This question consists of two slightly similar questions.
a) Let $ F$ be a field of positive characteristic. Let $ X$ be a smoooth proper geometrically connected $ F$ -scheme.
Let $ C$ be the category of local integral domains together with an isomorphism from the residue field to $ F$ .
Consider the (covariant?) functor from $ C$ to sets, sending a ring $ R$ to the set of $ R$ -isomorphism classes of smooth proper $ R$ -schemes whose special fiber, which is naturally an $ F$ -scheme, is isomorphic to $ X$ (or take values in groupoids if it helps).
Is this (co-)representable by something geometric? What if restrict to Noetherian rings, or DVR’s?
b) Let $ R$ be a DVR with the residue field $ F$ of positive characteristic. Consider the functor sending smooth proper geometrically connected $ F$ -schemes to the set of liftings to $ R$ -schemes as above. Is this (co-)representable by something geometric?