An Arithmetic Theory of Adjoint Plane Curves

Introduction. In classical algebraic geometry the adjoint curves to an irreducible plane curve are an essential tool in the study of the geometry on the curve. In this paper we shall give an algebro-arithmetic development of the theory of adjoint curves, and shall extend the classical results to irreducible plane curves with arbitrary singularities defined over arbitrary ground fields. Our definition of the adjoint condition at a given singular point of the curve is stated in terms of the conductor between the local ring of the point and its integral closure. The fundamental properties of the adjoint curves are then derived from corresponding properties of the conductor. The single deepest and most important property of the adjoint curves is that, on a curve of order m, the adjoint curves of order m — 3 cut out the complete canonical series. This property is equivalent to the fact that the degree of the fixed component of the adjoint series is twice the number of conditions which the adjoint curves impose on the curves of sufficiently high order('). We shall give two distinct and independent proofs of this proposition. The first proof is a direct one, based upon a detailed analysis of the singularities of the given curve. This analysis, to which part I is devoted, applies equally well to algebraic number fields, and our treatment will include this case with that of algebraic function fields of one variable. The second proof is more indirect, depending upon the Riemann-Roch theorem and a generalization of the classical representation theorem of the differentials of the first kind. This proof, which will be given in part II, holds only for plane curves whose function field is separably generated over the ground field.