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On gonality automorphisms of p-hyperelliptic Riemann surfaces

Sobre automorfismos de gonalIDad de superficies de Riemann p-hiperelípticas

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Abstract

A compact Riemann surface X of genus g > 1 is saID to be a p-hyperelliptic if X admits a conformal involution ρ for which X/ρ has genus p. This notion is the particular case of so called cyclic (q, n)-gonal surface which is defined as the one admitting a conformal automorphism δ of order n such that X/δ has genus q. It is known that for g > 4 p + 1, ρ is unique And so central in the automorphism group of X. We give necessary And sufficient conditions on p And g for the existence of a Riemann surface of genus g admitting commuting p-hyperelliptic involution ρ And (q, n)-gonal automorphism δ for some prime n And we study its group of automorphisms And the number of fixed points of δ. Furthermore, we deal with automorphism groups of Riemann surfaces admitting central automorphism with at most 8 fixed points. The condition on the small number of fixed points of such an automorphism is justified by the study of p-hyperelliptic surfaces.

Resumen

Una superficie de Riemann compacta X de género g > 1 se dice p-hiperelíptica si X admite una involución conforme ρ, tal que X/ρ tiene género p. Las superficies p-hiperelípticas son un caso particular de las superficies (q, n)-gonales cíclicas que se definen como aquellas superficies que admiten un automorfismo conforme δ de orden q y de modo que X/δ tiene género q. En este trabajo nos restringiremos al caso en que q es un número primo mayor que 2. Es un hecho conocIDo que si g > 4 p+1, la involución ρ es única y central en el grupo de automorfismos de X. Obtenemos condiciones necesarias y suficientes sobre p y g para la existencia de superficies de Riemann de género g que admiten una involución p-hiperelíptica y un automorfismo (q, n)-gonal que conmutan. Se determina la presentación de un cociente de los grupos de automorfismos de las superficies de Riemann que admiten un automorfismo (q, n)-gonal que sea central y con 8 puntos fijos como máximo. Esta restricción sobre el número de puntos fijos se justifica por el estudio anterior de las superfices que son a la vez p-hiperelípticas y (q, n)-gonales cíclicas.

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  1. Institute of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952, Gdańsk, Poland

    Ewa Tyszkowska

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  1. Ewa Tyszkowska
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Submitted by José María Montesinos Amilibia

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Tyszkowska, E. On gonality automorphisms of p-hyperelliptic Riemann surfaces. Rev. R. Acad. Cien. Serie A. Mat. 104, 87–96 (2010). https://doi.org/10.5052/RACSAM.2010.09

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  • DOI: https://doi.org/10.5052/RACSAM.2010.09

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