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Nef vector bundles on a quadric surface with the first Chern class $$(2, 1)$$. (English) Zbl 1433.14041
In the paper under review, the author pursues the classification of rank $$r$$ nef vector bundles $$\mathcal{E}$$ on a smooth quadric surface $$Q_2\subset\mathbb{P}^3$$ over an algebraicly closed field of characteristic zero in terms of the value of the determinant $$\bigwedge^r\mathcal{E}\cong\mathcal{O}_{Q_2}(a,b)$$ of $$\mathcal{E}$$. It is well-known that for such a nef bundle $$\mathcal{E}$$ it holds that $$a,b\geq 0$$ and that $$a=0$$ or $$b=0$$ implies that $$\mathcal{E}$$ splits as a direct sum of line bundles. The first non-trivial case, namely $$a=b=1$$, was classified by T. Peternell et al. [Lect. Notes Math. 1507, 145–156 (1992; Zbl 0781.14006)] using strongly the fact that in this case $$\mathbb{P}(\mathcal{E})$$ is a Fano manifold.
In this paper the author gives a complete classification of nef bundles for the next open case $$a=2$$ and $$b=1$$ where $$\mathbb{P}(\mathcal{E})$$ is no longer a Fano manifold. As a by-product, he proves that all of them are indeed globally generated. The main tool used in this classification is A. I. Bondal’s results [Math. USSR, Izv. 34, No. 1, 23–42 (1990; Zbl 0692.18002); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 1, 25–44 (1989)] on the structure of the derived category of coherent sheaves on $$Q_2$$.
##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14N30 Adjunction problems 14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
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##### References:
 [1] I. Assem, D. Simson, A. Skowroński, Elements of the representation theory of associative algebras. Vol., 1 volume 65 of London Mathematical Society Student Texts. Cambridge Univ. Press 2006. MR2197389 Zbl 1092.16001 · Zbl 1092.16001 [2] E. Ballico, S. Huh, F. Malaspina, Globally generated vector bundles on a smooth quadric surface. Sci. China Math. 58 (2015), 633-652. MR3319928 Zbl 1314.14086 · Zbl 1314.14086 [3] A. I. Bondal, Representations of associative algebras and coherent sheaves. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 25-44. English translation in Math. USSR-Izv. 34 (1990), 2-42. MR992977 Zbl 0692.18002 · Zbl 0692.18002 [4] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry. Oxford Univ. Press 2006. MR2244106 Zbl 1095.14002 · Zbl 1095.14002 [5] M. Kashiwara, P. Schapira, Categories and sheaves. Springer 2006. MR2182076 Zbl 1118.18001 · Zbl 1118.18001 [6] R. Lazarsfeld, Positivity in algebraic geometry. II. Springer 2004. MR2095471 Zbl 1066.14021 · Zbl 1093.14500 [7] M. Ohno, Nef vector bundles on a projective space or a hyperquadric with the first Chern class small. Preprint 2014, arXiv:1409.4191 [math.AG] [8] M. Ohno, H. Terakawa, A spectral sequence and nef vector bundles of the first Chern class two on hyperquadrics. Ann. Univ. Ferrara Sez. VII Sci. Mat. 60 (2014), 397-406. MR3275418 Zbl 06407046 · Zbl 1408.14133 [9] T. Peternell, M. Szurek, J. A. Wiśniewski, Numerically effective vector bundles with small Chern classes. In: Complex algebraic varieties (Bayreuth, 1990), volume 1507 of Lecture Notes in Math., 145-156, Springer 1992. MR1178725 Zbl 0781.14006 · Zbl 0781.14006 [10] M. Szurek, J. A. Wiśniewski, Fano bundles of rank 2 on surfaces. Compositio Math. 76 (1990), 295-305. MR1078868 Zbl 0719.14028 · Zbl 0719.14028
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