On the family of Riemann surfaces with tetrahedral group action

Abstract

This is the second of our series of papers to solve Mutsuo Oka's problems concerning our polyhedral construction of degenerations of Riemann surfaces. Oka posed globalization problem of our degenerations and determination problem of the defining equation of a Riemann surface appearing in our construction—which is equipped with the standard tetrahedral group action (i.e. topologically equivalent to the tetrahedral group action on the cable surface of the tetrahedron). A joint work with S. Takamura solved the first problem. In this paper, we solve the second one—in an unexpected way: an algebraic curve with the standard tetrahedral group action turns out to be not unique: a sporadic one (hyperelliptic) and a 1-parameter family of non-hyperelliptic curves. We study their properties. At first glance they are `independent', but actually intricately connected—we show that at one special value in this family, a degeneration whose monodromy is a hyperelliptic involution occurs, and the sporadic hyperelliptic curve emerges after the stable reduction (hyperelliptic jump). This jumping phenomenon seems deeply related to the moduli geometry and is possibly universal for other families of curves with finite group actions. Based on this observation, we pose stably-connectedness problem.