A remark on generalized abundance for surfaces

Abstract

AbstractLet $$(X, \Delta )$$ ( X , Δ ) be a projective klt pair of dimension 2 and let L be a nef Cartier divisor on X such that $$K_X + \Delta + L$$ K X + Δ + L is nef. As a complement to the Generalized Abundance Conjecture by Lazić and Peternell, we prove that if $$K_X + \Delta $$ K X + Δ and L are not proportional modulo numerical equivalence, then $$K_X + \Delta + L$$ K X + Δ + L is semiample. An example due to Lazić shows that this is no longer true in any dimension $$n \geqslant 3$$ n ⩾ 3 .