Exactness of a sequence

Question

Denote $X$ a rational projective surface, $\left|-K_X\right| $ not empty and $F_0 \in \left|-K_X\right|$ . In this paper ((2) on page 7) the authors state the sequence \begin{align*} 0 \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}_X(n F_0) \longrightarrow \mathcal{O}_{F_0}(n F_0) \longrightarrow 0 \end{align*} is exact. This is not quite an exact sequence which arises from the sequence associated to ideal sheaves, and I fail to see that it is in fact exact.