Does Isomorphic Generic Fibre Imply Isomorphic Special Fibre for Smooth Morphisms?

ag.algebraic-geometrynt.number-theory

Let $X$ and $Y$ be regular integral Noetherian schemes. Assume that $X$ and $Y$ are smooth and proper over a base scheme $S=Spec R$, where $R$ is a discrete valuation ring.

If $X$ and $Y$ have isomorphic generic fibres, is it also the case that their special fibres are isomorphic?

Remarks:

  1. The answer is yes when $X$ and $Y$ are abelian schemes (this follows from the theory of the Néron model). In general though it is not the case that morphisms between the generic fibres extend to the special fibre.

  2. I am also particularly interested in case where $R$ is the localisation of $\mathbb{Z}$ at some prime, and hence the generic fibres are smooth proper varieties over $\mathbb{Q}$.

Best Answer

Here is an example showing the answer is no:

Start with $Z =\mathbb{P}^2_R$, $R$ an arbitrary dvr. Let $P$ be a section of $Z \to Spec(R)$ and let $W$ be the blowup of $Z$ along the image of the section (so both fibres are $\mathbb{P}^2$ with a point blown up). Let $Q$ be a section of $W \to Spec(R)$ whose image does not intersect the exceptional divisor of the first blow up and let $Q'$ be a section whose image intersects the exceptional divisor only in the special fibre. Let $X$ be the blow up of $W$ along $Q$ and $Y$ the blow up of $W$ along $Q'$.

The generic fibres of $X$ and $Y$ are isomorphic since they are both just $\mathbb{P}^2$ over some field blown up in two distinct (rational) points and any two points on $\mathbb{P}^2$ over a field are "the same". That the special fibres are not isomorphic can be seen by considering curves of self intersection $-2$:

The special fibre of $X$ has none since we have blown up two distinct points, so the only curves with negative self intersection are three $(-1)$-curves i.e. the two exceptional divisors and the strict transform of the line joining the two points. The special fibre of $Y$ does have one; this is the strict transform in the second blowup of the exceptional divisor of the first blowup. (The second blow up changes the $(-1)$-curve into a $(-2)$-curve since we blow up a point on it.)

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