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Note in regard to surfaces in space of four dimensions, in particular rational surfaces*

Published online by Cambridge University Press:  24 October 2008

H. F. Baker
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To a non-singular algebraic surface in space of five dimensions there can generally (the Veronese surface, of order 4, and cones are exceptional) be drawn, from an arbitrary point, a finite number of chords. If such a surface be projected from a point into space of four dimensions, there will, therefore, in general, be a certain number of points upon the resulting surface, at which two sheets of this surface, with distinct tangent planes, have an isolated common point. Such points have been called improper double points. We consider an algebraic surface ψ, in space of four dimensions [4], with no other multiple points than such double points, which we shall call accidental double points. The chords of the surface ψ, drawn from an arbitrary point O of the space [4], form a surface, or conical sheet, of which a general generator meets the surface in two points. The locus of these points is a curve which we shall call the chord curve. This curve has an actual double point at each of the accidental double points of ψ There will also, generally, be a certain number of points of the surface which are points of contact of tangent planes of the surface passing through O (and therefore also points of contact of tangent lines through O, these tangent lines being generally tangent lines of the chord curve).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 1 , January 1932 , pp. 62 - 82
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

† Severi, , “Intorno ai punti doppi impropri”, Rend. Palermo, 15 (1901), 33.CrossRefGoogle Scholar

* Roth, , Proc. Camb. Phil. Soc., 25 (1929), 395.Google Scholar

* Cf. Edge, , Ruled Surfaces (1931), p. 76, § 88.Google Scholar

† Cf. Edge, pp. 202–206.

* Bordiga, , Memorie Lincei (4), 4 (1887), 182.Google Scholar

† White, , Proc. Camb. Phil. Soc., 21 (1923), 224.Google Scholar

‡ Clebsch, , Math. Annalen, 1 (1869), 253.CrossRefGoogle Scholar

§ Severi, , Memorie Torino (2), 52 (1903), 61.Google Scholar

∥ Semple, , Proc. Lond. Math. Soc. (2), 32 (1931), 374.Google Scholar

* Castelnuovo, , Memorie Torino (2), 42 (1892), 3.Google Scholar

* Professor Semple suggests another method. Suppose the surface projected into space [3], and assume that surfaces of order μ0 − 3 through the double curve of the projected surface give canonical sets on the plane sections of the surface. On the plane, the complete intersection with surfaces of order μ0 − 3 is represented by curves

and the canonical sets on the representing curves are given by curves c m−3 (k i − 1). Subtraction of curves of the latter character from curves of the former character gives the curve representing the double curve (and the chord curve).

* White, loc. cit., 226.

* That for the projected Del Pezzo surface in [4] we have d = 1 is the same fact as that remarked by Severi, with an enumerative proof to which the present is equivalent, that a single chord can be drawn from an arbitrary point to the Del Pezzo surface of order 5 in space [5]. Babbage, D. W. (Proc. Camb. Phil. Soc., 27 (1931), 399CrossRefGoogle Scholar) has recently given another proof of this, dependent on the fact that the four base points, in the plane representation of the surface, are the complete intersection of two conics; with a generalisation to a manifold J. A. Todd has proved the result with reference to special sets on the normal c 10, which is the intersection of the Del Pezzo surface with a quadric (as proved by F. Bath). Either of these proofs is intimately related with the remark that, in the plane representation of the surface, a line l of the plane corresponds to a rational cubic curve on the surface, the solid or space [3], which contains this curve, being the intersection of two primes intersecting the surface in two curves whose plane representations each consist of the line l and a conic through the four base points. Of such solids there are ∞2, one through an arbitrary point O of the space [5]; and the chord of the surface through O is the chord to the cubic curve in the solid. But another proof, of geometrical character, of the unique chord to the surface from an arbitrary point O, is as follows: It may be proved that if three general quadrics be put through this surface, their remaining intersection is a ruled cubic surface lying in space of four dimensions. It is easy to see, considering the enveloping cones from O, that four chords can be drawn from O to the complete intersection of the three quadrics; and then, that three of these have, for their two intersections with this locus, one point on and the other on the ruled cubic surface. There remains then one chord from O to the original . Other loci for which one chord can be drawn from an arbitrary point are: (1) the rational ruled normal quartic surface in [5]; (2) the V 9 [19], representing the planes of [5], as remarked by Segre (see Rao, C. V. H., Proc. Camb. Phil. Soc., 26 (1930), 72)CrossRefGoogle Scholar.

† White, loc. cit., 225.

* Severi, , Memoire Torino (2), 52 (1903), 61Google Scholar; Semple, loc. cit., 396.

* Castelnuovo, loc. cit.