  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.
