Exceptional Automorphisms of (generalized) Super-Elliptic SurfacesRiemann and Klein Surfaces, Symmetries and Moduli Spaces, in honour of Professor Emilio Bujalance (2013)
A super-elliptic curve S is a curve with a conformal automorphism g of prime order p such that S/〈g〉has genus zero. This generalizes the hyper-elliptic case p= 2. More generally, a cyclic n-gonal surface S has an automorphism g of order n such that S/〈g〉has genus zero. All cyclic n-gonal surfaces have tractable defining equations. Let A=Aut(S) and N be the normalizer of C=〈g〉in A. The structure of N can in principal be determined by the action of N/C on the sphere S/C which in turn can be determined from the defining equation. If the genus of S is sufficiently large in comparison to n, then A=N. For small genus, A\N may not be empty and, in this case, any automorphism h in A\N is called exceptional. The exceptional automorphisms of super-elliptic curves are known whereas the determination of exceptional automorphisms of all general cyclic n-gonal surfaces seems to be hard. In this talk we focus on generalized super-elliptic curves in which the projection of S onto S/C is fully ramified. Generalized super-elliptic curves are easily identified by their defining equations. In this talk we determine large classes of (generalized) super-elliptic curves with exceptional automorphisms. This is joint work with Aaron Wootton.
- Reimann surface,
- superelliptic curve,
- automorphism group
Publication DateJune 25, 2013
Citation InformationSean A Broughton and Aaron Wootton. "Exceptional Automorphisms of (generalized) Super-Elliptic Surfaces" Riemann and Klein Surfaces, Symmetries and Moduli Spaces, in honour of Professor Emilio Bujalance (2013)
Available at: http://works.bepress.com/allen_broughton/62/
Creative Commons License
This work is licensed under a Creative Commons CC_BY-NC-SA International License.