line in $\mathbb{P}^{N-1}$.

Finally, we describe $\bar{D}$. For the case where $X$ is a cone $C_{N-1}$, $\bar{D}$ must be a sum of two rulings. This can be shown by considering the plane $H_{v, L}\subset \mathbb{P}^{N-1}$ containing $v,L$, where $v$ is the vertex of $C_{N-1}$ and $L$ is a secant line of $X$ passing through $c$.

For the case where $X$ is $\mathbb{P}^2$ embedded in $\mathbb{P}^5$, we have $\bar{D}\cdot \mathcal{O}_{X}(1)=2$ since $D$ is a line in $\mathbb{P}^{N-1}$. Hence $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$.

For the case where $X$ is $\Sigma_d\hookrightarrow \mathbb{P}^N$, similarly as the former case we have $\bar{D}\cdot \mathcal{O}_{X}(1)=\bar{D}\cdot(\Delta_0+\frac{N+d-1}{2}\Gamma)=2$. Hence we have either $\bar{D}\in |\Delta_0+\Gamma|$, $N=d+3$, or $\bar{D}=\Delta_0$, $N=d+5$.

We first consider the cases where $c\in \bar{X}$.

Case I) $\bar{X}$ is $\mathbb{P}^2$ and $\pi^*(K_X+\Lambda)=\mathcal{O}_{\mathbb{P}^2}(1)$. Then $\bar{\Lambda}+\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$ and $pr_c|_{\bar{X}}$ is a fibration over $\mathbb{P}^1$. By the classification of Gorenstein slc singularities, we see that $\bar{\Lambda}\cdot\bar{D}$ is even. Hence we have either $\bar{\Lambda},\bar{D}\in  |\mathcal{O}_{\mathbb{P}^2}(2)|$ or $\bar{D}\in  |\mathcal{O}_{\mathbb{P}^2}(4)|$ and $\bar{\Lambda}=0$. Examples will be given later. We note further that $c\not\in\bar{D}$, otherwise it can not be glued into a Gorenstein slc singularity. Case II) $\bar{X}$ is $C_{N-1}$ and $c$ is the vertex. We see that $\bar{\Lambda}$ is two rulings and $\bar{D}\in \mathcal{O}_{C_{N-1}}(2)$.

Case III) $\bar{X}$ is 