On the family of cyclic trigonal Riemann surfaces of genus 4 with several trigonal morphisms.

*(English)*Zbl 1137.14306Summary: 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. A. F. Costa, M. Izquierdo and D. Ying [Manuscr. Math. 118, No. 4, 443–453 (2005; Zbl 1137.30013)] 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.