Orbits of Theta Characteristics
Abstract
The theta characteristics on a Riemann surface are permuted by the induced action of the automorphism group, with the orbit structure being important for the geometry of the curve and associated manifolds. We describe two new methods for advancing the understanding of these orbits, generalising existing results of Kallel & Sjerve, allowing us to establish the existence of infinitely many curves possessing a unique invariant characteristic as well as determine the number of invariant characteristics for all Hurwitz curves with simple automorphism group. In addition, we compute orbit decompositions for a substantial number of curves with genus $\leq 9$, allowing the identification of where current theoretical understanding falls short and the potential applications of machine learning techniques.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.09890
 arXiv:
 arXiv:2404.09890
 Bibcode:
 2024arXiv240409890B
 Keywords:

 Mathematics  Algebraic Geometry;
 14H37;
 14Q05;
 14H60;
 20J06
 EPrint:
 32 pages