A Note on Projective Line Bundles

Abstract

In a previous paper we introduced the notion of a projective bundle and the notion of a projective bundle.1 The term regular refers to a method of generating the bundles. We should like to classify all projective bundles, but must content ourselves at present (for reasons explained before) with attempting to classify only the projective bundles; at the beginning we have even limited ourselves to the case where the fibres are complex projective lines. The purpose of this paper is to describe several methods of generating projective line bundles and to point out that each of these methods gives rise only to bundles which are equivalent to projective line bundles. This kind of procedure will, of course, never show that every projective line bundle is equivalent to a one, but it does lend a certain increased motivation for the classification of the projective bundles. Let 9? be a closed Riemann surface, S2 the complex projective plane, y a complex analytic mapping of t into S2. Assume that 4 is not a constant mapping, and denote the image of T under ,u by e(TJ, ,4), or simply e; it is an algebraic curve. Each point of e is the center of a finite number of places, and all but a finite number of points are the center of exactly one place. The places of e are in (1, 1) correspondence with the points of V', the Riemann surface of C; there is a natural mapping v of the points of A' onto the points of e: v maps each point of T' onto the center of the corresponding place. This mapping v is also complex analytic. The mapping /u can be factored into a map X of J onto J?' and the map v, i.e.