On the sections of universal hyperelliptic curves

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves, when $g\ge 3$ and the image of the $\mathrm{\ell }$-adic cyclotomic character $G_k\to {\mathbb{Z}}^{\times }$ 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.