It follows from results in Teichmüller Theory that generic curves of type (g,n) in characteristic zero have only n rational points that come from the tautological points. Richard Hain gave an algebraic proof of the theorem. Extending his algebraic method to positive characteristics, we prove the analogous result for generic curves in positive characteristics. The primary tool used is the theory of weighted completion, which was developed by Richard Hain and Makoto Matsumoto. It linearises a profinite group such as arithmetic mapping class groups. In our case, the weighted completion connects topology and algebraic geometry in positive characteristics.