On holomorphic sections of Veech holomorphic families of Riemann surfaces

TL;DR

This paper establishes upper bounds on the number of holomorphic sections in Veech holomorphic families of Riemann surfaces, linking topological types and Veech group properties to geometric decompositions.

Contribution

It provides the first explicit bounds on holomorphic sections based on topological and group-theoretic data, connecting Veech groups with cylinder decompositions.

Findings

Upper bounds depend only on topological types

Relation established between Veech groups and cylinder moduli

Results contribute to understanding the structure of Veech families

Abstract

We give upper bounds of the numbers of holomorphic sections of Veech holomorphic families of Riemann surfaces. The numbers depend only on the topological types of base Riemann surfaces and fibers. We also show a relation between types of Veech groups and moduli of cylinder decompositions of flat surfaces.