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 $Y\_i$?



Finiteness of birational types for targets of algebraic fibrations
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.


1. 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)
2. Severi's theorem was generalized to the case when $Y$ is a surface of general type by Martin-Deschamps and Lewin-Ménégaux (two people :)...
3. ...and it was generalized to arbitrary dimension ($Y$ is still of general type) by Hacon and McKernan. (See Corollary 1.4 of [this paper](https://link.springer.com/article/10.1007/s00222-006-0504-1))


