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)

I don't think so. Consider the product of two isogeneous elliptic curves. This surface has infinitely many smooth elliptic fibrations. The basis are all isogeneous but I guess that they belong to infinitely many distinct binational equivalence classes.

Tags:

Ag.Algebraic Geometry

Birational Geometry

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