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Nef vector bundles on a projective space with first Chern class three. (English) Zbl 1451.14129
Given a nef vector bundle \(\mathcal{E}\) on a projective space \(\mathbb{P}^n\) it is well-known that \(c_1(\mathcal{E})\geq 0\). Nef vector bundles \(\mathcal{E}\) with \(c_1(\mathcal{E})\leq 2\) were classified by [T. Peternell et al., Lect. Notes Math. 1507, 145–156 (1992; Zbl 0781.14006)] analyzing the contraction morphisms of extremal rays. In particular, for \(n\geq 2\), \(\mathbb{P}(\mathcal{E})\) is a Fano variety. A different proof of the classification was obtained by M. Ohno [“Nef vector bundles on a projective space or a hyperquadric with the first Chern class small”, Preprint, arXiv:1409.4191] using the twists \(\mathcal{E}(d)\).
The paper under review deals with the next case, namely nef vector bundles \(\mathcal{E}\) on \(\mathbb{P}^n\) (over an algebraically closed field of characteristic zero) with \(c_1(\mathcal{E})=3\) are completely classified. In particular, one has \(0\leq c_2(\mathcal{E})\leq c_1(\mathcal{E})^2=9\). When \(c_2(\mathcal{E})<8\), the author proves that the nef vector bundles \(\mathcal{E}\) are globally generated. For \(c_2=8\) and \(9\), there exist examples of non-globally generated nef vector bundles on the projective plane.
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14F06 Sheaves in algebraic geometry
Full Text: DOI
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