Tangency and a ruled surface associated with a Hitchin system

TL;DR

This paper extends elliptic soliton theory to Hitchin systems by constructing a ruled surface and generalizing tangency conditions, revealing new relations between coverings and divisor singularities.

Contribution

It introduces a novel geometric framework for Hitchin systems, generalizing elliptic soliton concepts through ruled surfaces and tangency conditions.

Findings

Calculated the dimension of the moduli space of Hitchin covers with tangency conditions

Established a relation between coverings and divisor singularities

Generalized elliptic soliton theory to Hitchin systems

Abstract

We will generalize the Treibich-Verdier theory about elliptic solitons to a Hitchin system by constructing a particular ruled surface and we will propose a generalization of a tangency condition associated with elliptic solitons to a Hitchin system. In particular, we will calculate the dimension of the moduli space of Hitchin covers satisfying the tangency condition. With this new point of view, we will see a subtle relation between the characterizations of coverings and the singularities of divisors in a particular algebraic surface.