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The general surfaces in four dimensions with one or two apparent triple points

Published online by Cambridge University Press:  24 October 2008

J. W. Archbold
J. W. Archbold
Affiliation:
St John's College
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Extract

1. Let F be an irreducible surface in [4] with apparent triple points, t in number; let P be an arbitrary point in the space and p1,…,pt the trisecant lines which can be drawn through P. Then, if F lies on at least ∞1 cubic primals, any one of these which passes through P will contain p1,…,pt in consequence and the linear system will be compounded of the congruence of trisecant lines of F. By a well-known theorem it follows that the grade of the system is zero. The free surface of intersection Φ of two cubic primals will be ruled, having t generators, each a trisecant of F, passing through every point on it.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 1 , January 1932 , pp. 51 - 54
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

* Bertini, , Geom. proiettiva degli iperspazi, 2nd Edn. (1923), p. 277.Google Scholar

† Semple, , “On the surfaces of intersection of cubic primals”, Proc. Lond. Math. Soc. (2), 32 (1931), 369CrossRefGoogle Scholar. For proofs of the statements in the text about the intersections of two cubic primals, the reader is referred to this paper.

‡ Those surfaces for which t = 2 have been obtained otherwise by Marletts, , “Le superficie generali dell’ S 4 dotati di due punti tripli apparenti”, Rend. Palermo, 34 (1912), 179–186CrossRefGoogle Scholar. Those for which t = 1 have been obtained by Ascione, , Rend. Lincei (5), 6 (1897), 162Google Scholar and by Severi, , Rend. Palermo, 15 (1901), 33.CrossRefGoogle Scholar

§ For properties of this surface see Edge, , Theory of Ruled Surfaces, 131.Google Scholar

* In a paper entitled, “On special Cremona involutions and transformations”, communicated to the London Math. Soc.

† Severi proves the first two results otherwise, Rend. Palermo, 15 (1901), 33.Google Scholar

* Edge, loc. cit., § 128.