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Multiple canonical surfaces

Published online by Cambridge University Press:  24 October 2008

D. W. Babbage
D. W. Babbage
Affiliation:
Magdalene College
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Extract

1. If the canonical series of an algebraic curve of genus p is compounded of an involution of sets of points on the curve then the involution must be rational and of order two, and the canonical model is a repeated rational normal curve of order p − 1 with 2p + 2 branch points. An analogous question suggests itself for algebraic surfaces. Under what conditions is the canonical model of a surface, whose geometric genus is not less than two, a multiple surface, and what in this case are the properties of the simple surface on which the multiple surface is based? In this paper we give particular examples of multiple canonical surfaces and attempt to go some way towards the solution of the general problem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 3 , July 1934 , pp. 297 - 308
Copyright
Copyright © Cambridge Philosophical Society 1934

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References

† If the quadrics are put in correspondence with the planes of a solid S, then the sextic surface corresponds to a general cubic surface in S.

‡ Enriques, , Lezioni sulla teoria delle superficie algebriche (Padova, 1932), p. 346.Google Scholar

† Severi, , “Sulle relazioni che legano i caratteri invarianti di due superficie in corrispondenza algebrica”, Rend. Ist. Lomb. (2), 36 (1903), 495–511.Google Scholar

‡ Enriques, loc. cit. § 60.

† Supposing this system to exist, to be irreducible, and to have freedom at least two.

‡ Hurwitz, , “Ueber Riemann'sche Flächenmitgegebenen Verzweigungspunkten”, Math. Ann. 39 (1891), 1–61 (57)CrossRefGoogle Scholar; Math. Werke (Basel, 1932), 1, 379.Google Scholar

† Dr Du Val has given a topological proof of this suggestion (Proc. Cambridge Phil. Soc. 30 (1934), 309)Google Scholar. I am also greatly indebted to him for some general discussion and enlightenment on these matters.

† ψ may be a multiple surface, and it may also in particular close up into a curve or a point.

† Castelnuovo, , “Sulle superficie aventi il genere aritmetico negativo”, Rend. di Palermo, 20 (1905), 55–60CrossRefGoogle Scholar. For these surfaces p α = −1.

† If the branch surface of the double threefold is, in non-homogeneous coordinates, f (x, y, z) = 0, the double threefold may be regarded as the projection of the primal t 2=f(x, y, z) in [4]: and any double surface lying on the double threefold is transformable into the section of this primal by a cone.

† Although it has not been established that F 48* can always be obtained as a partial quadric section of V 3 it is sufficient for our purpose that it behaves as though it were thus obtained.

† Castelnuovo, , “Osservazioni intorno alla geometria sopra una superficie, Nota. II”, Rend. Ist. Lomb. (2), 24 (1891), 307–318.Google Scholar

‡ Cf. Enriques, loc. cit. p. 347.

§ Val, Du, Journal Lond. Math. Soc. 8 (1933), 11–18.CrossRefGoogle Scholar