Lang Maps and Harris’s Conjecture Preliminary Version

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 X(K) is not Zariski dense in X. As noted in [א], this implies that if X is a variety which only dominates a variety of general type then X(K) 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 L ⊃ K we have that the set of L rational points X(L) 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.