Let $X$ ba a smooth hyperelliptic curve of genus $g$, and let $f:X\rightarrow X$ be the hyperelliptic involution. Consider a $K3$ surface $S$ with an involution $g$ without fixed points. The quotient $F = S/g$ is an Eniques surface.
Now $f\times g:X\times S\rightarrow X\times S$ does not have fixed points, and the quotient $(X\times S)/(f\times g)\rightarrow F$ is a family of hyperelliptic curves parametrized by $E$ whose fibers are all isomorphic to $X$.
My question is: why isn't the family $(X\times S)/(f\times g)\rightarrow F$ trivial?