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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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