curve is $2C+B\in |\mathcal{O}_{\mathbb{P}^2}(2m+2k)|$, where $C,B$ are reduced curves of degree $m,2k$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(4-k-m)|$. $k=2,3$. $0<m\le 4-k$; ($p_g(X,\Lambda)=3$)

-   or $\Lambda=0$, $K_X^2=(K_{\bar{X}}+\bar D)^2=1$. $(\bar{X},\bar{D})$ is a normal Gorenstein stable log surface as in Thm \[delta-genus-one\] (6). $\bar D$ is a 2 section with $p_a(\bar D)=2$. $\tau$ is induced from the double covering $\bar D \to \mathbb{P}^1$. ($p_g(X)=2$)

We see that $\pi^*H^0(X,K_X+\Lambda)=H^0(\bar{X},\pi^*(K_X+\Lambda))$ and $(\bar{X},\bar D+\bar{\Lambda})$ is a stable log surface as in Thm \[delta-genus-one\].

First $\Phi_{|\pi^*(K_X+\Lambda)|}$ is not an embedding by Remark \[separateNonnormalCurve\]. Second $\Phi_{|\pi^*(K_X+\Lambda)|}$ is not composed with a pencil of genus 2 curves since $\bar D+\bar{\Lambda}\not= 0$.

If $\Phi_{|\pi^*(K_X+\Lambda)|}$ is composed with a pencil of elliptic curves, $\bar{D}+\bar{\Lambda}$ is either a 2-section or a sum of two elliptic fibers of the fibration map. For the former case, we see that $\bar{\Lambda}=0$ and $\bar{D}$ is a genus 2 curve by Remark \[separateNonnormalCurve\]. Moreover, $(K_{\bar X}+\bar D)^2=1$. $\tau$ is induced from the double map $\bar{D} \to \mathbb{P}^1$. For the latter case, we see that either $\bar{D}=F_1+F_2$ and $\bar{\Lambda}=0$, or $\bar{D}=F_1$ and $\bar{\Lambda}\not=0$, where $F_i$ is a fiber of the fibration map. Both cases can be excluded since the $F_1\cap F_2$ or $F_1\cap\bar{\Lambda}$ on $\bar{X}$ can not be glued into a Gorenstein 