论文标题
Wiman-Edge Monodromy的几何形状
Geometry of the Wiman-Edge monodromy
论文作者
论文摘要
Wiman-Edge铅笔是属属$ 6 $曲线的铅笔，通用成员具有自动形态组的交替组$ \ mathfrak {a} _5 $。有一个独特的平滑成员，Wiman Sextic，其自动形态组对称组$ \ Mathfrak {S} _5 $。 Farb and Looijenga证明了Wiman-Edge铅笔的单曲与Hilbert模块化组$ \ MATHRM {SL} _2（\ Mathbb {Z} [\ sqrt {5}]）$相称。在本说明中，我们通过一致条件Modulo $ 4 $和$ 5 $对单片的完整描述。一致性条件Modulo $ 4 $是新的，这回答了Farb-looijenga的问题。我们还表明，与单片相关的局部对称歧管的baily-borel紧凑型的平滑分辨率是一般类型的投影表面。最后，我们提供有关铅笔时期图的图像的新信息。
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.
