Some degenerations of a compact Riemann surface of genus 4

Abstract

A number of degenerations of a compact hyperelliptic Riemann surface of genus 4 are studied, using theta function techniques. 1. Let (5, T, A) be a hyperelliptic Riemann surface of genus 4 with a canonical homology basis. Then S has a representation as a two-sheeted cover of the sphere with ten branch points. We can arrange for the surface to have branch points over 0 , 1 , «>, 1/Xj, 1/X2> * * > IA7» where its real branch points other than 0 and 1 (if any) are all greater than 1 and in ascending order (see [2]). We obtain a concrete representation of (S, T, A) which we henceforth assume is that illustrated in Figure 1.