I'm doing my graduation thesis in algebraic geometry and I intend it prove Riemann Roch for curves at the end. I thought that I would like to expand it to include the general theorem for surfaces or even for general varieties.
My approach in the thesis does not include schemes. I only work over varieties (affine, projective, quasi-affine and quasi-projective).
I know some good resources where I can find the proof in case of surfaces (e.g. Fulton).
Now, I look for some resources the provide the general theorem (either for surfaces or for general varieties) and prove them so that I can understand the proof.
My only requirement is that the treatment does not use scheme-theoretic techniques (but using sheaf etc ... is welcome) since that would require me to change everything I've done so far, which would not be reasonable in the remaining time.
As I've been searching through the web, I noted that some references deal with the case of surfaces only over complex numbers (i.e. Riemann Surfaces) and restrict their treatment to that case and use things from complex geometry. Other references deal with the relative version due to Grothendick using schemes.
I look for something in between. That is, I want a reference that deals with the theorem in case of a surface over any algebraically closed field, or even better, the theorem in case of a variety over any algebraically closed field.
Can anyone point some resources (notes, books ... etc) that satisfy those requirements?
I've skimmed Zaraski book on algebraic surfaces from the thirties but it seems to be using techniques of linear systems, which seems outdated and are due to the Italian school. I'm aware of the text book "Topological methods in algebraic geometry" by Hirzebruch. But I'm not sure if its approach coincides with my goals. Would I need to read all of it? Can I skip something If I'm only interested in Riemann-Roch? if yes, which?