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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
##### Keywords:
Vector bundles; rank two; globally generated; projective space
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