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On the theorem of Riemann-Roch

Published online by Cambridge University Press:  24 October 2008

Edwin A. Maxwell
Edwin A. Maxwell
Affiliation:
Queens' College
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Extract

The theory of surfaces and of manifolds of higher dimension is greatly handicapped by the lack of exact information concerning the extension of the theorem of Riemann-Roch for curves. Allied to this is the difficulty of finding an expression for the arithmetic genus Pa of primals in terms of their defining elements. This paper seeks to examine certain particular applications of the theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 33 , Issue 1 , March 1937 , pp. 26 - 34
Copyright
Copyright © Cambridge Philosophical Society 1937

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References

* See, for example Todd, J. A., Journal Lond. Math. Soc. 9 (1934), 206.CrossRefGoogle Scholar

† E.g. Enriques, F., Lezioni sulla teoria delle superficie algebriche (Padova, 1932), p. 264.Google Scholar

‡ Baker, H. F., Proc. Lond. Math. Soc. (2), 12 (1912), 1–40,Google Scholar (19).

§ Severi, F., Rend. di Palermo, 28 (1909), 33–87.CrossRefGoogle Scholar

* Maxwell, E. A., Proc. Camb. Phil. Soc. 32 (1936), 185.CrossRefGoogle Scholar

† See Zariski, O., Algebraic surfaces (Berlin, 1936), 63–71.Google Scholar

‡ Segre, B., Mém. Acad. Royale de Belgique (2), 14 (1936).Google Scholar In particular, pp. 72–98.

§ If F = 0 is the given surface and X = 0 that of the surface of order x, then any surface through their intersection is PF + QX = 0, which meets F = 0 in X = 0 together with Q = 0.

* See, for example, Zariski, op. cit. p. 69.

† There are three excellent summaries available which give an account of the general ideas, each with detailed bibliography: Rosenblatt, A., Atti del congresso internazionale dei matematici, 4 (Bologna, 1928), 93–113;Google ScholarLefschetz, S., Géométrie sur les surfaces et les variétés algébriques (Paris, 1929), pp. 42–47;Google ScholarGodeaux, L., Questions non-résolues de géométrie algébrique (Paris, 1933).Google Scholar

* Op. cit. p. 72.

† See Zariski, op. cit. p. 68.

* If there were a of the system, not a prime section, it would give rise to a curve section of order k and genus ½(k − 1)(k − 2) in space of three dimensions, which is impossible. Hence each member is a prime section, so that r = n.

* It is clear from previous work that P g − P a > 0 here.

† Op. cit. p. 87.

* See Zariski, op. cit. p. 64.

† Todd, op. cit. p. 210. His p a is, of course, the number which we have denoted above by p 2.

‡ Todd, op. cit. p. 209, with N = n − 6.