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To give you some personal background: I am a ring theorist, and most of my research focus on invariant theory of noncommutative rings. Recently I became interested in a certain problem that requires a rather solid understanding of the birational classification of projective curves.

The most important thing about the introduction to the birational classification of curves I'm looking for is being short and/or concise, quick to absorb, and include a discussion of the Hurwitz-Riemann formula. If it is possible, I would prefer a source that: 1 - does not use unecessary heavy machinery from algebraic geometry and 2 - works over any algebraically closed field. About 2-, in fact, I'm okay with just algebraically closed fields of zero characteristic, since this is the case I need (but I don't know if restriction of the characteristic makes any signifficant difference in the classification).

I stress that 1- and 2- are optional. For instance, if the quickiest way to learn the subject is over $\mathbb{C}$ using some theory of Riemann surfaces, I am ok with that.

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    $\begingroup$ There are several books by Fulton, Miranda and others on algebraic curve theory which don't assume much background. The other comment is that the smooth projective curves are equivalent in a precise sense to function fields (fields of transcendence degree 1 over an alg. closed field). Some older books, such as by Chevalley, stress that point of view. Perhaps this might be more appealing to you, although some intuition gets lost. $\endgroup$
    – Donu Arapura
    Commented Nov 18, 2023 at 19:26
  • $\begingroup$ @DonuArapura thank you, I had completely forgotten Miranda's book. I actually like the function field point of view. I will have a look at chevalley's book $\endgroup$
    – jg1896
    Commented Nov 18, 2023 at 19:37
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    $\begingroup$ For curves it’s fairly simple. You can resolve all singulairites by taking the normalization and reduce to the smooth case. Then because any birational map of smooth projective curves extends to an isomorphism, you’re classifying projective curves up to isomorphism. All spelled out here: math.stackexchange.com/questions/190127/… . Saying much more requires more sophisticated tools. Riemann-Hurwitz is kind of incidental to this $\endgroup$
    – Vik78
    Commented Nov 19, 2023 at 2:31

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