June 8, 2009
First talk 8 June 2009.- Aaron Wootton
Title: Cyclic n-gonal surfaces - weakly malnormal actions
Abstract: A cyclic n-gonal surfaces S is a compact Riemann surface which admits an automorphism h such that the surface S/C has genus 0, where C=<h>. From the perspective of Galois theory, cyclic n-gonal surfaces are interesting since they correspond to cyclic extensions of the rational function field and hence a defining equation for such a surface is fairly straight forward to find. When the order n of h is prime, as in the hyperelliptic case, C is normal in the automorphism group A=Aut(S) if the genus g of S is sufficiently large. For small genus there are two infinite families of surfaces, and four exceptional surfaces in which C is not normal (such as Klein's quartic (n=7,g=3) and and Brings curve (n=5,g=4)), all of which can be determined by hand. In this talk we introduce the notion of weakly malnormal actions on surfaces which is precisely the condition needed to extend the theory to general cyclic n-gonal surfaces. Simple examples show that for composite n there are non-normal cyclic n-gonal actions for arbitrarily large genus.
Second talk 8 June 2009. - Allen Broughton
Title: Cyclic n-gonal surfaces - computational methods
Abstract:A cyclic n-gonal surfaces S, is a compact Riemann surface with an automorphism h such that the surface S/C has genus 0, where C = <h>. In certain circumstances (weakly malnormal actions, described in the previous talk) C is normal in the automorphism group A=Aut(S) if the genus g of S is sufficiently large. For small genus, when n is composite, unlike the case where n is prime, it is not feasible to determine the exceptional cases where C is not normal by hand and instead it is necessary to resort to computer classification. In this talk, we give an overview of the computational methods used to determine these exceptional cases, and a tabulation of results achieved to date.