William E. Lang writes in Examples of Surfaces of General Type with Vector Fields
In the next two lemmas, we show that $K_X$ and $\mathcal O_X(D)$ are linear compinations of [some other bundles] . These computations can be carried out in the Neron-Severi group of divisors modulo numerical equivalence, and we shall do so.
He than has results of the form $$\mathcal O_X(D) \simeq \mathcal O_X(bS) \otimes f^* L^{\otimes a}.$$ Is that equivalence only meant up to numerical equivalence? Or does he implicitly claim that those line bundles are actually isomorphic? The way he proves it is mainly by using intersection theory on the surface $X$, so I would guess anything is only valid up to numerical equivalence? Or is there some hidden feature (maybe the fact that $\dim X = 2$?) that I'm missing which gives "numerical equivalence" $\implies$ "linear equivalence"?
For the context: $X$ is a smooth projective surface over an algebraically closed field of characteristic $p$. $D,S \subset X$ are some given divisors, and $f: X \to C$ is a singular morphism to a smooth projective curve $C$, $\mathcal L$ some line bundle on $C$.