$C_{N-1}$ and $c$ is not the vertex. We show that this does not occur. By Cor \[lineIntersecting\] $pr_c|_{\bar{X}}$ is an isomorphism outside the ruling $l$ passing $c$. Hence $\bar{D}$ must be $l$. However this is impossible as the vertex of $C_{N-1}$ would not be glued into a Gorenstein slc singularity.

Case IV) $\bar{X}$ is $\Sigma_{d}$. We show that this does not occur as well. Similarly as in Case III), we see that $\bar{D}$ should be a ruling $\Gamma$ passing through $c$. Hence $\bar{D}\cdot \bar{\Lambda}=3$, which is impossible.

Case V) $\bar{X}$ is $\mathbb{P}^2$ embedding in $\mathbb{P}^5$. We show that this does not occur. Since by Cor \[lineIntersecting\], we see that $pr_c|_{\bar{X}}$ is an isomorphism outside $c$, which is impossible.

Next we consider the cases where $c\not\in \bar{X}$. We see $pr_c|_{\bar{X}}$ is a morphism of degree at most two by Cor \[lineIntersecting\] and it contracts no curve on $\bar{X}$.

Case VI) $pr_c|_{\bar{X}}$ is a double cover. In this case $\bar{X}$ is a quadric in $\mathbb{P}^3$. Denote $pr_c|_{\bar{X}}$ as $\delta$. The branch curve $B\in |\mathcal{O}_{\mathbb{P}^2}(2)|$. $K_{\bar{X}}+\bar{D}+\bar{\Lambda}=\mathcal{O}_{\bar{X}}(1)=\delta^*\mathcal{O}_{\mathbb{P}^2}(1)$. Hence $\bar{D}+\bar{\Lambda}\in |\delta^*\mathcal{O}_{\mathbb{P}^2}(3)|=|\mathcal{O}_{\bar{X}}(3)|$. $\bar{D}$ is the pre-image of a curve $C$ on $\mathbb{P}^2$ under $pr_c|_{\bar{X}}$. Denote $m$ as the degree of $C$. $m\le 3$. We have the 