surface.

Suppose that the ${\bf Q}$-divisor $L$ is nef and big and $K_X + L$ is nef. Let $F_0 = m (F_0)_{\rm red}$ be the multiple fiber. We set $m = 1$ and let $F_0$ be a general (smooth) fibre, if $\pi$ is multiple fibre free. Then $K_X \sim -(F_0)_{\rm red}$. Let $a > 0$. Consider the exact sequence: $$0 \rightarrow {{{\mathcal{O}}}}_X(K_X + \lceil aL \rceil - (F_{\rm 0})_{\rm  red}) \rightarrow
{{{\mathcal{O}}}}_X(K_X + \lceil aL \rceil) \rightarrow {{{\mathcal{O}}}}_{(F_{\rm 0})_{\rm red}}
(K_X + \lceil aL \rceil|_{(F_{\rm 0})_{\rm red}}) \rightarrow 0.$$

Let us find the condition for $aL - (F_{\rm 0})_{\rm red}$ to be nef and big. Note that $aL - (F_{\rm 0})_{\rm red} \sim aL + K_X = (a-1) L + (K_X + L)$. So $aL - (F_{\rm 0})_{\rm red}$ is nef and big if either

\(i) $a > 1$, or

\(ii) $a = 1$ and $K_X + L$ is nef and big.

Assume that either (i) or (ii) is satisfied. Then $H^i(X, K_X + \lceil aL \rceil - (F_{\rm 0})_{\rm red}) = 0 =
H^i(X, K_X + \lceil aL \rceil)$ for all $i > 0$, by Sakai’s vanishing for surfaces. For the integral divisor $M := K_X + \lceil aL 