亏格为3的Prym-Teichmüller曲线上的Orbifold点
Orbifold points on Prym-Teichmüller curves in genus three
摘要：
Prym-Teichmüller曲线$W_D(4)$构成了模空间$\mathcal{M}_3$中已知的本原Teichmüller曲线的主要例子。我们确定，对于每个非平方判别$D>1$，在$W_D(4)$的orbifold点的数量和类型。这些结果与Lanneau-Nguyen和Möller关于尖点数和欧拉示性数的公式一起，完成了亏格为3的Prym-Teichmüller曲线的拓扑刻画.确定轨道点的关键是分析$4$和$6$度的属3循环覆盖族，分支于$\mathbb{P}^1$的四个点。作为我们的研究的副产品，我们提供了一个明确的描述这两个家庭的雅可比和Prym-Torelli图像，以及相应的平面的描述。
英文摘要：
Prym-Teichmüller curves $W_D(4)$ constitute the main examples of known primitive Teichmüller curves in the moduli space $\mathcal{M}_3$. We determine, for each non-square discriminant $D>1$, the number and type of orbifold points in $W_D(4)$. These results, together with the formulas of Lanneau-Nguyen and Möller for the number of cusps and the Euler characteristic, complete the topological characterisation of Prym-Teichmüller curves in genus 3. Crucial for the determination of the orbifold points is the analysis of families of genus 3 cyclic covers of degree $4$ and $6$, branched over four points of $\mathbb{P}^1$. As a side product of our study, we provide an explicit description of the Jacobians and the Prym-Torelli images of these two families, together with a description of the corresponding flat surfaces.
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