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In this talk, we will start by explaining why a height inequality and a construction of non-degenerate subvariety together imply a rather uniform bound on the number of rational points on curves. Then we will digest this height inequality and explain how the “non-degeneracy” condition should be understood to be a bigness result of a suitable line bundle and thus make a (short) connection to adelic line bundles. We will finish the talk by a summary of some classical results in Arakelov geometry, which will be used in part 2 for more detailed discussion of adelic line bundles and adelic divisors over quasi-projective varieties.

DioG Learning Seminar: Uniform Mordell-Lang And Adelic Line Bundle, Part 1