Let $\mathcal{X}$ be an SNC model of $X$. The image of the evaluation map $\ev_{\mathcal{X}}:X\to\Div_{0}(\mathcal{X})_{\mathbf{R}}^{*}$ defined in Corollary 2.5 then admits the structure of a rational simplicial complex, defined as follows. Write the special fiber as $\mathcal{X}_{0}=\sum_{i\in I}b_{i}E_{i}$, where $b_{i}\in\mathbf{N}^{*}$ and $(E_{i})_{i\in I}$ are the irreducible components. Let $x_{E_{i}}\in X$ be the associated divisorial points and set $e_{i}:=\ev_{\mathcal{X}}(x_{E_{i}})\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}^{*}$. Recall from Definition 1.1 that for each $J\subset I$ the intersection $E_{J}:=\bigcap_{j\in J}E_{j}$ is either empty or a smooth irreducible $k$-variety. For each $J\subset I$ such that $E_{J}\neq\emptyset$ let $\hat{\sigma}_{J}\subset\Div_{0}(\mathcal{X})^{*}_{\mathbf{R}}$ be the simplicial cone defined by $\hat{\sigma}_{J}:=\sum_{j\in J}\mathbf{R}_{+}e_{j}$. These cones naturally define a (regular) fan $\hat{\Delta}_{\mathcal{X}}$ in $\Div_{0}(\mathcal{X})^{*}_{\mathbf{R}}$. Slightly abusively, we shall also denote by $\hat{\Delta}_{\mathcal{X}}$ the support of this fan, that is, the union of all the cones $\hat{\sigma}_{J}$. We then define the dual complex³³ 3 The dual complex is called the Clemens polytope in [KS06]. of $\mathcal{X}$ by

$$\Delta_{\mathcal{X}}:=\hat{\Delta}_{\mathcal{X}}\cap\left\{\langle\mathcal{X}_{0},\cdot\rangle=1\right\}.$$

Each $J\subset I$ such that $E_{J}\neq\emptyset$ corresponds to a simplicial face

$$\sigma_{J}:=\hat{\sigma}_{J}\cap\left\{\langle\mathcal{X}_{0},\cdot\rangle=1\right\}=\Conv\{e_{j}\mid j\in J\}$$

of dimension $|J|-1$ in $\Delta_{\mathcal{X}}$, where $\Conv$ denotes convex hull. This endows $\Delta_{\mathcal{X}}$ with the structure of a (compact rational) simplicial complex, such that $\sigma_{J}$ is a face of $\sigma_{L}$ iff $J\supset L$.