On the universal curve with unordered marked points in positive characteristic

Abstract

We study the relative pro-$\ell$ and continuous relative completions of the algebraic fundamental groups of universal curves over the moduli stack of curves with unordered marked points in positive characteristic. Using specialization and homotopy exact sequences, we compare the ordered and unordered settings and prove that the natural projection from the relative completion of the universal curve over the unordered moduli stack admits no section in positive characteristic. This yields a non-splitting result for the corresponding projection on algebraic fundamental groups. The present paper is a sequel to our earlier work in characteristic zero.