slc singularity on $X$.

If $\bar{X}$ is a double cover of $\mathbb{P}^2$, then $\bar{D}+\bar{\Lambda}\in |\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}^*\mathcal{O}_{\mathbb{P}^2}(4-k)|$, $k=2,3$. Remark \[separateNonnormalCurve\] indicates $\bar{D}=\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}^{-1}(C)$, where $C$ is a curve on $\mathbb{P}^2$. Denote $m$ as the degree of $C$. Then $\bar{\Lambda}\in |\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}^*\mathcal{O}_{\mathbb{P}^2}(4-k-m)|$. We have the following commutative diagram: $$\xymatrix{
 \bar{D}\ar@{^{(}->}[r]\ar[d] & \bar{X}\ar[d]^{\pi}\ar[rr]^{\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}} && \mathbb{P}^2\\
 D\ar@{^{(}->}[r] & X \ar[rru]_{\Phi_{|K_{X}+\Lambda|}}
 }.$$

We see that $X$ is also a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branch curve is $2C+B\in |\mathcal{O}_{\mathbb{P}^2}(2m+2k)|$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(4-k-m)|$.

Next we consider those irreducible non-normal Gorenstein stable log surfaces $(X,\Lambda)$ with $\Delta(X,K_X+\Lambda)=1$ and $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$.

We first notice that $\pi^*H^0(X,K_X+\Lambda)\subset H^0(\bar{X},\pi^*(K_X+\Lambda))$ is of codimension one. Hence it has a base point $c\in \mathbb{P}(H^0(\bar{X},\pi^*(K_X+\Lambda)))$. We have the following commutative diagram: $$\xymatrix{
      \bar{X} \ar@{^{(}->}[rrr]^<(0.3){\Phi_{|\pi^*(K_X+\Lambda)|}}\ar[d]^{\pi} & & & \mathbb{P}(H^0(\bar{X},\pi^*(K_X+\Lambda)))\ar@{-->}[d]^{pr_c}\\
X\ar@{-->}[rrr]^<(0.3){\Phi_{|K_X+\Lambda|}}& & & \mathbb{P}(\pi^*H^0(X,K_X+\Lambda))
  }.$$

We regard $\Phi_{|\pi^*(K_X+\Lambda)|}$ as an inclusion. To describe $pr_c|_{\bar{X}}$, we use the following theorem (see [@Ber06 Thm 2.1]):

\[linearsectionthm\] Let $X$ be a non-degenerate irreducible subvariety of $\mathbb{P}^r$ and $L$ be a linear subspace of $\mathbb{P}^r$ of dimension $s\le r$ such that $L\cap X$ is a 0-dimensional scheme $\zeta$. Then $\mathrm{length}\, \zeta \le \Delta(X,\mathcal{O}(1))+s+1$.

\[lineIntersecting\] Let $X$ be a non-degenerate irreducible subvariety of $\mathbb{P}^r$ with $\Delta(X,\mathcal{O}(1))=0$ and $L$ is a line intersecting $X$ 