On the period map for prime Fano threefolds of degree 10

2008/12/18 by O. Debarre, Olivier Debarre, A. Iliev +6 · 3 citations
Mathematics · #14C34 #14D20 #14E07 #14J30 #14J45 #14J50 #14J60 #14K30 #32G20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #math.AG #msc:14C34 #msc:14D20 #msc:14E07 #msc:14J30 #msc:14J45 #msc:14J50 #msc:14J60 #msc:14K30 #msc:32G20

paper · pdf · doi:10.48550/arxiv.0812.3670

arxiv created 2008/12/18 · openalex publication_date 2008/12/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We prove that the deformations of a smooth complex Fano threefold X with Picard number 1, index 1, and degree 10, are unobstructed. The differential of the period map has two-dimensional kernel. We construct two two-dimensional components of the fiber of the period map through 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.

Cited by

• Lagrangian interpretation of Abel-Jacobi mappings associated to Fano threefolds
• Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives
• On nodal prime Fano threefolds of degree 10

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