Teichmüller curves and Hurwitz spaces
FacultiesFakultät für Mathematik und Wirtschaftswissenschaften
LicenseCC BY 3.0 Unported
A Teichmüller curve is a curve embedded in the moduli space of smooth projective curves of genus g which is totally geodesic for the Teichmüller metric. In this thesis we construct a new class of Teichmüller curves, using a characterisation due to Martin Möller. This involves constructing a suitable one-dimensional family of smooth projective curves parametrised by the points of a Teichmüller. We show that our new Teichmüller curves are the last Teichmüller curves in a larger class of Teichmüller curves constructed by Irene Bouw and Martin Möller. About candidates for further Teichmüller curves not much is known. A starting point may be the following observation. The points of a Teichmüller curve correspond to curves with real multiplication by large totally real number fields. Jordan Ellenberg constructs three one-dimensional families with this property. However, in this thesis we show that, except for some special cases, Ellenberg"s families do not define Teichmüller curves. To do this, we interpret them as families over suitable Hurwitz spaces. We then describe a criterion to check whether a family of curves does not define a Teichmüller curve by studying the boundary of the associated Hurwitz space and apply this criterion for exclusion to Ellenberg"s families. We moreover show how to modify Ellenberg"s families by passing to an adapted Hurwitz space in such a way that the criterion for exclusion no longer holds. It remains open whether this modification indeed produces Teichmüller curves.
Subject HeadingsAlgebraische Kurve [GND]