Geometry of the Wiman-Edge monodromy

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.