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.