Multiple correspondences determined by the rational plane quintic curve

Abstract

The occurrence of rational curves in pairs is a well-known fact: thus, given a rational curve pp, of order n, in a space of p dimensions, there is uniquely determined, to within a collineation, a curve pn_p-l n of order n, in a space of n-p-1 dimensions, by requiring all hyperplane sections of either curve to be apolar to the hyperplane sections of the other.t We call two curves associated in this way conXugate curves. If the rational curve pp is regarded as the projection of the norm-curve pn in a space Sn of n dimensions, from an Sn_p_ln the interpretation of this fact is immediate. An Sn_l in Stl meets Pn in n points which may be regarded as given by a binary form of order n: dually, a point of Sn determines on pn a set of n points, which may be given by a second binary form. The condition of apolarity of the two forms is precisely the condition of incidence of point and Sn--l * All Sn lns having n-point contact with ptl meet Sn_p_l in the hyperplanes of a curve rn_p_l of class n. The curves obtainable by projection from S7t_ and section by Sn_l are conjugate curves. In this paper we shall deal with the case n = 5, p-2, the rational plane quintic. If our curve r5 is given parametrically by