an:05586505
Zbl 1177.14079
Roulleau, Xavier
Elliptic curve configurations on Fano surfaces
EN
Manuscr. Math. 129, No. 3, 381-399 (2009).
00251057
2009
j
14J29 14J45 14J50 14J70 32G20
surfaces of general type; cotangent sheaf; cubic threefold
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.
Lidia Stoppino (Univ. dell'Insubria)
Zbl 1180.14041