Non-Existence and Finiteness Results for Teichmüller Curves in Prym Loci

Abstract

The minimal stratum in Prym loci have been the first source of infinitely many primitive, but not algebraically primitive Teichmüller curves. We show that the stratum Prym(2,1,1) contains no such Teichmüller curve and the stratum Prym(2,2) at most 92 such Teichmüller curves.This complements the recent progress establishing general -but non-effective -methods to prove finiteness results for Teichmüller curves and serves as proof of concept how to use the torsion condition in the non-algebraically primitive case.

There are no primitive Teichmüller curves in Prym(2,2)

“…In this section we review, following Lanneau-Möller [4], how to construct the 92 candidate surfaces that could generate a Teichmüller curve.…”

“…Teichmüller curves are closed GL(2, R)-orbits in the moduli space of translation surfaces ΩM g that descend to isometrically-immersed algebraic curves in the moduli space of genus g Riemann surfaces M g . Lanneau-Möller [4] searched for geometrically primitive Teichmüller curves, i.e., those not arising from a covering construction, in a certain locus Prym(2, 2) of genus 3 translation surfaces with two cone-points called Prym eigenforms. In the spirit of McMullen's proof that the decagon is the unique primitive Teichmüller curve in ΩM 2 (1, 1) (see [5,Theorem 6.3]), they reduced their search to considering whether any of 92 candidate Prym eigenforms generate a Teichmüller curve.…”

“…Classifying the translation surfaces that generate Teichmüller curves, called Veech surfaces, is a central problem in Teichmüller dynamics. See Hubert-Schmidt [3] for an introduction to Veech surfaces, Möller [8] for a list of known examples of Teichmüller curves, Lanneau-Möller [4] for a history of classification results, and McMullen [7] for a general survey.…”

“…At another level, Lanneau-Möller [4] began searching for Teichmüller curves among translation surfaces having a certain Prym involution that generalizes the hyperelliptic involution in genus 2. (See Lanneau-Möller [4] for the definition.)…”

“…At another level, Lanneau-Möller [4] began searching for Teichmüller curves among translation surfaces having a certain Prym involution that generalizes the hyperelliptic involution in genus 2. (See Lanneau-Möller [4] for the definition.) They showed that there are no geometrically primitive Teichmüller curves in Prym(2, 1, 1) and initiated the proof for Prym(2, 2) that we finish in this note.…”