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The number of apparent double points of certain loci

Published online by Cambridge University Press:  24 October 2008

W. L. Edge
W. L. Edge
Affiliation:
Trinity College
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Extract

1. A sketch of the first part of this paper, together with some of the second part, was first written out in October last, shortly after the publication of Mr Babbage's paper, A series of rational loci with one apparent double point. It is well known that the Del Pezzo quintic surface the only non-ruled quintic surface in [5], has one apparent double point; Babbage shows that this surface is a member of a series of loci in [2n + 1], each of which has one apparent double point; he establishes the existence of these loci from their representations on flat spaces ∑n, these representations being analogous to the plane representation of by means of cubic curves through four fixed points.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 3 , July 1932 , pp. 285 - 299
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

* Babbage, , Proc. Camb. Phil. Soc., 27 (1931), 399.CrossRefGoogle Scholar

† Severi, , Palermo Rendiconti, 15 (1901), 33–51 (44).CrossRefGoogle Scholar

* Baker, , “Note in regard to surfaces in space of four dimensions, in particular rational surfaces”, Proc. Camb. Phil. Soc., 28 (1932), 62–82.CrossRefGoogle Scholar

† Mr Babbage has written a note on this second series of loci , the manuscript of which he has shown to me. He uses an induction argument to prove that has one apparent double point.

* This locus is rational and represented on [3] by means of cubic surfaces through an elliptic quartic curve; it is studied from this point of view by Semple, , Proc. Camb. Phil. Soc., 25 (1929), 158CrossRefGoogle Scholar.

* Segre, , “Sulle rigate razionali in uno spazio lineare qualunque”, Atti Torino, 19 (1884), 355–372 (360).Google Scholar

* Cf. Segre, , “Sulle rigate razionali in uno spazio lineare qualunque”, Atti Torino, 19 (1884), 355–372 (371).Google Scholar

* The projection of this surface from a point on to [4] figures in a recent paper by Semple, , Proc. Lond. Math. Soc., 32 (1931), 388–9.Google Scholar

† The genus of a curve of order μ which lies on a cubic scroll in [4] and meets each generator in k points is . See, for example, Edge, , The theory of ruled surfaces (Cambridge 1931), 15.Google Scholar

* I is a relative invariant for birational transformation. If two surfaces are birationally equivalent there is in general a certain number of fundamental points on each surface, i.e. points to each of which there correspond, in the birational correspondence, all the points of a curve on the other surface. If the values of the Zeuthen-Segre invariant for the two surfaces are I and I′ while the numbers of fundamental points are f and f′ respectively, then I + f = I′ + f′.

* Baker, loc. cit., 76.

† The condition that a plane cubic should have a double point is that its discriminant should vanish; and this discriminant is of degree 12 in the coefficients.