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arXiv · 1502.05381

Orbifold points on Prym-Teichmüller curves in genus three

David Torres-Teigell·Jonathan Zachhuber
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Abstract

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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David Torres-Teigell, Jonathan Zachhuber. 2015-11-24. Orbifold points on Prym-Teichmüller curves in genus three. https://doi.org/10.1093/imrn%2Frnw277

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