2020/12/31 by Stover, Matthew
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2012.15708
The Wiman-Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group \mathfrakA5. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group \mathfrakS5. Farb and Looijenga proved that the monodromy of the Wiman-Edge pencil is commensurable with the Hilbert modular group SL2(ℤ[√(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.