is a quadric in $\mathbb{P}^3$, and $\Lambda\in |\mathcal{O}_{\mathbb{P}^3}(4)|_X|$. ($p_g(X,\Lambda)=6$)

-   $X$ is $\mathbb{P}^2$ blown up at $k$ points (possible infinitely near), and $\Lambda\in |-2K_X|$. ($p_g(X,\Lambda)=10-k$, and $k=0,1,...,7$)

-   $X$ is a cone over an elliptic curve of degree $N$ in $\mathbb{P}^{N-1}$, and $\Lambda\in |-2K_X|$. ($p_g(X,\Lambda)=N$)

-   $X$ is obtained from $\tilde{X}$ by contracting a $(-n)$ curve $G$, where $n=p_g(X,\Lambda)-1$ and $\tilde{X}$ is an elliptic surface (possible singular) with $G$ as the rational zero section. Every elliptic fiber of $\tilde{X}$ is irreducible. $\Lambda$ is the image of a sum of two different elliptic fibers which admit at worst $A_n$ type singularities.

-   $X$ is a (possibly singular) Del Pezzo surface of degree 1, namely $X$ has at most canonical singularities and elliptic singularities, $-K_X$ is ample and $K_X^2=1$. The curve $\Lambda$ belongs to the system $|-2K_X|$, and $p_a(\Lambda)=2$. $p_g(X,\Lambda)=2$.

-   $\Lambda=0$, $|K_X+\Lambda|$ is composed with a pencil of genus $2$ curves. $X$ is canonically embedded as a hypersurface of degree 10 in the smooth locus of $\mathbb{P}(1,1,2,5)$. $p_g(X)=2$.

In Theorem 1.1(1) and Theorem 4.3(i), $k=1,2,3,4$ should be corrected by $k=2,3,4$. The case $k=1$ is excluded as $\Phi_{|K_X+\Lambda|}$ would be an embedding.

Non-normal Gorenstein stable log surfaces {#non-normal}
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Let $(X,\Lambda)$ be 