The discriminant locus is the image of the singular locus $\mathcal{C}_{ij}=\{z_{i}=z_{j}=0\}$ under the moment map. We shall focus on $\mathcal{C}_{01}$. The holomorphic moment coordinate $\eta$ depends only on $\Omega$ and the $T^{2}$ action, so $\eta=z_{0}z_{1}z_{2}$ as before and vanishes on $\mathcal{C}_{01}$. The symplectic moment coordinates are defined by $d\mu_{i}=-\iota_{\frac{\partial}{\partial\theta_{i}}}\omega$, and are normalised to be zero at $(z_{1},z_{2},z_{3})=0$. In particular since the Hamiltonian vector field $\frac{\partial}{\partial\theta_{1}}$ vanishes on $\mathcal{C}_{01}$, the moment $\mu_{1}$ must be the constant zero on $\mathcal{C}_{01}$. Furthermore $\mu_{2}>0$ on $\mathcal{C}_{01}$ by considering the weight of the remaining $S^{1}$ action at the fixed point, so the image of $\mathcal{C}_{01}$ is contained in $\mathfrak{D}_{1}\cup\{0\}$. The infinite volume condition and the formula

$$0\leq\int_{\mathcal{C}_{01}\cap\{\mu_{2}<m\}}\omega=-2\pi\int_{\mu_{2}=m}^{0}d\mu_{2}=2\pi m,\quad\forall m\geq 0,$$

ensure that $\mu_{2}$ stretches to infinity, so $\mathfrak{D}_{1}\cup\{0\}$ is the image of $\mathcal{C}_{01}$. Likewise the image of $\mathcal{C}_{02}$ is $\mathfrak{D}_{2}\cup\{0\}$ and the image of $\mathcal{C}_{12}$ is $\mathfrak{D}_{3}\cup\{0\}$. ∎