log surface $X_i$ with $\Delta(X_i,(K_X+\Lambda)|_{X_i})=0$. All the connecting curves of $X$ are lines.

Now we give a brief account of each section. In \[prelim\], we recall some definitions and facts about Gorenstein stable log surfaces. In \[zero-Delta-genus\], we recall the definition of Fujita’s $\Delta$-genus and include some results about normal Gorenstein stable log surfaces $(X,\Lambda)$ with $\Delta(X,K_X+\Lambda)=0$ or $1$. In \[non-normal\], we include some results about non-normal Gorenstein stable log surfaces. In \[nnorm\], we deal with the case that $X$ is non-normal and irreducible. In \[reducible-stable\], we deal with the case that $X$ is reducible. Finally we describe Gorenstein stable surfaces with $K_X^2= p_g-1$.

Acknowledgements: {#acknowledgements .unnumbered}
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I am grateful to Prof. Jinxing Cai and Prof. Wenfei Liu for their instructions. I would also thank the anonymous referee for helpful advices and suggestions.

Notations and conventions
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We work exclusively with schemes of finite type over the complex numbers $\mathbb{C}$.

-   A surface is a connected reduced projective schemes of pure dimension two.

-   By abuse of notation, we sometimes do not distinguish a Cartier divisor $D$ and its associated invertible sheaf $\mathcal{O}_X(D)$.

-   $\Sigma_d$ denotes a Hirzebruch surface, which admits a $\mathbb{P}^1$ fibration over $\mathbb{P}^1$. We denote $\Gamma$ 