Let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be the normalization of the blow-up of $\mathcal{X}$ along $\mathfrak{a}$ and let $D\in\Div_{0}(\mathcal{X}^{\prime})$ be the vertical Cartier divisor such that $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(D)$. The assumption implies that $\pi^{*}\mathcal{L}\otimes\mathcal{O}_{\mathcal{X}^{\prime}}(D)$ is also generated by its global sections, so that $\mathcal{L}+D$ is nef. The result follows since the model function $\log|\mathfrak{a}|$ is determined on $\mathcal{X}^{\prime}$ by $D$. ∎