Remarks on Rational Points of Universal Curves

Abstract

In this notes, we will give some remarks on the results in Rational points of universal curves by Hain. In particular, we consider the universal curves $\mathcal{M}_{g,n+1}\to \mathcal{M}_{g,n}$ and the sections of their algebraic fundamental groups.Comment: 14 pages. The section of the unipotent section conjecture in positive characteristic has been removed from this version. It will be a separate paper due to the nature of the conten