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On rank 2 vector bundles with \(c^ 2_ 1=10\) and \(c_ 2=3\) on Enriques surfaces. (English) Zbl 0766.14031
Algebraic geometry, Proc. US-USSR Symp., Chicago/IL (USA) 1989, Lect. Notes Math. 1479, 39-49 (1991).
[For the entire collection see Zbl 0742.00065.]
Let \(S\) be an Enriques surface, \(D\) an ample, nef divisor such that \(D^ 2=10\), which embeds \(S\) in \(\mathbb{P}^ 5\). If \(D\) is a ‘Fano polarization’ (i.e. \(DF\geq 3\) for all nef divisors \(F\) with \(F^ 2=0)\) then in this embedding \(S\) contains 20 cubic plane curves \(F_ i\).
The authors prove that any vector bundle of rank 2 on \(S\), with \(c_ 1(E)=D\) and \(c_ 2(E)=3\) arises from an extension \(0\to{\mathcal O}(D-F_ i)\to E\to{\mathcal O}(F_ i)\to 0\) or \(0\to{\mathcal O}(F_ i)\to E\to{\mathcal O}(D-F_ i)\to 0\). These extensions are trivial, unless either \(D\) or \(D+K_ S\) is a ‘Reye polarization’, that is, it induces an embedding of \(S\) inside some smooth quadric hypersurface of \(\mathbb{P}^ 5\). — The study of rank 2 bundles arising from these extensions gives a pattern for proving several results on the geometry of Enriques surfaces embedded in the Grassmannian of lines in \(\mathbb{P}^ 3\).
Reviewer: L.Chiantini (Roma)

14J28 \(K3\) surfaces and Enriques surfaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14J45 Fano varieties
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14E25 Embeddings in algebraic geometry