C}((K_X+\Lambda)|_{X_2})\}-2.
\end{split}$$ By Lemma \[restdim\] we see that $\Delta(X_2,(K_X+\Lambda)|_{X_2})+r_{X_2\to C}((K_X+\Lambda)|_{X_2})\ge 2$. Thus $\Delta(X,K_X+\Lambda)  \ge \Delta(X_1,(K_X+\Lambda)|_{X_1})$.

If ’=’ holds, then $$\begin{aligned}
  \max\{r_{X_1\to C}((K_X+\Lambda)|_{X_1}),r_{X_2\to C}((K_X+\Lambda)|_{X_2})\}= 2- \Delta(X_2,(K_X+\Lambda)|_{X_2}).
  \end{aligned}$$ Therefore, by Lemma \[restdim\], $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=2$, if $\Delta(X_2,(K_X+\Lambda)|_{X_2})=0$. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=1$, if $\Delta(X_2,(K_X+\Lambda)|_{X_2})=1$. Then other statements of (ii) follow from Lemma \[restdim\].

We then have some corollaries.

\[genus-like\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface. Assume $Y\subset X$ is a connected subsurface. Then we have $\Delta(X,K_X+\Lambda)\ge \Delta(Y,(K_X+\Lambda)|_{Y})$.

If $X_i\subset X$ is an irreducible surface connected to $Y$, $\Delta(Y,(K_X+\Lambda)|_{Y})\le \Delta(Y\cup X_i,(K_X+\Lambda)|_{Y\cup X_i})$ by Lemma \[2comps\]. Then by induction hypothesis, we have $\Delta(Y,(K_X+\Lambda)|_{Y})\le \Delta(X,K_X+\Lambda)$.

Let $(X,\Lambda)$ be a reducible Gorenstein stable log surface. Write $X=\bigcup X_i$. We say that $\Phi:=\Phi_{K_X+\Lambda}$ separates $X_i$ and $X_j$, if $\Phi(X_i\setminus X_i\cap X_j)\cap \Phi(X_j\setminus X_i\cap X_j)=\emptyset$.

\[globalsection\] Let $(X,\Lambda)$ be a connected reducible Gorenstein stable log surface with $X=X_1\cup X_2$ such that $X_1$ is connected and $X_2$ is irreducible. Then

-   We have the following commutative diagram: $$\xymatrix{
       0 \ar[r] & \ker \mathcal{R}_{X\to X_1} \ar[r]\ar[d]^{\cong} & H^0(X,K_X+\Lambda) \ar[r]^{\mathcal{R}_{X\to X_1}}\ar[d]^{\mathcal{R}_{X\to X_2}} &H^0(X_1,(K_X+\Lambda)|_{X_1})\ar[d]^{\mathcal{R}_{X_1\to X_1\cap X_2}}\\
       0 \ar[r] & \ker\mathcal{R}_{X_2\to X_1\cap X_2} \ar[r] & H^0(X_2,(K_X+\Lambda)|_{X_2}) \ar[r]^{\mathcal{R}_{X_2\to X_1\cap X_2}} &H^0(X_1\cap X_2,(K_X+\Lambda)|_{X_1\cap X_2})
       .
      }$$

-   If $\mathrm{im}\, \mathcal{R}_{X_1\to X_1\cap X_2}\subset \mathrm{im} \, \mathcal{R}_{X_2\to X_1\cap 