Point-curve correspondences

Abstract

Attempts have been made to extend to higher dimensional varieties the theory of correspondences developed for algebraic curves. So far efforts have been concentrated on ‘point-point’ correspondences (i.e. between two varieties of the same dimension such that a generic point of one corresponds to a 0-dimensional variety of the other), and even in the case of surfaces important problems, such as the base number for the correspondences, are still unsolved (cf. Hodge(3)). The purpose of these papers is to draw attention to, and study in the simpler cases, another class of correspondences, the ‘point-primal’ type, in which to a generic point of one variety corresponds a primal of the other. These correspondences provide at least as natural a generalization of the theory for curves as point-point correspondences, but have so far only been touched on by Severi(7, 8) in two simple cases. In this paper we give a number of general results for point-primal correspondences (mostly immediate generalizations of Severi's results), and embark on a detailed discussion of such correspondences, which we naturally call point-curve correspondences, between two surfaces, the chief result being the determination of the base in that case. Further results on the transformation of the cycles of a surface by a point-curve correspondence, the correspondences induced between curves of the two surfaces and, in the case where the surfaces coincide, the ‘united curve’ of the correspondence, as well as the question of correspondences of non-zero valency, will be dealt with in a later paper, where we shall also consider the case of correspondences between a curve and a surface discussed in Severi's paper(7).