The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group A5. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group S5. 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.

中文翻译：

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Wiman-Edge 单极几何的几何

Wiman-Edge 铅笔是属 6 曲线的铅笔，其类属成员具有自同构群交替群一种5. 有一个独特的光滑成员，Wiman 六序，具有自同构群的对称群小号5. Farb 和 Looijenga 证明了 Wiman-Edge 铅笔的单调性与 Hilbert 模群可通约SL2(ℤ[5]). 在这篇笔记中，我们通过同余条件模 4 和 5 对单偶性进行了完整的描述。同余条件模 4 是新的，这回答了 Farb-Looijenga 的问题。我们还表明，与单调相关的局部对称流形的 Baily-Borel 紧化的平滑分辨率是一般类型的射影曲面。最后，我们提供了有关铅笔周期图图像的新信息。