Point Sets and Allied Cremona Groups

Abstract

Part Ii of this account was devoted to a study of the invariants of a set Pk of n points in Sk under the group of permutations of the points. The set as a projective figure was mapped by a point P of a space lk(n-k-2) in which the permutation group appeared as a Cremona group Gnf!. In this part we shall consider the effect upon the set pk of certain Cremona transformations C in Sk These transformations C are of a special charatcer when k > 2 described hereafter by the term regular. They are determined essentially by their fundamental points alone and in all important particulars are entirely analogous to the ternary Cremona transformations. These regular Cremona transformations form the regular Cremona group in Sk. If one set of fundamental points of a regular transformation C be placed at P' there is determined a new set Pnk congruent to P' under C. The totality of sets P,' congruent in some order to a given set Pk is mapped in fk(n-k-2) by an aggregate of points P' which form a conjugate set under the extended group Gn, k of P'. This group Gn, k in contains Gw! as a subgroup and in general is infinite and discontinuous. The major part of this article is devoted to a study of this group. In ? 5 a group gn, k of linear transformations which is isomorphic with Gn, k iS introduced. The new group brings to light properties both of Gn, k and of regular transformations in Sk. An interesting result is a determination of all types of regular transformations with a single symmetrical set of fundamental points. MIost of the types are well knob n but some are novel. In ? 6 another group en, k of linear transformations, also isomorphic with Gn, k, iS defined, which is particularly effectixe for a discussion of the infinite groups Gg,2, G8, 3, and G9,5. The close relation between the associated sets P7 and Q-k-2 which appeared