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

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.