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 $1$.



Picard number of a general fiber of a fiber contraction
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](https://doi.org/10.1093/imrn/rnv173)*, Int. Math. Res. Notices (2016) 2016 (7): 2026-2067.


