On the period map for prime Fano threefolds of degree 10

TL;DR

This paper studies the deformation theory and period map of prime Fano threefolds of degree 10, revealing the structure of the fibers and their relation to conic and line transformations.

Contribution

It demonstrates that deformations of these threefolds are unobstructed and characterizes two-dimensional components of the period map fiber, linking them to conic varieties and moduli spaces.

Findings

Deformations are unobstructed.

The kernel of the period map differential is two-dimensional.

Two components of the period map fiber are constructed and described.

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.