We consider some more or less classical hypersurfaces in projective space,
known to be birational to a quotient of the unit ball in the corresponding
dimension by an arithmetic subgroup. We are interested in understanding the
intersection of each such hypersurface with its Hessian from the point of
view of arithmetic groups. In addition to unifying certain results found pr
eviously in the literature, we compute for four of these hypersurfaces the
Hessian as well as its intersection with the hypersurface.