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On the family of cyclic trigonal Riemann surfaces of genus 4 with several trigonal morphisms. (English) Zbl 1137.14306
Summary: 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.

##### MSC:
 14H15 Families, moduli of curves (analytic) 30F10 Compact Riemann surfaces and uniformization
Zbl 1137.30013
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