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Elliptic curve configurations on Fano surfaces. (English) Zbl 1177.14079
A Fano surface is the Hilbert scheme of lines of a smooth cubic threefold of \(\mathbb P^4\). It is a smooth surface of general type with irregularity \(5\) and globally generated cotangent sheaf.
The author studied [Geom. Dedicata 142, 151–171 (2009; Zbl 1180.14041)] the cotangent map of these surfaces. In this paper he investigates the elliptic curves contained in a Fano surface.
The elliptic curves in a surface of general type are an obstruction for the ampleness of the cotangent sheaf, and in the case of Fano surfaces these curves are proved here to be the only obstruction. All the possible configurations of such curves are classified, i.e. their intersection matrix and a plane model of any of them are determined. Moreover, the author constructs a subgroup of the automorphism group of the surface which classifies completely these configurations.
The number of such curves is related to the Picard number. These results, as the author notes in the Introduction, imply that the ampleness of the cotangent bundle, and the geometric properties of the cotangent map, vary non-trivially in the moduli space of Fano surfaces.
Eventually the author applies is knowledge of Fano surfaces to construct a cubic threefold whose intermediate Jacobian is isomorphic – as a polarized abelian variety – to a product of elliptic curves.

MSC:
14J29 Surfaces of general type
14J45 Fano varieties
14J50 Automorphisms of surfaces and higher-dimensional varieties
14J70 Hypersurfaces and algebraic geometry
32G20 Period matrices, variation of Hodge structure; degenerations
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