Three related quartic curves in four dimensions

Abstract

1. It has been shown by Pieri, and independently by James, that the trisecant planes of a quartic curve in [4] which meet a line meet also another quartic curve intersecting the former in six points. James generalises the theorem and shows that trisecant planes of a quartic curve C, which meet a quartic curve C1 having six points in common with C, also meet another quartic curve C2, and that the relation between the three curves is symmetrical. The object of this note is to give a simple proof of this theorem and to discuss the representation by which this proof is effected. The method used is analogous to that of Pieri; an interesting differential method is adopted by C. Segre who shows that the foci of the first and second orders, of any linear system of ∞2 planes in [4] of which two planes pass through a point, determine a conic and five points respectively, in any plane of the system: the trisecant planes of C meeting C1 give a particular case of such a system.