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I read that Riemann started studying the so-called Riemann's surfaces in the second half of the 19th century, introducing tools like meromorphic functions and meromorphic 1-forms. The culmination of that theory can be considered the Riemann-Roch theorem, quantifying the possible meromorphic functions existing on a compact Riemann surface with some constraints expressed in terms of divisors.

Today, most Algebraic Geometry books introduce varieties defined over an arbitrary field. Compact Riemann surfaces, if mentioned, appear as a particular case $K = \mathbb{C}$.

My main question here is stated in the title, but here are related sub-questions that could guide the answer:

  • Who first started to define and study algebraic varieties and were those people aware of the results about Riemann surfaces?
  • Was the equivalence between compact Riemann surfaces and projective algebraic curves over $\mathbb{C}$ already known, or suspected, then?
  • How did people come up with the idea of using things like 1-forms to study varieties over an arbitrary field?
  • Also, with some knowledge of logic, one can justify that some interesting results over $\mathbb{C}$ might be proven algebraically (knowing that the theory of algebraically closed fields of characteristic 0 is complete). Was that known/influential when people started studying Algebraic Geometry?
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Hermann Weyl's book on Riemann surfaces was instrumental in the development of the Riemann-Roch theorem and its generalisations. The relevant chapter in Weyl's book is chapter 17: The single-valued functions on F (Riemann surface) as a subclass of additive and multiplicative functions on F^ ("Klassenfläche''). Riemann-Roch's theorem (see https://mathscinet.ams.org/mathscinet-getitem?mr=69903). The generalisations include the Hirzebruch-Riemann-Roch theorem https://en.wikipedia.org/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem, a result of central importance in algebraic geometry. Notably, Felix Klein took an active interest in the preparation of Weyl's book on Riemann surfaces, corroborating his exceptional modernism. Note that Weyl was Klein's mathematical great-grandson; see https://www.mathgenealogy.org/id.php?id=7373

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  • $\begingroup$ Unfortunately I don't have access to the document in your first link. I understand Weyl formalised results on Riemann surfaces. Who was the first to adapt some of the concepts — functions field, 1-forms — to study curves over general fields? $\endgroup$
    – Weier
    Commented Jun 20, 2023 at 9:19
  • $\begingroup$ The review includes a table of contents of Weyl's book. Here is the relevant part:: "(17) The single-valued functions on F (Riemann surface) as a subclass of additive and multiplicative functions on F^ (``Klassenfläche''). Riemann-Roch's theorem. (18) Abel's theorem. (19) The algebraic function field. (20) Uniformization. (21) Riemann surface and noneuclidian geometry. Fundamental regions. Poincaré's θ-series. (22) Conformal self-mapping of a Riemann surface." Note that in your Title, you asked a more general question. $\endgroup$
    – Mikhail Katz
    Commented Jun 20, 2023 at 9:23
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Algebraic geometry is much older than Riemann surfaces. Algebraic curves (of second degree, over the field of real numbers) where studied in great detail by Apollonius in 2 BC, and algebraic varieties (not only curves but of higher dimension!) occur in the book of Diophantus in 2 AD. The idea of rational parametrization of these varieties is due to him.

At the time when Riemann wrote his thesis, algebraic geometry was already a well developed subject, with many deep results (over the fields of real, complex and rational numbers). Some examples of these deep results are Newton's classification of cubics over the real field, the so-called Puiseux series (really discovered by Newton, Bezout's theorem, Poncelet's theorem, and much more.

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  • $\begingroup$ I understand algebraic curves where studied before, but when did people start studying the field of functions on a curve, or the 1-forms for instance? Was Riemann's surface theory influential regarding these? $\endgroup$
    – Weier
    Commented Jun 20, 2023 at 7:53
  • $\begingroup$ @Weier The function field is also in Weyl's book in chapter 19 (see my answer hsm.stackexchange.com/a/15521/604). $\endgroup$
    – Mikhail Katz
    Commented Jun 20, 2023 at 8:58
  • $\begingroup$ Alexandre, historians are generally not so thrilled about attributing modern concepts in algebraic geometry to classical authors such as Apollonius and Diophantus. Those authors certainly had marvellous insights, but extreme care has to be taken here to avoid presentism. $\endgroup$
    – Mikhail Katz
    Commented Jun 20, 2023 at 9:01
  • $\begingroup$ @Mikhail Katz: there are various historian and their opinions differ. See, for example the work of Izabella Bashmakova who was the main specialist on Diophantus. Few other historians had sufficient background in mathematics and really read Diophantus. $\endgroup$
    – Alexandre Eremenko
    Commented Jun 20, 2023 at 12:07
  • $\begingroup$ Yes, I am familiar with Bashmakova's work, but she presents such claim with great care in an effort to avoid presentism, and stays away from sweeping claims. I happen to be sympathetic to her point of view, but note that this is a minority opinion. One can detect germs of certain ideas that blossomed later on (in the 19th and 20th centuries), but this is very far from the claim that algebraic geometry was done by Apollonius, sorry. $\endgroup$
    – Mikhail Katz
    Commented Jun 20, 2023 at 12:19

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