$X$ be an irreducible non-normal Gorenstein stable surface with $\Delta(X,K_X)=1$. Then $X$ is one of the followings:

-   $X$ is a double cover of $\mathbb{P}^2$. The branch curve is $2C+B$, where $C$,$B$ are reduced curves of degree $4-k$, $2k$. $k=2,3$. ($p_g(X)=3$)

-   $X$ is obtained from a log surface $(\bar{X},\bar{D})$ by gluing the 2-section $\bar{D}$. $(\bar{X},\bar{D})$ is a normal Gorenstein stable log surface as in Thm \[delta-genus-one\] (6). ($p_g(X)=2$)

-   $\bar{X}$ is $\mathbb{P}^2$. $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. ($p_g(X)=2$)

-   $\bar{X}$ is a quadric in $\mathbb{P}^3$. $\bar{D}\in |\mathcal{O}_{\bar{X}}(3)|$. ($p_g(X)=3$)

reducible Gorenstein stable log surfaces with $\Delta(X,K_X+\Lambda)=1$ {#reducible-stable}
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In this section we consider reducible Gorenstein stable log surfaces. They are glued by irreducible ones along some connecting curves.

\[restsecions\] Let $X$ be a connected $S_2$ scheme of pure dimension, $\mathcal{L}$ be an invertible sheaf such that $\dim \mathrm{Bs} |\mathcal{L}|<\dim X-1$ and $C$ be a subscheme of codimension 1. Then $$\begin{aligned}
r_{X\to C}(\mathcal{L})=\dim <\Phi_{|\mathcal{L}|}(C)>+1,\end{aligned}$$ where $<\Phi_{|\mathcal{L}|}(C)>$ is the projective subspace of $|\mathcal{L}|^*$ spanned by $\Phi_{|\mathcal{L}|}(C)$.

Denote $\mathbb{P}:=|\mathcal{L}|^*$. We have the following commutative diagram: $$\xymatrix{
      H^0(\mathbb{P},\mathcal{O}_{\mathbb{P}}(1))\ar[rr]^<(0.2){\mathcal{R}_{\mathbb{P}\to \Phi_{|\mathcal{L}|}(C)}}\ar[d]^{\cong}_{\Phi_{|\mathcal{L}|}^*} & & H^0(\Phi_{|\mathcal{L}|}(C),\mathcal{O}_{\mathbb{P}}(1)|_{\Phi_{|\mathcal{L}|}(C)})\ar@{^{(}->}[d]_{\Phi_{|\mathcal{L}|}^*}  \\
    H^0(X,\mathcal{L})\ar[rr]^{\mathcal{R}_{X\to C}}& & H^0(C,\mathcal{L}|_C)
   }.$$ Therefore $r_{X\to C}(\mathcal{L})=r_{\mathbb{P}\to \Phi_{|\mathcal{L}|}(C)}(\mathcal{O}_{\mathbb{P}}(1))=\dim <\Phi_{|\mathcal{L}|}(C)>+1$.

Still we call a curve $C$ on a demi-normal scheme $X$ a *[line]{} if 