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 |
References |
Additional Information
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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Additional Information
Olivier Debarre
Affiliation:
Département de Mathématiques et Applications, UMR CNRS 8553, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris cedex 05, France
MR Author ID:
55740
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
MR Author ID:
291751
ORCID:
0000-0001-6235-454X
Email:
Laurent.Manivel@ujf-grenoble.fr
Received by editor(s):
January 9, 2009
Received by editor(s) in revised form:
February 15, 2011
Published electronically:
August 15, 2011