Height Estimates for Equidimensional Dominant Rational Maps

Let φ: W→V be a dominant rational map between quasi-projective varieties of the same dimension. We give two proofs that h_V (φ(P))≫h_W(P) for all points P in a nonempty Zariski open subset of W. For dominant rational maps φ: Pⁿ→Pⁿ, we give a uniform estimate in which the implied constant depends only on n and the degree of φ. As an application, we prove a specialization theorem for equidimensional dominant rational maps to semiabelian varieties, providing a complement to Habegger’s recent theorem on unlikely intersections.