Matthew Stover

The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group ${A}_{5}$. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group ${S}_{5}$. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group ${\mathrm{\text{SL}}}_{2}(\mathbb{Z}[\sqrt{5}])$. 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.