the proper transformation $\bar{C}$ is a line on the normalization $\bar{X}$ of $X$.*

\[restdim\] Let $(X,\Lambda)$ be an irreducible Gorenstein stable log surface with a reduced curve $C$ on it. Then

-   if $X$ is normal and $\Delta(X,K_X+\Lambda)=0$, then $r_{X\to C}(K_X+\Lambda)\ge2$. Moreover ’=’ holds if and only if $C$ is a line on $X$.

-   if $X$ is normal and $\Delta(X,K_X+\Lambda)=1$, then $r_{X\to C}(K_X+\Lambda)\ge1$.

    Moreover, ’=’ holds if and only if $|K_X+\Lambda|$ is composed with a pencil and $C$ is a fiber on $X$.

    If $|K_X+\Lambda|$ is not composed with a pencil, then $r_{X\to C}(K_X+\Lambda)\ge2$ and ’=’ holds if and only if $\Phi_{|K_X+\Lambda|}(C)$ is a line in $|K_X+\Lambda|^*$.

-   if $X$ is non-normal and $\Delta(X,K_X+\Lambda)=1$, then $r_{X\to C}(K_X+\Lambda)\ge1$. Moreover, ’=’ holds if and only if $|K_X+\Lambda|$ has a base point, and $C$ is a line passing through the base point.

\(i) and (ii) follows from Thm \[gepg-2\], Thm \[delta-genus-one\] and Lemma \[restsecions\]. To prove (iii), we note that $H^0(X,K_X+\Lambda)\cong \pi^* H^0(X,K_X+\Lambda)$, which corresponds to the space of hyperplane sections passing through the base point $c$ of $\pi^* H^0(X,K_X+\Lambda)$ in $ \mathbb{P}:=|\pi^*(K_X+\Lambda)|^*$. Denote $pr_c\colon \mathbb{P}\dashrightarrow \mathbb{P}':=|K_X+\Lambda|^*$ as the projection from the point $c$.

We have the following commutative diagram: $$\xymatrix{
  \bar{C}\ar@{^{(}->}[r]\ar[d] 