Q}$-divisor $L$.

As mentioned above, we assume that $X = \mathbb{F}_d$ for some $d \ge 1$. Let $L$ be a nef and big effective [**Q**]{}-divisor such that $K_X+L$ is nef. If ${\text{\rm Supp}}(L)$ does not contain the negative curve $C$, then $E : = \lceil L \rceil - L$ is effective and nef; so $\lceil L \rceil = L + E$ is nef and big and $K_X + \lceil L \rceil = K_X + L + E$ is nef; then the Serre duality and Riemann-Roch theorem for Cartier divisor imply that $h^0(X, K_X + \lceil L \rceil) \ge
\frac{1}{2} \lceil L \rceil (K_X + \lceil L \rceil)
+ \chi({{\mathcal{O}}}_X) \ge 0 + 1$. Theorefore, we may assume that ${\text{\rm Supp}}(L)$ contains $C$.

Write $L = \sum_i c_i C_i + \sum f_j F_j$ where $C_1 = C$, the $C_i$’s are distinct horizontal components and $F_j$’s are distinct fibres, where $c_i > 0$, $f_j > 0$.

Suppose on the contrary that $H^0(X, K_X+ \lceil L
\rceil)=0$. Then by Lemma \[treelem\], $L_{\rm red}$ is a connected rational tree. Hence one of the following cases occurs:

Case (i). $L = c_1C_1 + \sum_{j=1}^k f_j F_j$ ($k \ge 0$),

Case (ii). $L = \sum_{i=1}^k c_i 