On the sections of universal hyperelliptic curves

Abstract

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves CHg,n/k→Hg,n/k, when g≥3 and the image of the ℓ-adic cyclotomic character Gk→Z× is infinite. Furthermore, we will study the nonabelian phenomena associated to the universal hyperelliptic curves. For example, we will show that the section conjecture holds for CHg/k→Hg/k and the unipotent analogue of the conjecture holds for the pointed cases. This work is an extension of Hain's original work on the rational points of universal curves to the hyperelliptic case.