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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.
##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14F06 Sheaves in algebraic geometry
##### Keywords:
nef vector bundles; Fano bundles; spectral sequences
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##### References:
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