Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T07:41:02.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Degenerate surfaces in higher space

Published online by Cambridge University Press:  24 October 2008

L. Roth
L. Roth
Affiliation:
Clare College
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.