Hyperelliptic odd coverings

Abstract

We investigate a class of odd (ramification) coverings $C\to {\mathbb{P}}^1$ where $C$ is hyperelliptic, its Weierstrass points maps to one fixed point of ${\mathbb{P}}^1$ and the covering map makes the hyperelliptic involution of $C$ commute with an involution of ${\mathbb{P}}^1$. We show that the total number of hyperelliptic odd coverings of minimal degree $4g$ is $\left(\genfrac{}{}{0ex}{}{3g}{g-1}\right)2^{2g}$ when $C$ is general. Our study is approached from three main perspectives: if a fixed effective theta characteristic is fixed they are described as a solution of a certain class of differential equations; then they are studied from the monodromy viewpoint and a deformation argument that leads to the final computation.

15 pages, comments very welcome