2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/213174We 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.Algebraic Geometry14J30; 14J45; 14C34; 14E07; 14J50; 14J60; 14D20; 14K30; 32G20text