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Height Estimates for Equidimensional Dominant Rational Maps
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Let φ: W→V be a dominant rational map between quasi-projective varieties of the same dimension. We give two proofs that hV (φ(P))≫hW(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.
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