We work over fields of characteristic 0.

Let **X** be a variety of general type defined over a number field **K**. A well
known conjecture of S. Lang [L] states that the set of rational
points is not Zariski - dense in **X**. As noted in [], this implies
that if **X** is a variety which only * dominates* a variety of general type
then is still not dense in **X**.

J. Harris proposed a way to quantify this situation [H1]: define the
* Lang dimension* of a variety to be the maximal dimension of a variety of
general type which it dominates. Harris conjectured in particular that if the
Lang dimension is 0 then for * some* number field we have that
the set of **L** rational points is dense in **X**. The full statement of
Harris's conjecture will be given below (Conjecture 2.3).

The purpose of this note is to provide a geometric context for Harris's
conjecture, by showing the existence of a universal dominant map to a variety
of general type, which we call * the Lang map.*

Fri Dec 15 14:16:28 EST 1995