On rational surfaces, II

Abstract

In § 1 of the present paper, we introduce the notion of a virtual linear system on a non-singular projective surface and we clarify the theories o f infinitely near points, of divisors a n d o f linear system with preassigned base conditions. We introduce in § 2 the notions of a numerical types and of non-special points with respect to Cremona transformations. They play important roles in § 3 in order to prove characterizations and existence theorems o f exceptional curves of the first kind and of Cremona transform ations. In 4 , we introduce the notion of an abnormal curve, and in § 5 we give some remarks on superabun-dance o f a complete virtual linear system on a projective plane S. We add some remarks in § 6 on the case where the number of base points is at most 9. The recent paper "On rational surfaces, I" in the last volume o f our memoirs is quoted as Part I in th e present paper. The notations and terminology in Part I are preserved in this paper, except for that the symbol { } for the total transform o f a divisor is changed to () ; see § 1. We recall here that an S denotes always a p rojective p lan e. A curve will mean a positive divisor on a su rface. A divisor c on a surface F is identified with a divisor c ' on a surface F ' if c = / mi ci and c'= / rn 1 c a n d if ci and c ; are irreducible and are identical with each other as point sets (iden-tification of points is made by natural birational transformations). 1. Virtual linear system. Let F be a non-singular projective surface and let B be the fam ily o f non-singular projective surfaces which are birational with F by natural transformations.