The singular fibre of the Hitchin map

TL;DR

This paper characterizes the singular fiber of the Hitchin map for rank 2 Higgs bundles with fixed determinant on a Riemann surface, especially when the spectral curve has an A_{m-1} singularity, proving its connectedness.

Contribution

It provides a detailed description of the singular fiber of the Hitchin map for spectral curves with A_{m-1} singularities, establishing its connectedness.

Findings

The singular fiber is explicitly described for A_{m-1} singularities.

The singular fiber is proven to be connected.

The work extends understanding of the Hitchin fibration's fibers in singular cases.

Abstract

Given any line bundle L of positive degree, on a compact Riemann surface, let $M_L^\Lambda$ be the moduli space of L-twisted Higgs pairs of rank 2 with fixed determinant isomorphic to $\Lambda$ and traceless Higgs field. We give a description of the singular fibre of the Hitchin map $H:M^L_\Lambda\to H^0(L^2)$, when the corresponding spectral curve has any singularity of type $A_{m-1}$. In particular, we prove directly that this fibre is connected.