A closed Riemann surface which is a 3-sheeted regular covering of the Riemann sphere is called cyclic trigonal, and such a covering is called a cyclic trigonal morphism. Accola showed that if the genus is greater or equal than 5 the trigonal morphism is unique. Costa-Izquierdo-Ying found a family of cyclic trigonal Riemann surfaces of genus 4 with two trigonal morphisms. In this work we show that this family is the Riemann sphere without three points. We also prove that the Hurwitz space of pairs (X, f), with X a surface of the above family and f a trigonal morphism, is the Riemann sphere with four punctures. Finally, we give the equations of the curves in the family.
Una superficie de Riemann que es una cubierta regular de 3 hojas de la esfera se llama cíclica trigonal, y la cubierta un morfismo trigonal. Accola probó que el morfismo trigonal es único si el género de la superficie es mayor o igual que 5. Costa-Izquierdo-Ying encontraron una familia de superficies de Riemann de género 4 cíclicas trigonales con varios morfismos trigonales. En este trabajo demostramos que dicha familia es, en efecto, la esfera de Riemann con tres punzamientos. Además demostramos que el espacio de Hurwitz de pares (X, f), con X una surperficie en la familia anterior y f un morfismo trigonal, es la esfera de Riemann con cuatro punzamientos. Finalmente encontramos las ecuaciones de las curvas en la familia.
@article{urn:eudml:doc:41666,
title = {On the family of cyclic trigonal Riemann surfaces of genus 4 with several trigonal morphisms.},
journal = {RACSAM},
volume = {101},
year = {2007},
pages = {81-86},
zbl = {1137.14306},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41666}
}
Costa, Antonio F.; Izquierdo, Milagros; Ying, Daniel. On the family of cyclic trigonal Riemann surfaces of genus 4 with several trigonal morphisms.. RACSAM, Tome 101 (2007) pp. 81-86. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41666/