A remark on generalized abundance for surfaces

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