Affine curves on which every point is a set-theoretic complete intersection

Abstract

We consider here those reduced and irreducible affine algebraic curves on which every point is the set-theoretic complete intersection of the given curve and a hypersurface. These curves, for historical reasons explained in Section 1, are called “prefactorial.” Our joint research in this area stems from our common interest in a certain result due to Mm-thy and Pedrini 1121: over algebraically closed ground fields of characteristic 0, prefactorial curves are necessarily rational and nollsingular. Presented here are the results of our efforts to understand prefactoriaIity in the absence of “algebraically closed” and/or “characteristic 0.” Our methods are mainly algebraic, depending largely on an analysis of the ‘grayer-Vietoris sequence of the conductors’ (as in 112)) and on certain “well-known”’ arithmetic-geometric facts which we learned from a paper of Rosen [16j (and which seem not to be so readily avaitable elsewhere in the literature). It is also of interest to examine pr~fa~toriality from the viewpoint of the classical theory of aigebraic curves: Section 4 contains some of the product of our tllinking along those lines.