Hyperelliptic odd coverings

Abstract

We investigate a class of odd (ramification) coverings C → P 1 where C is hyperelliptic, its Weierstrass points maps to one fixed point of P 1 and the covering map makes the hyperelliptic involution of C commute with an involution of P 1 . We show that the total number of hyperelliptic odd coverings of minimal degree 4g is 3g g−1 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.