Lifting automorphism to a family Suppose that $\pi : Y \to X$ is a flat projective family, and that $f : X \to  X$ is an automorphism.  Is there any way to tell whether $f$ lifts to an automorphism of $Y$?  I would be happy even with the case of a $\mathbb P^1$-bundle, though a more interesting case might be a minimal elliptic fibration (perhaps isotrivial, since the fibers over the orbit of a point must all be isomorphic).
 A: If I got right you question your are asking whether given an automorphism $f:X\rightarrow X$ there is an automorphism $F:Y\rightarrow Y$ such that $\pi\circ F = f\circ \pi$. You see that in order to have this $F$ has to map fibers of $\pi$ to fibers of $\pi$. 
Perhaps the following examples, coming from moduli of curves, may be useful to you. Let $Y$ be a Del Pezzo surface of degree $5$ and let $X$ be $\mathbb{P}^1$. Note that we can interpret $Y$ as $\overline{M}_{0,5}$ and $X$ as $\overline{M}_{0,4}$. We have five forgetful maps $\overline{M}_{0,5}\rightarrow\overline{M}_{0,4}$ forgetting one of the marked points. Each one of these is a flat projective morphism. Now, $Aut(X) = Aut(\mathbb{P}^1) = PGL(2)$ and $Aut(Y) = Aut(\overline{M}_{0,5}) = S_5$, the symmetric group on five elements. Therefore the general automorphism of $X$ does not lift to $Y$.
In a similar way you can consider a forgetful morphism $\overline{M}_{1,3}\rightarrow\overline{M}_{1,2}$. You have that $\overline{M}_{1,2}$ is a toric surface and $Aut(\overline{M}_{1,2}) = (\mathbb{C}^{*})^{2}$ while $Aut(\overline{M}_{1,3})\cong S_3$. Again, the general automorphism does not lift.
