Transitivity on Weierstrass points

Abstract

An automorphism of a Riemann surface will preserve its set of Weierstrass points. In this paper, we search for Riemann surfaces whose automorphism groups act transitively on the Weierstrass points. One well-known example is Klein’s quartic, which is known to have 24 Weierstrass points permuted transitively by it’s automorphism group, PSL(2,7) of order 168. An investigation of when Hurwitz groups act transitively has been made by Magaard and Volklein [19]. After a section on the preliminaries, we examine the transitivity property on several classes of surfaces. The easiest case is when the surface is hyperelliptic, and we find all hyperelliptic surfaces with the transitivity property (there are infinitely many of them). We then consider surfaces with automorphism group PSL(2,q), Weierstrass points of weight 1, and other classes of Riemann surfaces, ending with Fermat curves. Basically, we find that the transitivity property property seems quite rare and that the surfaces we have found with this property are interesting for other reasons too.