Geometry of the Wiman-Edge monodromy

TL;DR

This paper provides a detailed description of the monodromy of the Wiman-Edge pencil, including new congruence conditions, and explores the geometric properties of the associated compactification and period map.

Contribution

It introduces a new congruence condition modulo 4 for the monodromy and fully characterizes the monodromy group of the Wiman-Edge pencil.

Findings

Monodromy described by congruence conditions modulo 4 and 5.

The compactification is a projective surface of general type.

New insights into the period map for the pencil.

Abstract

The Wiman-Edge pencil is a pencil of genus $6$ curves for which the generic member has automorphism group the alternating group $\mathfrak{A}_5$. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group $\mathfrak{S}_5$. Farb and Looijenga proved that the monodromy of the Wiman-Edge pencil is commensurable with the Hilbert modular group $\mathrm{SL}_2(\mathbb{Z}[\sqrt{5}])$. In this note, we give a complete description of the monodromy by congruence conditions modulo $4$ and $5$. The congruence condition modulo $4$ is new, and this answers a question of Farb-Looijenga. We also show that the smooth resolution of the Baily-Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.