Abstract
Since the beginnings of the development of the theory of algebraic surfaces, geometers have tried to prove a generalization of the Riemann–Roch theorem for algebraic curves (see Zariski’s account in [195, Chap. IV], as well as Gray ’s article [84]). Max Noether formulated such a generalization in his 1886 paper [144]. But he could only prove an inequality (I use the notations and the explanations from [32, Chaps. 7 and 35])
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Notes
- 1.
This means that |C| is contained in the canonical system |K|.
- 2.
This is the arithmetic genus of the curve C, which may be computed using the adjunction formula stated in Theorem 37.2.
- 3.
This is the number of varying intersection points of two general curves of the system, that is, the number of such points which are not base points of the system.
- 4.
A little later, it became usual to denote this group rather by \(H^{1}(\mathcal{O}_{X}(C))\), if X is the ambient surface. Here \(\mathcal{O}_{X}(C)\) is the now standard notation for the sheaf denoted by F C in Serre’s letter to Borel.
References
G. Castelnuovo, F. Enriques, Sur quelques récents résultats dans la théorie des surfaces algébriques. Math. Ann. 48, 241–316 (1897)
J. Gray, The Riemann–Roch theorem and geometry, 1854–1914, in Proceedings of the International Congress of Mathematicians, vol. III (Documenta Mathematica, Bielefeld, 1998), pp. 811–822
M. Noether, Extension du théorème de Riemann–Roch aux surfaces algébriques. C.R. Acad. Sci. Paris 103, 734–737 (1886)
J.-P. Serre, Un théorème de dualité. Commun. Math. Helv. 29 (1), 9–26 (1955). Republished in Œuvres I (Springer, New York, 2003), pp. 292–309
J.-P. Serre, Lettre à Armand Borel du 16 avril 1953, in Œuvres I (Springer, New York, 2003), pp. 243–250
O. Zariski, Algebraic Surfaces (Springer, Berlin, Heidelberg, 1935). Reprinted in 1971 with appendices by S.S. Abhyankar, J. Lipman and D. Mumford
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Popescu-Pampu, P. (2016). Serre and the Riemann–Roch Problem. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_44
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