Point sets and allied Cremona groups. II
Arthur B. Coble · Transactions of the American Mathematical Society · 1916
It is not the purpose of this article to develop specific facts concerning definite types of transformations C in Si.For these, treatises such as Clebsch-Lindemann, Leçons sur la géométrie, vol.II, sec.I, chap.IX; Doehlemann, Geometrische Transformationen, part II; and Sturm, Die Lehre von den geometrischen Verwandschaf ten, part IV, are available.So far as the author is aware the following points concerning C in Si are novel: (a) the definition and use in § §1, 2 of a definite mutual order of the F-points of C'1 and C; (6) the proof in §3 that, for a general point set, congruence does not imply projectivity except in the few particular cases enumerated in §2; (c) the development of the group G",2 in §3, of its algebraic relation to the set PI, and of the group en,i in §6; (d) the association of C in Sí with regular transformations in hyperspace; (e) the determination in §6 of all types of C in Si with 9 or fewer F-points; and (/) the invariants of P\ and P\ under G7,2 and G9,2 respectively ( §8).In order to present these matters adequately it has been necessary to use, and occasionally convenient to re-prove, some known facts whose origin it is not easy to ascertain.The most original and comprehensive advance in this field is due to S. Kantor in the attempt to determine all types of finite Cremona groups.This work has been perfected by A. Wiman, Zur Theorie der endlichen Gruppen . . ., Mathematische Annalen, vol.48 (1897), p. 195, where full references to the articles of Kantor and others are given.The mappings in §4 from the plane of PI, P,, and P¡ to respectively the cubic surface, the quartic curve, and the space sextic of genus four on a quadric cone date back to Clebsch and Noether and constantly reappear in later papers (cf.Wiman, loc.cit.).Articles along these general lines have been published by Snyder, these Transactions, vol.11 (1910), p. 371 and vol.12 (1911), p. 354; American Journal of Mathematics, vol.33 (1911), p. 327.The notion of a regular transformation in spaces of three or more dimensions appears to be new, as well as the entire discussion of G*,», gn.k, and e",k (k > 2) which is based on this notion.But particular regular transformations occur frequently in the literature.Thus for the Geyser involution determined by PI cf.Sturm, loc.cit.; and for the involution of order 15 determined by P? cf.Conner, American Journal of Mathematics, vol.38 (1916).* This transformation has been employed by S. Kantor in his crowned memoir of 1883: Theorie des transformations périodiques univoques: Naples (De Rubertis), 1891, p. 293.t Cf.Clebsch-Lindemann, loc.cit., vol.Ill, p. 451.* A similar table is given by Kantor, 1. c, p. 280 which includes a non-existent type, m = 11, aj = 5, a¡ = 3. * The product is properly taken if the F-points of Ctt are included in the set Ql.