CORRECTION TO THE PAPER “FRACTIONAL INDICES OF LOG DEL PEZZO SURFACES”

Abstract

P ROOF . By Proposition 1.3 the cone of effective curves on X is generated by finitely many curves. From the formula K~ = n*Kx + Y^aiFi it follows that the divisor ~MK~ is effective for some positive integer Μ. Let C be an irreducible reduced curve with C < 0. We now carry out the following procedure: if / Φ C is an exceptional curve of genus 1 and C • I < 1, then we contract this curve. We repeat this procedure several times until we obtain a morphism / : X > 5" with the following properties: S is a nonsingular surface, C{ — / (C ) is a nonsingular curve and for any curve Ι Φ Cx of genus 1 we have Cx • I > 2. If S = P 2 or F n , η < Ν, then C 2 > Ν and C 2 > c\ k > N k. If C > 3 , then C > 3 k . Now suppose S φ Ρ 2 , F n and c\ < 4 . We prove that the divisor 2KS + Cx is numerically effective. The cone of effective curves on the surface S is generated by finitely many curves; let {E^ be a minimal system of generators. If Ks · £J. > 0 and Et φ C, , then {2KS + Cx ) Ei> 0. If Ks • Ei < 0 and Ei φ C, , then Ej is an exceptional curve of genus 1 and {2KS + C, )·£ ,• > 0. Finally, {2KS + C,) · C{ = 4/ 7 Q(C, ) 4 C >0 . Thus, the divisor 2KS + Cx is numerically effective and the divisor —MKS =