On the sections of universal hyperelliptic curves

TL;DR

This paper provides an algebraic proof for the sections of universal hyperelliptic curves for genus at least 3, explores nonabelian phenomena, and extends Hain's work to hyperelliptic cases.

Contribution

It offers a new algebraic approach to determine sections of universal hyperelliptic curves and verifies the section conjecture in this context.

Findings

Section conjecture holds for universal hyperelliptic curves without marked points.

Unipotent analogue of the section conjecture holds for pointed hyperelliptic curves.

Extension of Hain's work to hyperelliptic cases.

Abstract

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves, when $g\geq 3$ and the image of the $\ell$-adic cyclotomic character $G_k\to \Z^\times$ 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 the universal hyperelliptic curve without marked points and the unipotent analogue of the conjecture holds for the pointed case. This work is an extension of Hain's original work to the hyperelliptic case.