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Some surfaces containing triple curves

Published online by Cambridge University Press:  24 October 2008

L. Roth
L. Roth
Affiliation:
Clare College
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Extract

The present paper is devoted to a consideration of a type of surface which appeared incidentally in some recent work and which seems worth separate notice for its own intrinsic interest. The surface in question, which lies in space of four dimensions, is one that possesses a triple curve containing pinch-points and quadruple points, and as such is a particular case of a general surface in [4] with the same projective characters. The results given here are illustrative of those contained in the papers referred to, and with few exceptions the main facts which are used in the present discussion are quoted from the same source. In the section dealing with rational surfaces some novel results are obtained in connection with the degeneration of a plane curve into two portions, one of which is counted twice.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 3 , July 1932 , pp. 300 - 310
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

* “On plane-forms in four dimensions”, Proc. Lond. Math. Soc. (2), 33 (1931), 115Google Scholar; “Ruled forms in four dimensions”; “Some formulae for primals in four dimensions”. These last are to be published in the Proceedings of the London Mathematical Society.

† See Proc. Camb. Phil. Soc., 26 (1930), 43.Google Scholar

‡ In the third of the papers referred to above.

* We assume here that F has the generality of the projection from [5] of a threefold without accidental singularities; in this case it is evident that Γ can have no improper nodes.

* It appears that, when a surface of general character in [4] acquires a triple curve of the character described (so that μ 1 remains the same) there is a diminution τ in the number of apparent pinch-points, and the class μ 2 is unchanged.

† That the two curves have only simple intersections at these points is clear from a particular example; thus in § 11 I, t = 1, k = 3, t 1 = 2, i = 0.

* Clebsch, , Math. Annalen, 1 (1869), 253CrossRefGoogle Scholar; a recent and exhaustive treatment of rational surfaces, from a different point of view, is given by Baker, , Proc. Camb. Phil. Soc., 28 (1932), 62CrossRefGoogle Scholar.

† Zeuthen, , Math. Annalen, 4 (1871), 26–9.Google Scholar

‡ Due to Cremona (quoted by Clebsch, op. cit.).

§ Clebsch and Baker, op. cit.

* Zeuthen, , Math. Annalen, 3 (1871), 150.CrossRefGoogle Scholar

* Del Pezzo, , Rend. di Palermo, 1 (1887), 241Google Scholar.

† Except for n=3, 4, in which cases there are 27 and 16 respectively.

‡ Castelnuovo, , Rend. di Palermo, 4 (1890), 73.CrossRefGoogle Scholar

§ The sextic type was first considered by Bordiga, , Memorie dei Lincei (4), 3 (1887), 182Google Scholar; the quintic type was discussed by Clebsch, op. cit., while the class as a whole was first studied by Castelnuovo, , Atti di Torino, 25 (1890), 3.Google Scholar

* See Roth, , Proc. London Math. Soc. (2), 33 (1931), 115Google Scholar (referred to as P.F.).

† A general discussion of ruled forms has been given by the writer in a paper to be published by the London Mathematical Society.

‡ P.F. pp. 120–22. We denote such a form by (n, a).

* It is erroneously stated in P.F. § 2·7 that Γ contains only three lines.

† The surface contains three simple lines also (P.F. § 1·2), but may contain more, as shown by Ex. I.

* For n=8, however, the form is not ruled; the space section is the second species of octavic described in § 8, which contains no lines. These forms have been discussed recently by Todd, J. A., Proc. Lond. Math. Soc. (2), 33 (1932), 328CrossRefGoogle Scholar, q.v. for other references to the subject.