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The surfaces whose prime-sections contain a

Published online by Cambridge University Press:  24 October 2008

J. Bronowski
J. Bronowski
Affiliation:
Jesus College
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Extract

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 2 , 30 April 1934 , pp. 170 - 177
Copyright
Copyright © Cambridge Philosophical Society 1934

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References

* Castelnuovo, G., Rend. di Palermo, 4 (1890), 73.CrossRefGoogle Scholar

* If υ is a prime number, then a pencil of elliptic curves of order υ on a rational surface is represented in the plane, either by a pencil R of cubics, or by a Halphen pencil of curves of order 3υ with nine υ-ple base-points (not in general position). In the present case, however, υ = 3, it is easily shown that the Halphen pencil gives rise to surfaces having a unode: they are in fact surfaces discussed by P. Du Val, Proc. London Math. Soc. (2), 35 (1933), 1. Hence the only non-singular surfaces F′ arise from the pencil R of cubics given in the text.

* Bronowski, J., Proc. Camb. Phil. Soc., 29 (1933), 482–3.Google Scholar

† Segre, C., Atti Ace. Torino, 21 (1885), 95.Google Scholar

* Palatini, F., Giorn. di mat., 38 (1900), 315.Google Scholar

* Enriques, F., Rend. Lincei (5), 23 (1914), 206, 291.Google Scholar

* The interesting surfaces F which have b = 1 have been discussed by Val, P. Du, Rend. Lincei (6), 15 (1932), 276.Google Scholar