A geometrical application of binary syzygies

Abstract

t. The present paper is devoted to a geometrical application of certain kinds of syzygies connected with binary forms of odd order. If we represent the forms with which we deal by sets of points on a rational curve, the application is to the problem of determining other curves of the same type whose intersections with the given curve are either original sets themselves, or new sets having simple invariant relations to the old. The type of rational curve which we shall use is that having but one multiple pOillt, which then, for a curve of order m, must be an ( m 1 )-fold point, and equivalent to 21(m-1 )(m 2) double points. Such a cllrve we shall designate for brevity as a JONQUIARES curve, and the symbol J(7n) will be used in referring to it. We proceed to consider the intersections distinct from the ulultiple point itself of two such curves having the same multiple point. Let the order of the second curve be n; then through the multiple point pass R 1 branches. The further points of -intersection are then mn ( xn 1 ) ( n 1 ) xn + n 1 in number. On the other hand, if the first curve is given, the number of conditions which the second can fulfil is 2n. This may be seen readily by choosing the multiple point as a vertex of the reference triangle, whereupon the equation of the J(n) reduces to 2n + 1 terms. The ulost interesting case occurs when the llumber of free intersections, m + n 1, is greater by one than the number of conditions that can be imposed on the second curve; that is, when n nb 2. In this case we have an involution, Ilm-3, set up on the first curve. Interpreting two well-known theorems relating to involutions, we have the following geometrical properties, which will be found usefu] in what follows: t) There eacist on a J(m) eacactly 2rn 3 points having the property that a J(m-2)s with the same rnultiple point as the J(m)s can be drawn to have (2m 3 )-point contact in each with the J(m) a 2) These 2m 3 points all lie on a J(m-2) with the same multiple point.