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Article Dans Une Revue Journal of Algebraic Geometry Année : 2012

On the period map for prime Fano threefolds of degree 10

Résumé

We study, after Logachev, the geometry of smooth complex Fano threefolds X X with Picard number 1 1 , index 1 1 , and degree 10 10 , and their period map to the moduli space of 10-dimensional principally polarized abelian varieties. We prove that a general such X X has no nontrival automorphisms. By a simple deformation argument and a parameter count, we show that X X is not birational to a quartic double solid, disproving a conjecture of Tyurin. Through a detailed study of the variety of conics contained in X X , a smooth projective irreducible surface of general type with globally generated cotangent bundle, we construct two smooth projective two-dimensional components of the fiber of the period map through a general X X : one is isomorphic to the variety of conics in X X , modulo an involution, another is birationally isomorphic to a moduli space of semistable rank- 2 2 torsion-free sheaves on X X , modulo an involution. The threefolds corresponding to points of these components are obtained from X X via conic and line (birational) transformations. The general fiber of the period map is the disjoint union of an even number of smooth projective surfaces of this type.

Dates et versions

hal-03808048 , version 1 (10-10-2022)

Identifiants

Citer

Olivier Debarre, Atanas Iliev, Laurent Manivel. On the period map for prime Fano threefolds of degree 10. Journal of Algebraic Geometry, 2012, 21 (1), pp.21-59. ⟨10.1090/S1056-3911-2011-00594-8⟩. ⟨hal-03808048⟩
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