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ORBIFOLD POINTS ON PRYM–TEICHMÜLLER CURVES IN GENUS $4$

Part of: Curves Differential topology Special properties

Published online by Cambridge University Press:  22 May 2017

David Torres-Teigell  and
Jonathan Zachhuber
David Torres-Teigell
Affiliation:
Fachrichtung Mathematik, Universität des Saarlandes, Campus E24, 66123 Saarbrücken, Germany (torres@math.uni-sb.de)
Jonathan Zachhuber
Affiliation:
FB 12 – Institut für Mathematik, Johann Wolfgang Goethe-Universität, Robert-Mayer-Str. 6–8, D-60325 Frankfurt am Main, Germany (zachhuber@math.uni-frankfurt.de)
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Abstract

For each discriminant $D>1$, McMullen constructed the Prym–Teichmüller curves $W_{D}(4)$ and $W_{D}(6)$ in ${\mathcal{M}}_{3}$ and ${\mathcal{M}}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_{D}(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau–Nguyen and Möller, complete the topological characterisation of all Prym–Teichmüller curves and determine their genus. The study of orbifold points relies on the analysis of intersections of $W_{D}(6)$ with certain families of genus $4$ curves with extra automorphisms. As a side product of this study, we give an explicit construction of such families and describe their Prym–Torelli images, which turn out to be isomorphic to certain products of elliptic curves. We also give a geometric description of the flat surfaces associated to these families and describe the asymptotics of the genus of $W_{D}(6)$ for large $D$.

Keywords

flat surfacesTeichmüller curvesorbifold pointscyclic covers

MSC classification

Primary: 14H10: Families, moduli (algebraic)
Secondary: 14H40: Jacobians, Prym varieties 14H52: Elliptic curves 57R18: Topology and geometry of orbifolds 54F65: Topological characterizations of particular spaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 4 , July 2019 , pp. 673 - 706
Copyright
© Cambridge University Press 2017 

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