorem says that such a bundle \(E\) always decomposes as a direct sum of two line bundles if \(n\geq 4\), whereas if \(n=3\) and \(E\) is indecomposable, then the pair of its Chern classes \((c_1, c_2)\) is one of the following eight pairs: \((2,2), (4,5), (4,6), (4,7), (4,8), (5,8), (5,10), (5,12)\). The authors describe the curves \(C\) obtained from a general section of \(E\) in the Serre correspondence. Moreover they prove that there exists a rank two globally generated vector bundle on \(\mathbb P^3\) with Chern classes in the list above in all cases except \((5,12)\), when the problem of the existence remains open.

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M15 Grassmannians, Schubert varieties, flag manifolds
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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