X_2$ is the connecting curve. Write $(K_X+\Lambda)|_{X_i}:=(\pi^*(K_X+\Lambda))|_{X_i}=K_{X_i}+C+\Lambda|_{X_i}$.

Let $\mathcal{R}_{X_i\to C}\colon H^0(X_i,(K_X+\Lambda)|_{X_i})\to H^0(C,(K_X+\Lambda)|_C)$ be the restriction map. Then we have the following fiber product diagram of vector spaces $$\xymatrix{
    H^0(X, K_X+\Lambda) \ar[r]\ar[d] & H^0(X_1,(K_X+\Lambda)|_{X_1})\ar[d]^{\mathcal{R}_{X_1\to C}}\\
H^0(X_2,(K_X+\Lambda)|_{X_2})\ar[r]^{-\mathcal{R}_{X_2\to C}}& H^0(C, (K_X+\Lambda)|_C)
   }.$$

Moreover, denote $r_{X_i\to C}((K_X+\Lambda)|_{X_i}):=\dim\mathcal{R}_{X_i\to C}(H^0(X_i,(K_X+\Lambda)|_{X_i}))$. We have $$\begin{aligned}
h^0(X, K_X+\Lambda)\le& h^0(X_1,(K_X+\Lambda)|_{X_1})+h^0(X_2,(K_X+\Lambda)|_{X_2})\\
&-\max\{r_{X_1\to C}((K_X+\Lambda)|_{X_1}),r_{X_2\to C}((K_X+\Lambda)|_{X_2})\}.\end{aligned}$$

irreducible Non-normal Gorenstein stable surfaces with $\Delta(X,K_X+\Lambda)=1$ {#nnorm}
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Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface and $\bar{X}$, $\pi$, $D$, $\bar{D}$, $\tau$ defined as before. Since $H^0(X,K_X+\Lambda)\cong \pi^*H^0(X,K_X+\Lambda) \subset H^0(\bar X,\pi^*(K_X+\Lambda))$, we have $\Delta(X,K_X+\Lambda)\ge \Delta(\bar{X},\pi^*(K_X+\Lambda))$, and ’=’ holds if and only if $\pi^*H^0(X,K_X+\Lambda) = H^0(\bar X,\pi^*(K_X+\Lambda))$.

\[nonnorm&irred\] Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface.

Then $\Delta(X,K_X+\Lambda)\ge 1$.

We only need to show that the case $\Delta(X,K_X+\Lambda)=\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$ does not occur. First, $h^0(X,K_X+\Lambda)=h^0(\bar{X},\pi^*(K_X+\Lambda))$ implies every section in $H^0(\bar{X},\pi^*(K_X+\Lambda))$ is $\tau$-anti-invariant restricting to $\bar{D}$. Hence $H^0(\bar{X},\pi^*(K_X+\Lambda))$ can not separate points of $\bar{D}$ by Remark \[separateNonnormalCurve\]. However, $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$ implies $\pi^*(K_X+\Lambda)$ is very ample by Cor \[gepg-2\], therefore it will separate points of $\bar{D}$, a contradiction.

\[delta1,1\]

Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface as before. Assume $\Delta(X,K_X+\Lambda)=\Delta(\bar{X},\pi^*(K_X+\Lambda))=1$. Then

-   either $X$ is a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branched 