Geometry of the Wiman-Edge pencil and the Wiman curve

摘要

The Wiman-Edge pencilis the universal family C-t, t is an element of B of projective, genus 6, complex-algebraic curves admitting a faithful action of the icosahedral group A(5). The curve C-0, discovered by Wiman in 1895 (Ueber die algebraische Curven von den Geschlecht p=4,5 and 6 welche eindeutige Transformationen in sich besitzen) and called theWiman curve, is the unique smooth, genus 6 curve admitting a faithful action of the symmetric group S-5. In this paper we give an explicit uniformization of B as a non-congruence quotient Gamma of the hyperbolic plane H, where Gamma