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arXiv · 2012.15708

Geometry of the Wiman-Edge monodromy

Matthew Stover
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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.

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Matthew Stover. 2020-12-31. Geometry of the Wiman-Edge monodromy. https://arxiv.org/abs/2012.15708

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