I have a morphism $f: \mathscr{F} \rightarrow \mathscr{G}$ of invertible sheaves in a smooth variety $X$, or equivalently, a global section $s \in H^{0}(X, \mathscr{F}^{\vee} \otimes \mathscr{G})$, which induces a morphism $\sigma: \mathcal{O}_{X} \rightarrow \mathscr{F}^{\vee} \otimes \mathscr{G}$, the multiplication by $s$. Denote by $Z(\sigma)$ the zero locus of $\sigma$, that is, the zero locus of $s$. I am thinking if the isomorphism $\mathcal{O}_{X}(Z(\sigma)) \cong \mathscr{F}^{\vee} \otimes \mathscr{G}$ holds, when $f$ is surjective. In other words, $\mathcal{O}_{X}/\ker(\sigma) \cong \mathcal{O}_{X}(Z(\sigma))$?
Thanks.