Approaching the Riemann-Roch Theorem for algebraic curves I am using "Algebraic Curves: An Introduction to Algebraic Geometry" by William Fulton as a guidline for approaching the Riemann-Roch Theorem for algebraic curves. I have two questions:

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*What chapters of the book should I go through in order to fully understand and prove the theorem? Or, is all of it necessary?

*Is there any other source that you would recommend for approaching the theorem?

I should also mention that I am a postgraduate student.
Thanks in advance!
 A: Several proofs are available.
If you are interested in a short algebraic proof, I can recommend S. Lang, Introduction to algebraic and Abelian functions, Addison-Wesley, Reading, MA, 1972. First 25 pages of this little book
give you a proof of the Riemann-Roch theorem. Prerequisite
is several chapters of Lang's Algebra, not too much, and he gives exact references to the places in Algebra that are needed.
This is a modern, algebraic proof, which goes back to Dedekind and Weber (their original article is also a good source, btw, there are several English translations).
Another proof, based on complex analysis, and which goes back to the original Riemann's proof is much simpler, given some basic properties of Abelian integrals. It can be found in Hurwitz-Courant, Funktionentheorie (unfortunately available only in German and Russian), or in the remarkable article by E. Ghys,
Six lecons autour des surfaces de Riemann, written on an intuitive level, similar to Riemann's own paper. This requires no prerequisite at all, begins with explanation of what is a conformal map, and its genesis in Cartography. It is freely available on E. Ghys web page.
