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.