  We prove sharp bounds on the discriminants of stable rank two sheaves on
surfaces in three-dimensional projective space. The key technical ingredient is
to study them as torsion sheaves in projective space via tilt stability in the
derived category. We then proceed to describe the surface itself as a moduli
space of rank two vector bundles on it. Lastly, we give a proof of the
Bogomolov inequality for semistable rank two sheaves on integral surfaces in
three-dimensional projective space in all characteristics.
