** 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.

Fri Dec 15 14:16:28 EST 1995