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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:

  1. What chapters of the book should I go through in order to fully understand and prove the theorem? Or, is all of it necessary?
  2. 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!

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    $\begingroup$ There is a nice self-contained adelic proof probably due to Tate but written up well by Belmans anagrams-seminar.github.io/grothendieck-duality/lecture-1.pdf $\endgroup$
    – Pulcinella
    Commented Aug 1, 2022 at 15:30
  • $\begingroup$ @Pulcinella Unfortunately I am not advanced enough for Serre duality, I was looking for something a bit more elementary. Thank you anyway! $\endgroup$
    – J.Spil
    Commented Aug 1, 2022 at 15:36
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    $\begingroup$ I don't think you are going to find a reasonable route to Riemann-Roch that doesn't go through Serre Duality (for curves). We had a previous question about different approaches to RR here mathoverflow.net/questions/253090 . I'll also promote some notes that I am proud of math.lsa.umich.edu/~speyer/631_2014/RiemannRoch.pdf . (Not posting this as an answer because it doesn't address the question about Fulton's book.) $\endgroup$
    – David E Speyer
    Commented Aug 1, 2022 at 17:10
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    $\begingroup$ @SamHopkins If that is a response to me, the adelic approach definitely proves Serre duality. The observation that $H^1(C, \mathcal{O}(D))$ can be computed adelically is Prop 17 in the notes Pulcinella linked above, and Serre duality (for curves) is proved at the end of section 2.2. $\endgroup$
    – David E Speyer
    Commented Aug 1, 2022 at 19:20
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    $\begingroup$ To answer question 1, at least assuming you come with no prior experience in AG, I would say pretty much all of it is necessary. The book is quite streamlined to cover specifically those topics which are needed for the elementary proof of RR presented. $\endgroup$
    – Wojowu
    Commented Aug 3, 2022 at 11:31

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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.

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  • $\begingroup$ I second Lang's proof as a starting point. My impression is that it is similar to the adelic proof, but it uses Weil's "ring of repartitions", which is essentially adeles without completions. As such it can be quicker to grasp, avoiding some of the technical overhead which accompanies the adeles. $\endgroup$
    – PrimeRibeyeDeal
    Commented Sep 2, 2022 at 17:32

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