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The number of contact primes of the canonical curve of genus p

Published online by Cambridge University Press:  24 October 2008

W. G. Welchman
Affiliation:
Sidney Sussex College

Extract

1. It is known, from the theory of the Riemann theta-functions, that the canonical series of a general curve of genus p has 2P−1 (2P − 1) sets which consist of p − 1. points each counted twice. Taking as projective model of the curve the canonical curve of order 2p − 2 in space of p− 1 dimensions, whose canonical series is given by the intersection of primes, we have the number of contact primes of the curve. The 28 bitangents of a plane quartic curve, the canonical curve of genus 3, have been studied in detail since the days of Plücker. The number, 120, of tritangent planes of the sextic curve of intersection of a quadric and a cubic surface, the canonical curve of genus 4, has been obtained directly by correspondence arguments by Enriques. Enriques also remarks that the general formula 2P−1 (2p −1) is a special case of the formula of de Jonquières, which was proved, by correspondence methods, by Torelli§.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1930

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References

* See, for instance, Baker, Abelian Functions (1897), Chap. X.

Enriques, Teoria geometrica delle equazioni e delle funzioni algebriche, Vol. III (1924), p. 469.Google Scholar

Enriques, ibid. p. 395.

§ Torelli, R., Rend. Circolo Mat. di Palermo, 21 (1906), 5865.CrossRefGoogle Scholar

| Severi, , Geometria algebrica, I, 1 (1926), p. 233.Google Scholar