  We calculate the Picard group of generic (very general) spectral varieties
living in the total space of a very ample line bundle over an algebraically
closed field $k$ of odd characteristics or characteristic 0. We follow the
strategy of Ravindra and Srinivas [RS06,RS09] via formal Picard groups. As an
application, we calculate the generic fibers of Hitchin systems over a smooth
quintic surface of Picard number 1.
