Let $S subset mathbb P^3$ be a smooth projective surface (over complex numbers). Let $C$ be a smooth hyperplane section. Let $Delta$ be a non-zero effective divisor on $S$ such that $h^1(mathcal O_S(nC+Delta))=0, h^1(mathcal O_S(nC-Delta))=0$ for all $n in mathbb Z$. Then my question is the following :

In this situation can we say that: $h^1(mathcal O_S(m Delta))=0$ for $m geq 2$? Can we impose any condition so that this happens?

Any help from anyone is welcome.