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On a family of constructs in higher space

Published online by Cambridge University Press:  24 October 2008

C. G. F. James
C. G. F. James
Affiliation:
Trinity College
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Extract

The constructs in question are represented by the vanishing of all determinants of two rows and columns drawn from a matrix of r + 1 rows and s + 1 columns, where r can be taken less than or equal to s. They are thus the multiple constructs of highest dimension on the well-known family of constructs generated by projective systems of linear spaces. In the present note the constructs are considered from a somewhat different point of view, the principal applications being to the determination of directrix loci of minimum order for the systems of spaces which they contain, and to the re-proof and extension of a known theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 23 , Issue 1 , February 1926 , pp. 50 - 67
Copyright
Copyright © Cambridge Philosophical Society 1926

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References

* Cf. Richmond, H. W. and Bath, F., Proc. Camb. Phil. Soc. vol. 22 (1924), §6(r = 1).Google Scholar

† [a] denotes a linear space of dimension a.Google Scholar

‡ For example the present and examples quoted in “Complexes of conies and the Weddle surface,” Proc. Camb. Phil. Soc. vol. 22 (1924), p. 5 (small print). Such constructs may be referred to as section-normal. In the paper quoted this property was proved only for r equal to unity, but from the present construction the extension to any value of r is immediate.Google Scholar

* Mat. Ann. vol. 26, p. 56.Google Scholar

† It is readily seen that each [j + 1] arises once only in this construction.Google Scholar

* See Bertini, , Geometria proiettiva degli spazi…, Ed. 2, Messina 1923, chap. 14, p. 538.Google Scholar

† [x] as a numerical symbol denotes as usual the greatest integer contained in the positive number x.Google Scholar

‡ See Severi, , “Sopra alcune singolarita delle curve…,” Mem. Torino (2), vol. 51 (1902), p. 81.Google Scholar

* This formula is not new, but the present proof I believe is. See Segre, C., “Gli ordine delle varietà…,” Atti Lincei (5), vol. 9, ii (1900), p. 253Google Scholar

* The theorem as now worded summarises the results of my paper, “Extensions of a theorem of Segre's and their natural position in space of seven dimensions,” Proc. Camb. Phil. Soc. vol. 21 (1923), pp. 664–84. See also Id. vol. 22, p. 24, for references.Google Scholar

† Generally R(ab)a denotes a regulus whose lines belong to a system each line of which is named “a,” and the directrices belong to a system ”b,“ the various symbols R, a and b being possibly qualified by accents, suffices, etc. Q (ab) denotes the quadric base of these reguliGoogle Scholar.

† The system is thus represented on the tangent planes of a quadric, and therefore also on the points of a quadric such that those through a point are represented by points on a line of one regulus. This result, read off from the equation of the construct, was the starting-point of my other method of proof. “Extensions, etc.” loc. cit.Google Scholar

‡ Loc. cit.Google Scholar

* Cf. James, , loc. cit.Google Scholar

* By the lemma, § 8, namely in the first place .Google Scholar

† Cf. James, , “Extensions…,” loc. cit. § 15.Google Scholar

* Of order namely here . See Segre, loc. cit.