The singular fibre of the Hitchin map

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