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Some surfaces containing triple curves, II

Published online by Cambridge University Press:  24 October 2008

L. Roth
L. Roth
Affiliation:
Clare College
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Extract

1. A general threefold in [5] has an apparent double surface whose projection on to a [4] is a surface which contains a triple curve and which is characterised by the fact that it has no improper nodes; such surfaces have been considered in a previous paper. In the present note we consider a class of surfaces in [5] which possess triple curves and project into surfaces in [4] having improper nodes: these are the surfaces which represent the chords of a general curve of ordinary space upon a quadric primal Ω of [5]. For the representation of a line congruence of [3] by the points of a surface on Ω, reference may be made to a previous note, the results of which are used in the present work.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 2 , May 1933 , pp. 178 - 183
Copyright
Copyright © Cambridge Philosophical Society 1933

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References

* “Some surfaces containing triple curves”, Roth, , Proc. Camb. Phil. Soc., 28 (1932), 300CrossRefGoogle Scholar [referred to as S].

† The surfaces which represent the pairs of points of a curve, or the points of two curves, have been discussed from the point of view of birational transformation by Severi, , Atti di Torino, 39 (1903)Google Scholar, and De Franchis, , Rendiconti di Palermo, 17 (1903), 104.CrossRefGoogle Scholar

‡ “Some properties of line congruences”, Roth, , Proc. Camb. Phil. Soc., 27 (1931), 190.CrossRefGoogle Scholar

* For this and other formulae quoted, reference may be made to a previous paper, Proc. Camb. Phil. Soc., 25 (1929), 390.Google Scholar

† There is a curve of order a on F, for all points of which the two curves are coincident.

‡ This is the number of tangents to C t which lie on Ω and is found as follows. The tangents to C t generate a scroll of order r t, meeting Ω in a residual curve of order 2r t − 2t, which must consist of the required tangents, since a line which meets Ω in three points lies on it altogether.

* Cayley, Collected Papers, v, 187; this is reproduced in Salmon-Rogers, Analytic Geometry of Three Dimensions (1915), ii, 97.

† Salmon-Rogers, op. cit.

* De Franchis, op. cit.; the irregularity of F is equal to p, so that the only regular surfaces of this type are rational, corresponding to rational curves C.

† Cf. S, equation (6).

‡ The sextic projection on [3] from two points of itself has been considered by Scorza, , Annali di Mat. (3), 16 (1909), 255CrossRefGoogle Scholar. For the octavic surface see Scorza, , Annali di Mat. (3), 17 (1910), 281.CrossRefGoogle Scholar

* Castelnuovo, , Atti di Torino, 25 (1890), 695Google Scholar.

† For the representation of a rational surface having a triple curve and for the nomenclature see S, 304–5.