Proof. Let and let be the induced map. The map is dominant and has irreducible general fiber. We claim that Z is a variety of general type. If either of is generically finite (and therefore birational) then this is clear. By Viehweg's additivity theorem ([V1], Satz III), it suffices to show that the generic fiber of is of general type. This follows since the fibers of sweep . (Specifically, let d be the dimension of the generic fiber of . Choose a general codimension-d plane section , then is generically finite and dominant, therefore is of general type, therefore the generic fiber of is of general type.)
Proof. Given an extension let be a dominant rational map. By the lemma above with there exists a variety Z and a dominant rational map with irreducible general fiber dominating both and . By maximality and since the general fibers of are irreducible, is birational. The map gives the required dominant rational map.
Proof of the theorem. Using Stein factorization we may restrict attention to maps with irreducible general fibers. As above, let , and let be an extension such that . We need to show that can be descended to k.
First, we may assume that K is finitely generated over k, since both Y and require only finitely many coefficients in their defining equations.
Next, we descend to an algebraic extension of k. Choose a model B for K, and a model for Y. We have a dominant rational map over B. There exists a point with finite, such that is a variety of general type of dimension l and such that the rational map exists. The lemma above shows that is birational to Y. Alternatively, this step follows since by theorems of Maehara (see [Mor]) and Kobayashi - Ochiai (see [MD-LM]) the set of rational maps to varieties of general type is discrete, therefore each is birationally equivalent to a map defined over a finite extension of k.
We may therefore replace K by an algebraic Galois extension of k, which we still call K. Let . For any we have a rational map . Applying lemma 1.3 we obtain a birational map There are open sets over which these maps are regular isomorphisms, giving rise to descent data for to k.
Is there a way to describe the fibers of the Lang map ? A first approximation is provided by the following:
Proof. Let be the generic point and let be the lang map of the generic fiber. Let be a model of . By definition, the generic fiber of M is of general type, therefore by Viehweg's additivity theorem M is of general type, and by definition M is birational to .
We will see that the answer is yes, if one assumes the following inspiring conjecture of higher dimensional classification theory:
This conjecture allows us to ``construct'' the Lang map ``from above'':
The proof is obvious.