Given a simple integral affine $3$-manifold with singularities $(B,\Delta,\mathscr{A})$, all of whose negative vertices are straight (i.e. locally affine isomorphic to Example 3.12), a localized thickening of $\Delta$ is given by the data $(\Delta^{\blacklozenge},\{D_{p^{-}}\}_{p^{-}\in\mathcal{N}})$ where:

• (i)

  $\Delta^{\blacklozenge}$ is the closed subset obtained from $\Delta$ after replacing a neighborhood of each negative vertex with a shape of the type depicted in Figure 17. This replacement takes place in the plane corresponding to $\{x_{1}=0\}$ of the local model, Example 3.12.

• (ii)

  $\mathcal{N}$ is the set of negative vertices and for each $p^{-}\in\mathcal{N}$, $D_{p^{-}}$ is a submanifold of $B$, homeomorphic to a disk and containing the codimension $1$ component of $\Delta^{\blacklozenge}$ around the negative vertex $p^{-}$. Moreover, $D_{p^{-}}$ is contained in the plane $\{x_{1}=0\}$. We depict $D_{p^{-}}$ as the gray area in Figure 17.