Teichm\"uller curves in hyperelliptic components of meromorphic strata

TL;DR

This paper classifies Teichmüller curves in hyperelliptic components of meromorphic strata, showing that outside specific cases, such curves cannot be algebraic due to boundary intersection constraints.

Contribution

It provides a complete classification of Teichmüller curves in these hyperelliptic components, introducing a non-existence criterion based on boundary intersection analysis.

Findings

Non-existence of algebraic Teichmüller curves outside specific Hurwitz covers

Complete classification of Teichmüller curves in hyperelliptic components

Boundary intersection criterion as a key tool

Abstract

We provide a complete classification of Teichm\"uller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichm\"uller curves intersect the boundary of the moduli space we derive a contradiction to the algebraicity of any candidate outside of Hurwitz covers of strata with projective dimension one and Hurwitz covers of zero residue loci in strata with projective dimension two.