case $X$ itself might not even be $\mathbb{Q}$-Gorenstein.

Normal Gorenstein stable log surfaces with small $\Delta$-genus {#zero-Delta-genus}
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Let $X$ be a variety and $\mathcal{L}$ be an ample line bundle on it. Fujita introduced several invariants for such polarized varieties. One important of them is the $\Delta$-genus $\Delta(X,\mathcal{L}):=\mathcal{L}^{\dim X}-h^0(X,\mathcal{L})+\dim X$.

\[gepg-2\] Let $(X,\Lambda)$ be a normal irreducible Gorenstein stable log surface. Then $\Delta(X,K_X+\Lambda)\ge 0$.

Moreover if ’=’ holds, then $(X,\Lambda)$ is one of the followings:

-   $X$ is $\mathbb{P}^{2}$, $\mathcal{O}_X(K_X+\Lambda) =\mathcal{O}_{\mathbb{P}^{2}}(1)$ and $\Lambda\in|\mathcal{O}_{\mathbb{P}^{2}}(4)|$;

-   $X$ is $\mathbb{P}^{2}$, $\mathcal{O}_X(K_X+\Lambda)=\mathcal{O}_{\mathbb{P}^{2}}(2)$ and $\Lambda\in|\mathcal{O}_{\mathbb{P}^{2}}(5)|$;

-   $X$ is $\Sigma_d$, $K_X+\Lambda\equiv \Delta_0+\frac{N+d-1}{2}\Gamma$ and $\Lambda\in|3\Delta_0+\frac{N+3d+3}{2}\Gamma|$; ($N-d-3\ge0$ is an even number);

-   $X$ is a singular quadric $C_2$ in $\mathbb{P}^{3}$, $\mathcal{O}_X(K_X+\Lambda)=\mathcal{O}_{C_2}(1)=\mathcal{O}_{\mathbb{P}^{2}}(1)|_{C_2}$ and $\Lambda\in|\mathcal{O}_{C_2}(3)|$;

-   $X$ is a cone $C_{N-1}\hookrightarrow \mathbb{P}^{N}$, $\mathcal{O}_X(K_X+\Lambda)=\mathcal{O}_{\mathbb{P}^{N}}(1)|_{C_{N-1}}$ and the proper transformation $\bar{\Lambda}$ in the minimal resolution $\Sigma_{N-1}$ is linearly equivalent to $2\Delta_0+2N\Gamma$. ($N>3$)

\[delta-genus-one\] Let $(X,\Lambda)$ be a normal Gorenstein stable log surface with $(K_X+\Lambda)^2= p_g(X,\Lambda)-1$.

Then $(X,\Lambda)$ is one of the followings:

-   $X$ is a double cover of $\mathbb{P}^2$. $\Phi_{|K_X+\Lambda|}\colon X\to \mathbb{P}^2$ is the double covering map. The branch curve $B\in |\mathcal{O}_{\mathbb{P}^2}(2k)|$ is a reduced curve which admits curve singularities of lc double-covering type, and $\Lambda\in |{\Phi_{|K_X+\Lambda|}}^*\mathcal{O}_{\mathbb{P}^2}(4-k)|$. ($p_g(X,\Lambda)=3$, and $k=2,3,4$)

-   $X$ 