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Some formulae for surfaces in higher space

Published online by Cambridge University Press:  24 October 2008

L. Roth
L. Roth
Affiliation:
Clare College
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Extract

The formulae of this paper are chiefly concerned with multiple secants and tangents to surfaces in spaces of dimension four to seven; they are obtained by the functional method of Cayley and Severi, which was first applied to scrolls by C. G. F. James. The majority of the results have already been given for scrolls in James's third paper; but whereas the functional equations in this case involve only two variables, those of the present paper contain four, and the work is considerably more complicated. It is also to be noted that, though the formulae for scrolls might perhaps have been expected to follow as particular examples from the corresponding formulae for surfaces, this is not always the case. The reason for the discrepancy is not always clear, although in some instances theoretical reasons for disagreement can be assigned.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 25 , Issue 4 , October 1929 , pp. 390 - 406
Copyright
Copyright © Cambridge Philosophical Society 1929

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References

† Trans. Camb. Phil. Soc. 23 (1925), p. 201;Google ScholarJournal Lond. Math. Soc. 1 (1926), p. 218;Google ScholarProc. Lond. Math. Soc. (2), 27 (1928), p. 513Google Scholar. The last paper will be referred to as J.

‡ Rend, di Palermo, 25 (1901), p. 33Google Scholar. This paper will be referred to as Severi I.

† “Jacobian surfaces of quadrics in four dimensions,” Proc. Camb. Phil. Soc. 25 (1929), pp. 268–71.CrossRefGoogle Scholar

‡ Memorie di Torino (2), 52 (1903), §§ 20, 23. This paper will be referred to as Severi II.

§ Rend, dei Lincei (1915), p. 877.

‖ The general theory of these equations is given by Severi, , Mem. di Torino (2), 51 (1902), p. 81.Google Scholar

† It also gives the deduction kl which must be made in order to find the correct value of μ in any given case.

‡ Proc. Lond. Math. Soc. (2), 24 (1925), p. 359;Google Scholaribid. 28 (1928), p. 161.

† Kalkü;l der Abzählenden Geometrie; the formulae obtained by James for the intersection of lline-systems are also employed.

† The line constant appropriate to each problem is denoted by the corresponding suffix.

† Since six generators pass through any point of f.

† Since generators of one system of the cone meet F5 in general plane cubics (Severi I).

† Rend, di Palermo, 1 (1887), pp. 241–271.CrossRefGoogle Scholar