On rational surfaces I. Irreducible curves of arithmetic genus $0$ or $1$

W e are to prove a fundamental theorem on irreducible curves o f arithmetic genus 0 or 1 on a non-singular rational surface and to show applications of the theorem to the followings :(1) The classification of non-singular rational surfaces which have no exceptional curves of the first kind.(2) The classification of rational ruled surfaces.(3) Factorization o f Cremona plane transformations.(4) Classification of projective surfaces of degree d which are not in any projective space of dimension d -1 .Results assumed to be known : Besides elementary facts on surfaces, we need to know ( i) irreducible exceptional curves of the first kind (see Zariski [10, Part I I ] ) and (ii) the genus formula o f curves on non-singular surfaces (see § 1, (7), (I,,)).As for the definition of surfaces, we shall employ that in the sense of Zariski [10 ].