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Involutions of pairs of points in three dimensions determined by cubic surfaces
Published online by Cambridge University Press: 24 October 2008
Extract
1.1. In a linear space Sr, of r dimensions, we may consider involutory transformations determined by linear systems of primals whose freedom is r and whose grade, that is, the number of free intersections of r primals of the system, is two; we shall be concerned only with complete systems, systems of all the primals (of given order) satisfying certain fundamental linear conditions, for example, containing a point, or a curve, or touching a plane at a given point. The primals of such a system Φ that pass through a point P form a system ∞r−1; any r linearly independent primals of this subsidiary system will meet in one other point Q outside the base elements, and Q is common to every primal of Φ through P; further, all the primals of Φ that contain Q must contain P. We thus have an involutory transformation of the space Sr.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 3 , July 1931 , pp. 404 - 420
- Copyright
- Copyright © Cambridge Philosophical Society 1931
References
* This primal clearly belongs to the Jacobian of the ∞r primals of the system Φ, since all the primals of Φ that pass through a point of the Jacobian have a fixed tangent line at that point, i.e. their two free intersections coincide; but the Jacobian may contain a further part. See De Paolis, quoted below.
* Snyder, , Bull. Amer. Math. Soc. 30 (1920), 106.Google Scholar
† Geiser, , Journal für Math. 67 (1867), 83.Google Scholar
‡ DePaolis, , Memorie Lincei (4), 1 (1885), 576–608.Google Scholar
§ Semple, , Proc. Camb. Phil. Soc. 25 (1929), 145–167.CrossRefGoogle Scholar
* Del Pezzo, , Rend. Ace. Napoli, 25 (1886), 176.Google Scholar
* Segre, , Mem. Acc. Torino (2), 39 (1883), 3.Google Scholar
* Loc. cit.
* This involution, and the two following, are considered by Snyder, , Trans. Amer. Math. Soc. 21 (1920), 52–78Google Scholar, but from a different point of view.
† See, for example, Baker, , Principles of Geometry, Vol. iv, p. 234.Google Scholar
* For a fuller discussion of the involution, its branch surface and surface of self-corresponding points, see Snyder, loc cit.
* This involution is discussed by Romano, Sopra una transformazione doppio del terzo ordine, Avolo, 1906, and later by Snyder, loc. cit. They show that the branch surface in a double space mapping the involution is the Eummer surface, as for the Geiser involution determined by quadrics through 6 points. Mr Todd has pointed out to me that this system of cubic surfaces can in fact be transformed birationally into a system of quadrics with 6 base points, namely, by means of the Cremona transformation determined by the homaloidal system of cubic surfaces passing through the lines m and having a node, at a point P 1 (and therefore containing the 3 transversals from P 1 to pairs of the lines m). The 6 base points will arise from the remaining points P and from the 3 planes through P 1 and the lines m.
† Encyk. der Math. Wiss. iii C 7, p. 958.Google Scholar