On the Monodromy and Period Map of the Winger Pencil

TL;DR

This paper studies the Winger pencil of genus ten curves with icosahedral symmetry, analyzing its monodromy group and period map, revealing deep connections to Hilbert modular surfaces and arithmetic groups.

Contribution

It provides a modern analysis of the Winger pencil, showing its monodromy group is a finite index subgroup of SL_2(Z[√5]) and describing its period map on a Hilbert modular surface.

Findings

Monodromy group is a finite index subgroup of SL_2(Z[√5])

Period map maps the pencil to a curve on a Hilbert modular surface

Contains essentially all smooth genus ten curves with icosahedral symmetry

Abstract

The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and a three times a conic). This pencil was first considered in a note by R.~M.~Winger in 1925 and is nowadays named after him. We gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here consider the monodromy group and the period map naturally defined by the icosahedral symmetry. We showed that this monodromy group is a subgroup of finite index in $\SL_2(\Zds[\sqrt{5}])$ and the period map brings the Winger pencil to a curve on the Hilbert modular surface $\SL_2(\Zds[\sqrt{5}])/\Hds^2$.