  In this paper, we prove that the admissible canonical bundle of the universal
family of curves is a big adelic line bundle, and apply it to prove a uniform
Bogomolov-type theorem for curves over global fields of all characteristics.
This gives a different approach to the uniform Mordell-Lang type of result of
Dimitrov-Gao-Habegger and Kuhne. The treatment is based on the recent theory of
adelic line bundles of Yuan-Zhang.
