Picard Groups of Spectral Varieties and Moduli of Higgs Sheaves

We study moduli spaces of Higgs sheaves valued in line bundles and the associated Hitchin maps on surfaces. We first work out Picard groups of generic (very general) spectral varieties which holds for dimension of at least 2, i.e., a Noether--Lefschetz type theorem for spectral varieties. We then apply this to obtain a necessary and sufficient condition for the non-emptyness of generic Hitchin fibers for surfaces cases. Then we move on to detect the geometry of the moduli spaces of Higgs sheaves as the second Chern class varies.