I am looking for an (advanced) book on algebraic curves that treats positive characteristic, something beyond Hartshorne Chapter IV. I am looking for material close to Geometry of Algebraic Curves by Arbarello et al, but where positive characteristic is also treated.
All algebraic curves books I have seen rely heavily on the fact that things are being done specifically over the complex numbers. (I am starting to think that maybe it just not true that the analogous results hold in positive characteristic?)