TITLE: Isomorphic schemes over DVR
QUESTION [10 upvotes]: Let $S, S'$ be flat schemes over a DVR. Their generic fibers are isomorphic and their special fibers are isomorphic as well.
Does that imply $S$ and $S'$ isomorphic? If not, what can go wrong?
Thanks in advance!

REPLY [3 votes]: If we are all pitching in our favorite counterexamples, consider a smooth, projective morphism $\pi:S\to \text{Spec}(R)$ over a DVR whose generic fiber is the Hirzebruch surface $\mathbb{P}^1\times \mathbb{P}^1$ and whose special fiber is the Hirzebruch surfaces $\Sigma_2 = \mathbb{P}_{\mathbb{P}^1}(\mathcal{O}(-1)\oplus \mathcal{O}(+1))$.  For such a family, there is an integer invariant; roughly the order of vanishing at the closed point of the DVR of the "modulus" of the moduli space of Hirzebruch surfaces.  More precisely, it is the length of the torsion sheaf $R^1\pi_* \textit{Hom}_{\mathcal{O}_S}(\Omega_\pi,\mathcal{O}_S)$.  This integer invariant can take on any positive integer value.  Thus, there exist families $S$, $S'$ as above such that the invariant is $1$, respectively $2$.