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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.

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
Full Text: arXiv