Hurwitz equivalence of braid monodromies and extremal elliptic surfaces

TL;DR

This paper establishes a connection between ribbon graphs and modular group subgroups to generate large families of non-Hurwitz equivalent braid monodromies, leading to many topologically distinct algebraic surfaces.

Contribution

It introduces a method to construct exponentially many non-Hurwitz equivalent braid factorizations and applies this to produce numerous distinct extremal elliptic surfaces and related objects.

Findings

Constructed exponentially large families of non-Hurwitz equivalent braid factorizations.

Generated exponentially many topologically distinct extremal elliptic surfaces.

Linked ribbon graphs with modular group subgroups to classify algebraic objects.

Abstract

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $\Gamma$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\it topologically} distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces.