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Suppose in the last step of a MMP, we obtain a Mori fiber space $f: X \to Z$, and let $F$ be a general fiber of $f$, then is the Picard number $\rho(F)$ of $F$ equal to $1$? Notice that the relative Picard number $\rho(X/Z)=1$ because I only assume to contract an extremal ray.

My feeling is that $\rho(F)$ may not be one, but I don't have a good example. Notice that if $X$ is toric, the fiber always has Picard number $1$.

I would also appreciate examples (if any) of fibration (with connect fibers) $f: X \to Y$ such that $\rho(X/Z)=1$ but the general fiber does not have Picard number $1$.

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  • $\begingroup$ Although you already have a definitive answer, let me just mention that is not difficult to construct examples like the ones you ask for in the last paragraph. For example, take a general pencil of quadric surfaces in $\mathbf P^3$ and blow up the base locus. Then you get a fibration $X \rightarrow \mathbf P^1$ where $X$ has Picard number 2, hence $\rho(X/\mathbf P^1)=1$, but the general fibre is a smooth quadric, hence has Picard number 2. $\endgroup$
    – Bertie
    Sep 5, 2017 at 8:58
  • $\begingroup$ You can produce a more extreme example if you do the same construction but now starting with a pencil of cubic surfaces in $\mathbf P^3$; again the relative Picard number is 1, but the Picard number of the fibre is now 7. $\endgroup$
    – Bertie
    Sep 5, 2017 at 9:07

1 Answer 1

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I do not think so, because of the following result.

Proposition. A smooth del Pezzo surface $F$ can be realised as the general fibre of a Mori fibre space if and only if it is not isomorphic to the blow-up of $\mathbb{P}^2$ in one or two points.

This shows that all values $\rho(F) \in \{1, \ldots, 9\}$ with $\rho(F) \neq 3$ are realized.

See Theorem 1.4 in

Fano Varieties in Mori Fibre Spaces, Int. Math. Res. Notices (2016) 2016 (7): 2026-2067.

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