TITLE: Finiteness of birational types for targets of algebraic fibrations
QUESTION [8 upvotes]: Let $X$ be a smooth projective variety. A fibration is a surjective map with connected fibers between projective varieties. Is it true that there's a finite number of birational equivalence classes of projective varieties (say, with representatives $Y_1,\ldots, Y_n$) such that if $X\to Y$ is a fibration then $Y$ is birational to some $Y_i$?

REPLY [6 votes]: As Jorge already pointed out this is too much to hope for. On the other hand, if you can put some restriction on $Y$, then there are results in this direction.

A theorem of Severi implies that if you restrict $Y$ to be a curve of genus at least $2$, then this is true. (Severi's theorem is slightly more general, requiring only that the map is dominant)
Severi's theorem was generalized to the case when $Y$ is a surface of general type by Martin-Deschamps and Lewin-Ménégaux (two people :)...
...and it was generalized to arbitrary dimension ($Y$ is still of general type) by Hacon and McKernan. (See Corollary 1.4 of this paper)