This is the local model for a fibration over a neighborhood of a component of $\Delta_{a}$. Consider $Y$ and $\Sigma$ as in Example 2.8. Now think of making a small perturbation of $\Sigma$ just in a neighborhood of the “figure eight” –i.e. where the three cylinders forming $\Sigma$ are joined together– and leaving the rest unchanged. A generic perturbation will be such that, near the fibre over $b_{0}$, $\Sigma$ will intersect the fibres of $P:Y\rightarrow B$ in isolated points. Then $P(\Sigma)$ will have the shape of a $3$-legged amoeba. One then constructs the bundle $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}=Y-\Sigma$ with Chern class $c_{1}=1$ and compactifies it to $\pi:X\rightarrow Y$. The total fibration is $f=P\circ\pi$.

We can give an explicit construction of a fibration of this type, following ideas in [6]§4. Consider $(\mathbb{C}^{\ast})^{2}$ with the $T^{2}$ fibration $\Log:(v_{1},v_{2})\mapsto(\log|v_{1}|,\log|v_{2}|)$. Let $Y=\mathbb{R}\times(\mathbb{C}^{\ast})^{2}$ and $P$ be the fibration

where $t\in\mathbb{R}$ and $v=(v_{1},v_{2})\in(\mathbb{C}^{\ast})^{2}$. Define a surface $\Sigma^{\prime}$ in $(\mathbb{C}^{\ast})^{2}$ to be

and view it as a surface in $\{0\}\times(\mathbb{C}^{\ast})^{2}\subset Y$. Clearly $P(\Sigma^{\prime})$ is $\{0\}\times\Log(\Sigma^{\prime})$ and one can compute that it has the shape depicted in Figure 4. Images by $\Log$ of algebraic curves in $(\mathbb{C}^{\ast})^{2}$ are known in the literature as amoebas, and this explains the name we gave to the components of $\Delta_{a}$.

As a surface in $\mathbb{C}^{2}$, $\Sigma^{\prime}$ intersects $\{v_{1}=0\}$ in $q_{1}=(0,0,-1)$ and $\{v_{2}=0\}$ in $q_{2}=(0,-1,0)$. One can see that in a small neighborhood of $q_{1}$ one can twist $\Sigma^{\prime}$ slightly, so to make it coincide, in a smaller neighborhood, with $\{v_{2}=-1\}$. Similarly one can twist $\Sigma^{\prime}$ near $q_{2}$, so to make it coincide with $\{v_{1}=-1\}$. Finally, when $|v_{1}|$ and $|v_{2}|$ are both big, we can twist $\Sigma^{\prime}$ so to coincide with $\{v_{1}+v_{2}=0\}$. Let $\Sigma$ be this new twisted version of $\Sigma^{\prime}$. A schematic description of these twistings is described in Figure 5, where $\Sigma^{\prime}$ is the light-colored diagonal line and $\Sigma$ is the over-imposed twisted dark line. It is clear that $P(\Sigma)=\{0\}\times\Log(\Sigma)$ will have the shape of a $3$-legged amoeba whose legs have been pinched to $1$-dimensional segments toward the ends, as depicted in the right-hand side of Figure 5 (Mikhalkin [25] also defines a similar construction and calls this shape a localized amoeba). The bundle $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}=Y-\Sigma$ with Chern class $c_{1}=1$ and its compactification $\pi:X\rightarrow Y$ can again be constructed. The fibration is $f=P\circ\pi$ and $\Delta=P(\Sigma)$.

We give a description of the fibration over the codimension $1$ part of $\Delta$. One can see that the fibres of $\Log$ over a point in the interior of the amoeba intersect $\Sigma$ in two distinct points. These two points come together to a double point as the base point approaches the boundary of the amoeba. If $p_{1}$ and $p_{2}$ are two points on $T^{2}$ –which may coincide– then the singular fibres of $f$ look like $S^{1}\times T^{2}$ after $S^{1}\times\{p_{j}\}$ is collapsed to a point. This behavior is topologically the same as the one conjectured by Joyce [21] for special Lagrangian $T^{3}$ fibrations. Moreover, the singularities of the fibres are modeled on those of an explicit example of a special Lagrangian fibration with non-compact fibres (cf. Joyce [21]§5).

In view of Proposition 2.4 and Remark 2.5, the total space $X$ in this example is diffeomorphic to the one in Example 2.8, although the fibrations differ. In both cases the singularities of the fibres occur along the intersection of the critical surface $\Sigma$ with the fibres of $P$. But the intersections happen in a different way. In Example 2.8 they occur either along circles, or along a figure eight. Here they occur along circles when the fibre is over a point in the codimension $2$ part of $\Delta$ and as isolated points when the fibre is over a point in the codimension $1$ part. As argued by Joyce, the isolated singularities are more generic in certain sense (cf. [21]§3). A schematic description of the fibration over the codimension $1$ part of $\Delta$ is depicted in Figure 6. It can be compared with Figure 2. We remark that over the codimension $2$ part of $\Delta$, the fibration has the same topology of the generic singular fibration of Example 2.7. It follows that the monodromy around the legs is same as the monodromy of Example 2.8, i.e. it is represented by the matrices (3).