In this paper we prove a version of Grothendieck's section conjecture for the
restriction of the universal complete curve over M_{g,n}, g > 4, to the
function field k(M_{g,n}) where k is, for example, a number field. In this
version, the fundamental group of the closed fiber is replaced by its ell-adic
unipotent completion when n > 1.Comment: 60 pages. This is a minor revision of the previous version. To appear
in JAM