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A representation of [k]'s of [m] by points of [(m − k) (k + 1)]

Published online by Cambridge University Press:  24 October 2008

T. G. Room
T. G. Room
Affiliation:
St John's College
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Extract

The algebraical foundation of this representation is very simple: it depends on the theorem

If a coordinate system in [m] is given, then in a general [k] of the [m] a unique set of k + 1 points can be found whose coordinates form the scheme

where h = m − k − 1, and n = (m − k)(k + 1) = kh + m.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 3 , 30 July 1933 , pp. 331 - 346
Copyright
Copyright © Cambridge Philosophical Society 1933

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References

* Throughout this paper the Greek suffixes take the following ranges of values:

α = 0, …, m; β = 0, …, n; γ = 0, …, h − 1; δ = 0, …, h; ε = 0, …, k. a+hε is a single suffix; where double suffixes have to be used the two parts are separated by a comma, as e.g. z λ, μ in § 1.

† James, , Proc. Camb. Phil. Soc. 21 (1922), 664Google Scholar, where the representation is used without explicit statement of it in this form. Semple 1, Phil. Trans. Roy. Soc. A, 228 (1929), 350Google Scholar, where other references are given.

* Semple 2,Proc. London Math. Soc. (2), 30 (1930), 507.Google Scholar

† Babbage, , Proc. Camb. Phil. Soc. 28 (1932), 423CrossRefGoogle Scholar. In this paper the representation is considered as being one of planes of [4], rather than of lines, and, the representation being required only to establish a particular theorem, it is not discussed in detail. The writer is indebted to Mr Babbage for some suggestions in regard to this present paper.

‡ Babbage, loc. cit.

§ Chordal [k + 1] = [k + 1] meeting a manifold in k+2 points.

∥ These manifolds are discussed in Segre, , Encyk. d. Math. Wiss. iii C 7, p. 825Google Scholar, where references are given.

* Segre, loc. cit.

† When t = 1, i.e. for the normal rational curve in [q], the symbol r q is used.

* The symbol ]r[ denotes the linear ∞r family of primes dual to the family of points forming a [r], i.e. the family that passes through a [n − r − 1].

† Cf. Segre, loc. cit.

* A solution not for the λ's themselves, but for their ratios.

† I.e. R is a manifold R hn−h+1.

‡ If h > m − h + 1, i.e. if k < h − 2, this is a cone with a [h − k − 3]- vertex.

* If k < h, the last member of the series is the cone representing [k]'s which meet a given S k in a point, namely K (k) of order .

† If k < h, these “generators” more than fill the [n], i.e. the “cone” is not such in the usual sense.

* If the matrix contained general linear functions, the apparent dimension of this manifold would be n − k (h + i). This number is negative if k ≥ h, and i > 1, i.e. we have more equations than it is possible to solve, and there are no actual points of the manifold.

* This is easily verified from the parametrie form 3.1 for R.

† Segre, loc. cit., ch. viii.

‡ Segre, loc. cit., p. 729.

* Both cones are of type (| h + 1, k + 1|1, [n]). It should be noticed, however, that, although Γ is covered by the cones G in the same way as is the [n] by the cones K, this representation is not obtainable by a projection, since primes in [n] do not represent prime sections of Γ.

† Cf. Babbage, loc. cit., p. 421, who quotes the theorem from Brown, L. M., Journ. Lond. Math. Soc., 5 (1930), 168CrossRefGoogle Scholar. The process adopted in the present paper is to some extent a generalisation of that adopted by Brown in § 2 (p. 172).

‡ Semple 1.