-F_Y) \ne 0\quad\text{for all }m\in\mathcal S.$$ Consider the exact sequence $$0 \to \OO_Y(-F_Y) \to \bigwedge^r\mathcal E\to \mathcal Q \to 0.$$ Since $\OO_Y(-F_Y)$ is saturated in $\bigwedge^r\mathcal E$, the sheaf $\mathcal Q$ is torsion free, and hence $\tilde F_Y=c_0(\mathcal Q)$ is pseudoeffective by Theorem \[thm:CP00\]. From the above exact sequence, there exists a positive integer $\ell$ such that $\ell K_Y\sim\tilde F_Y-F_Y$.  From , for every $m\in\mathcal S$ we obtain a