Take as a symplectomorphism $\Phi$ the linear map

$$\Phi(u_{1},u_{2})=\frac{1}{\sqrt{2}}\left(u_{1}-u_{2},u_{1}+u_{2}-\sqrt{2}\right).$$

(34)

Then the fibration resulting from Proposition 5.4 can be written explicitly in the coordinates of the total space. We obtain:

$$f(z_{1},z_{2},z_{3})=\left(\frac{1}{2}\left(|z_{1}|^{2}-|z_{2}|^{2}\right),\log\frac{1}{\sqrt{2}}\left|\gamma-z_{3}\right|,\log\frac{1}{\sqrt{2}}\left|\gamma+z_{3}-\sqrt{2}\right|\right),$$

(35)

where $\gamma$ is as in (33). It is not difficult to see that $\Phi\circ\Gamma_{0}$ sends $\Sigma$ to the surface in $(\mathbb{C}^{\ast})^{2}$ given by

$$\Sigma^{\prime}=\{v_{1}+v_{2}+1=0\},$$

which is, topologically, a pair of pants. Then the discriminant locus is

$$\Delta=\{0\}\times\Log(\Sigma^{\prime}),$$

which has the shape in Figure 4. This example is topologically conjugate to the one in Example 2.9, before the surface $\Sigma^{\prime}$ has been twisted. For the discussion of the topology of the fibres in this example we refer to Example 2.9.