Point-curve correspondences

Abstract

In this paper we consider the properties of a point-curve correspondence between two (generally distinct) surfaces F and G, paying special attention to the idea of ‘induced’ and ‘extended’ correspondences. We also outline the theory of point-curve corre spondences between a curve and a surface, which is not only of interest in itself, but is of importance because such correspondences arise, as do correspondences between curves, as induced correspondences of point-curve correspondences between two surfaces. The results depend chiefly on Lefschetz's theory of correspondences between algebraic curves(1) and the properties of intersections of cycles on an algebraic surface given by the author (2), as well as on the preceding paper of this series (3). Use is also made of a paper by Severi (4) on the self-correspondences on a variable curve of an algebraic surface. The theory of correspondences on a single surface will be further developed in the next paper on this subject.