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Some Quartic Primals and Associated Cremona Transformations of Four-Dimensional Space

Published online by Cambridge University Press:  24 October 2008

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

The object of this paper is to draw attention to a series of five types of rational quartic primal in [4]. Two of these are already known. The greater part of this paper deals with one of the other three types of primal and with a new symmetrical quarto-quartic Cremona transformation of [4] determined by a homaloidal system of primals of this type passing through a certain surface of order nine. It is hoped in a subsequent paper to continue the investigation into the remaining two types of primal.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 1 , January 1936 , pp. 12 - 22
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

* Todd, J. A., Proc. Camb. Phil. Soc. 29 (1933), 52–68.CrossRefGoogle Scholar

* Todd, J. A., Proc. Camb. Phil. Soc. 26 (1930), 323–33.CrossRefGoogle Scholar

* Severi, F., Atti di Torino, 36 (1901), 74–93.Google Scholar For the actual expression quoted here see Pascal, E., Repertorium der höheren Mathematik, II, 2 (1922), 910.Google Scholar

† Room, T. G., Proc. Camb. Phil. Soc. 27 (1931), 518–37.CrossRefGoogle Scholar

‡ The ∞1 lines on a quartic primal in [4] form a scroll of order 320, Encyk. der Math. Wiss. III C 7, 937.Google Scholar

* Clebsch, A., Math. Ann. 1 (1869), 253–316.CrossRefGoogle Scholar

† For this and other formulae quoted see Baker, H. F., Principles of Geometry (Cambridge), 6 (1933)Google Scholar, Ch. iv.

‡ H. F. Baker, loc. cit. pp. 263–5.

§ Each double conic of F 15 is accidental in that it consists of two distinct conics of the surface which are coincident in space. The two conics are exceptional in the Nöther sense, being transformable into simple points of F 9, one on each sheet at an accidental double point.

* The postulation to primals of sufficiently high order ρ of a surface of order μ0 in [4] which has numerical genus p n and sectional genus p is ½μ0ρ(ρ+1)−ρ(p−1)+p n+1−d, where d is the number of accidental double points of the surface; see H. F. Baker, loc. cit. p. 263.

† Nöther, M., Annali di Mat. (2), 5 (1871), 163–71.CrossRefGoogle Scholar

‡ This admits of an easy analytical verification by considering the case of a double line meeting a quadruple line.

§ H. F. Baker, loc. cit. pp. 257–63.

* On a general quintic surface f 1 with a double twisted cubic γ1 there are no twisted cubics which meet γ1 in three points only. If there is such a cubic it is met again in three points by a quadric through γ1 and therefore its plane representation is by a nodal cubic. γ2, γ3 must therefore be represented in the way shown.