Q pY ' " -X >(X,Y) ,niX,y ) be such a d e s i n g u l a r i z a t i o n a) (X,Y) f:(X',Y') there can prove Proposition.
For every seH°(E) the section SFi^s is either zero, or vanishes on a curve Fi+D(s)eI(gs(A) I for some D(s)e[A-Fil. Since the map H°(E) --~ H°(C3s(A-Fi)) is surjective and IA-Fi] has no base points, we find that E is generated by its global sections outside the curve F i. Now let us show that the same E can be also represented as an extension (*-) o ~ es(F~) - , Z --,Gs(A-Fj) - , 0 for any j ;~ i, Ij-il ;~ 10. Then, repeating the argument from above we obtain that E is generated by global sections outside Fj.
This contradiction shows that D m cannot take negative values on N E K. Henceforth the linear function Dm takes negative value on some extremal ray R c __ NE(S) \ N E K (see ). According to the classification of extremal rays on a surface (see ) there are only three possibilities: (i) R =IR+[E], where E is a ( - D - c u r v e ; (ii) R =IFI+[L], where L is a fibre of a geometrically ruled surface ; (i~) R =llq+[L], where L c IP2 is a line. (D+mKs) = E . D - m < 0. D < m - 1 . (D'-Z') > 2 for any irreducible component Z' c D' of genus 0.