# Non-unique completion of a flat family of smooth projective varieties

Let $$\mathbb{k}$$ be an algebraically closed field of characteristic 0. Denote $$S=\mathrm{Spec}\:\mathbb{k}[t]$$, $$U=\mathrm{Spec}\:\mathbb{k}[t, t^{-1}]$$, $$Z=\mathrm{Spec}\:\mathbb{k}[t]/(t)$$.

What is the minimum integer $$n$$ such that there exist smooth projective morphisms of relative dimension $$n$$ $$X\rightarrow S$$, $$Y\rightarrow S$$ such that there is no $$Z$$-isomorphism $$X_{Z}\rightarrow Y_{Z}$$ but there is an $$S$$-morphism $$X\rightarrow Y$$ inducing a $$U$$-isomorphism $$X_{U}\rightarrow Y_{U}$$?

• I don't think this is possible. The morphism $X_Z\rightarrow Y_Z$ would contract some subvariety, hence it would not induce an isomorphism on cohomology, while your hypotheses imply that it does.
– abx
May 28 '20 at 16:26
• @abx how does your cohomology argument work? Is it for coherent cohomology of the structure sheaf?
– user145520
May 28 '20 at 17:35
• No, for cohomology with rational (if you are over $\Bbb{C}$) or $\ell$-adic coefficients.
– abx
May 28 '20 at 18:56

I believe that this never happens. The reason is as follows. The morphism $$f:X\to Y$$ is a small birational morphism (it contracts some divisors on the special fiber) and so by standard arguments $$Y$$ is not $$\mathbb Q$$-factorial which is a contradiction as $$Y$$ is smooth.
To see that $$Y$$ is not $$\mathbb Q$$-factorial, let $$A$$ be a relatively ample divisor, then if $$f_*A$$ is $$\mathbb Q$$-Cartier and so $$A=f^*f_*A$$ (as $$f$$ is small), but $$f^*f_*A$$ can not be relatively ample.
Far reaching generalizations of this argument are in https://arxiv.org/pdf/0901.0389.pdf. Counterexamples when $$Y$$ has mild singularities are in Remark 4.4 of that paper (dim 2) and in Section 4 of Wilson's paper (dim 3).
• You can also get this out of the purity theorem in SGA 2 (this gives a way to prove a version of the result over a mixed characteristic DVR as well as over $k[t]_{\langle t \rangle}$). Jun 6 '20 at 4:16