For $G = GL\_n$, it is known that the generic fibers of the Hitchin fibration are the Picard stacks of line bundles on the corresponding spectral curves and the duality of Hitchin fibrations in this case amounts to the self-duality of the Jacobian of an algebraic curve. Note however that these statements are valid on an open subset of the Hitchin base. My question is about this open subset, See S4.2, S4.4 and Cor 4.5 of Bezrukavnikov-Braverman for an overview (and all notation that is used in the next paragraph): <https://arxiv.org/pdf/math/0602255.pdf>


Prop 4.3 of that paper states that there is a non-empty open subset $Hitch\_n^0$ of $Hitch\_n$ over which $pr\_1$ is smooth. How can we explicitly write down the open subset $Hitch\_n^0$ in this context? The proof of Prop 4.3 is unclear to me at this stage, so any clarifications would be appreciated (in particular, I'm looking for an explicit description of this open subset). Note that $Hitch\_n$ is explicitly defined in S4.1.



Duality of Hitchin fibrations in type A
The relevant open subset is simply the open subset where the spectral curve is smooth. Thus it is the nonvanishing locus of some discriminant polynomial.


The Hitchin fiber is the moduli space of bundles on the spectral curve, generically free of rank one, whose pushforward to the base curve is locally free of rank $n$.


For a smooth curve, the only way a generically free bundle can fail to be locally free is if it contains torsion, in which case its pushforward is never locally free, so the Hitchin fiber is simply the space of line bundles on the spectral curve generically free of rank one, i.e. the Jacobian, and thus is smooth.


However, for a non-smooth curves, one will need to compactify the Jacobian to get the Hitchin fiber, and this will introduce singularities.


