Hyperelliptic odd coverings

Abstract

We investigate a class of odd (ramification) coverings C → ℙ1 where C is hyperelliptic, its Weierstrass points map to one fixed point of ℙ1 and the covering map makes the hyperelliptic involution of C commute with an involution of ℙ1. We show that the total number of hyperelliptic odd coverings of minimal degree 4g is \(\left({\matrix{{3g} \cr {g - 1} \cr}} \right){2^{2g}}\) 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.