Transactions of the American Mathematical Society, Vol. 24 (Classic Reprint)

Two sets of n points ordered with respect to each other, the one, Pk, in a linear space Sk, determined by the equations (up1) =0, (wp2)=0, -, (upn} =0, and the other Q -, in a linear space Sn-k-2, determined by the equations (up1)=0, (vq1)=0 .., (vqn)=0, are called associated sets if the factors of proportionality in the coordinates of the points can be so chosen that an identity in u, v exists of the following form: (1) (up1 )(vq1) +(up2) (vq2 )+ ... + ( upn ) (vqn) = 0. This relation, obviously mutual, between the two sets is such that either set uniquely defines the other to within projective modifications. Some general properties of such sets have been given by the writer. A characteristic algebraic property of two associated sets is that complementary determinants formed from the matrices of the coordinates of the two sets of points when taken so that (1) is satisfied are proportional. A characteristic geometric property is the following: On k+ 3 of the points of P there is a unique rational norm curve N k upon which the k+ 3 points determine a set of k+ 3 parameters; on the complementary set of n - k - 3 points of Q there is a pencil of linear spaces Sn-k-3 whose members on the remaining k+ 3 points determine a set of k+ 3 parameters; these two sets of k+ 3 parameters are projective. Unless k - n - k - 2 the associated sets are in spaces of different dimension. Conventional methods of passing from one space to another are the process of mapping the space of lower dimension upon that of higher dimension, and the process of projecting from the space of higher dimension upon the Read before the American Mathematical Society at Evanston, Dec. 29, 1922. This investigation has been pursued under the auspices of the Carnegie Institution of Washington, D. C. A. B. Coble, Point sets and allied Cremona groups (I), these Transactions, vol. 16 (1915), p. 155, in particular 1