Geometry of the Winger Pencil

TL;DR

This paper studies genus 10 curves with icosahedral symmetry, revealing their moduli space structure as a Deligne-Mumford stack related to Winger's plane sextic pencil, with a detailed orbifold analysis.

Contribution

It characterizes the moduli space of genus 10 icosahedral curves as a Deligne-Mumford stack linked to Winger's sextic pencil, replacing the unstable member with a smooth curve.

Findings

Moduli space is a Deligne-Mumford stack with two copies of Winger's pencil

Each orbifold has genus zero with four orbifold points

The fine moduli space is a finite cover of ,4

Abstract

We investigate the moduli of genus 10 curves that are endowed with a faithful action of the icosahedral group $\mathcal{A}_5$. We show among other things that this has the structure of a Deligne-Mumford stack whose underlying coarse moduli space essentially consists of two copies of the pencil of plane sextics that was introduced by Winger in 1924, but with the unique unstable member (a triple conic) replaced by a smooth non-planar curve. The orbifold defined by any member has genus zero and comes with 4 orbifold points. We show that by numbering the points, we get a fine moduli space whose base is naturally a finite cover of $\Mcal_{0,4}$.