Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On the period map for prime Fano threefolds of degree $ 10$


Authors: Olivier Debarre, Atanas Iliev and Laurent Manivel
Journal: J. Algebraic Geom. 21 (2012), 21-59
DOI: https://doi.org/10.1090/S1056-3911-2011-00594-8
Published electronically: August 15, 2011
MathSciNet review: 2846678
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Abstract: We study, after Logachev, the geometry of smooth complex Fano threefolds $ X$ with Picard number $ 1$, index $ 1$, and degree $ 10$, and their period map to the moduli space of 10-dimensional principally polarized abelian varieties. We prove that a general such $ X$ has no nontrival automorphisms. By a simple deformation argument and a parameter count, we show that $ 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$, 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$: one is isomorphic to the variety of conics in $ X$, modulo an involution, another is birationally isomorphic to a moduli space of semistable rank-$ 2$ torsion-free sheaves on $ X$, modulo an involution. The threefolds corresponding to points of these components are obtained from $ 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.


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Olivier Debarre
Affiliation: Département de Mathématiques et Applications, UMR CNRS 8553, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris cedex 05, France
Email: Olivier.Debarre@ens.fr

Atanas Iliev
Affiliation: Seoul National University, Department of Mathematics, Seoul 151-747, Korea
Email: ailiev@snu.ac.kr

Laurent Manivel
Affiliation: Institut Fourier, Université de Grenoble I et CNRS, BP 74, 38402 Saint-Martin d’Hères, France
Email: Laurent.Manivel@ujf-grenoble.fr

DOI: https://doi.org/10.1090/S1056-3911-2011-00594-8
Received by editor(s): January 9, 2009
Received by editor(s) in revised form: February 15, 2011
Published electronically: August 15, 2011

American Mathematical Society