Let $\mathcal{X}$ be a regular model and let $\mathfrak{a}$ be a vertical ideal sheaf on $\mathcal{X}$. For each rational number $c>0$ the multiplier ideal of $\mathfrak{a}^{c}$ is the vertical ideal sheaf of $\mathcal{X}$ defined as

$$\mathcal{J}(\mathfrak{a}^{c}):=\pi_{*}\mathcal{O}_{\mathcal{X}^{\prime}}\left(K_{\mathcal{X}^{\prime}/\mathcal{X}}-\lfloor c\,D\rfloor\right)$$

where $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ is a vertical blow-up with $\mathcal{X}^{\prime}$ SNC such that $\pi^{-1}\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}$ is locally principal and $D\in\Div_{0}(\mathcal{X}^{\prime})$ is the corresponding effective Cartier divisor.