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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)

##### MSC:
 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