Vanishing thetanulls for some dihedral and cyclic coverings of Riemann surfaces

Abstract

Let W_g→W_z be a ramified p-sheeted covering of Riemann surfaces of genus g and z, (z>0) where p is an odd prime. Assume that the Galois group is either dihedral or cyclic. Assume, moreover, that the covering is full; that is, there us an integral divisor E, of degree 2r on W_z which lifts to be canonical on W_g. Then g=rp+1, where r≥1. Clearly, W_g admits 2^2z half-canonical linear series of dimension at least r−z arising from divisors on W_z whose double is E. Theorem 1 Of these 2^2z half-canonical linear series u_z (=2^z−1(2^z−1)) have dimension at least r−z+1. Theorem 2 Let W_g (g=3r+1, r≥3) admit four half canonical linear series, three of dimension r−1, and one of dimension r, whose sum is bi-canonical, where the half-canonical linear series of dimension r is unique. Then W_g is a full elliptic-trigonal Riemann surface. (This characterizes the cases z=1, p=3, g≥10).