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Degenerate 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

1. In a previous note the author has examined the systems of tangent planes to a degenerate surface in S3 consisting of n planes, regarded as the limit of a general surface of the same order. It is well known that a pair of space curves which are the limiting form of a non-degenerate curve must have a certain number of intersections; hence, if a surface in higher space degenerates into a number of surfaces, these must intersect in curves of various orders. In the present paper we consider the nature of the envelope to a surface consisting of two general surfaces of S4 having in common a single curve of general character, the degenerate surface being regarded as the limit of some surface of general type. The same conclusions hold for a similar degenerate surface in Sr (r>4).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 1 , January 1933 , pp. 88 - 94
Copyright
Copyright © Cambridge Philosophical Society 1933

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References

* Roth, , Proc. Camb. Phil. Soc. 28 (1932), 45.CrossRefGoogle Scholar

† Severi-Löffler, , Vorlesungen über Algebraische Geometrie (1921), 353.Google Scholar

‡ Chisini, , Rendiconti Lincei (5), (26)1, (1917), 543.Google Scholar

§ Chisini, op. cit.

* In general f c, and therefore C, will have a number of triple points; this possibility is excluded by Chisini, but the corresponding result is incorporated in what follows.

* A tangent to a surface of a pencil in a point of the base-curve is tangent also to the consecutive surface of the pencil. See Enriques-Chisini, Teoria geometrica delle equazioni, 1, 179.

* See Severi, , Memorie Accad. di Torino, 52 (1903), § 20.Google Scholar

† Note that (4) may be written as ε1 = ε0 (n + n′ − 2) − i − i′ − X.

* Severi, , Rendiconti di Palermo, 17 (1903), 33.Google Scholar

* Unless otherwise stated, a “line” lying on a scroll is a generator, not the directrix.

* For the nomenclature see a previous paper, Proc. Camb. Phil. Soc. 28 (1932), 300.Google Scholar

† This surface is the intersection of a quadric and a quartic primal in S 4, residual to two planes; it is represented by means of plane quintics having a common triple point and ten simple base-points. It is thus a surface normal in S 4 having an improper node.