---
abstract: 'In this paper, we classify Gorenstein stable log surfaces with $(K_X+\Lambda)^2=p_g(X,\Lambda)-1$.'
address: 'Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084'
author:
- Jingshan Chen
title: ' Gorenstein stable log surfaces with $(K_X+\Lambda)^2=p_g(X,\Lambda)-1$'
---

Introduction
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KSBA stable (log) surfaces are the two-dimensional analogues of stable (pointed) curves. They are the fundamental objects in compactifying the moduli spaces of smooth surfaces of general type.

In general, stable (log) surfaces are difficult to classify. We may first focus on Gorenstein ones, i.e. $K_X$ (resp. $K_X+\Lambda$) being Cartier. In [@LR13], Liu and Rollenske give several inequalities for the invariants of Gorenstein stable log surfaces. One important inequality among them is the stable log Noether inequality $(K_X+\Lambda)^2\ge p_g(X,\Lambda)-2$ (see [@LR13 Thm 4.1]). This can be rephrased as $\Delta(X,K_X+\Lambda)\ge 0 $, where $\Delta$ is Fujita’s $\Delta$-genus.

Gorenstein stable log surfaces with $\Delta(X,K_X+\Lambda)=0$ have been classified in [@LR13]. Normal Gorenstein stable log surfaces with $\Delta(X,K_X+\Lambda)=1$ have been classified in [@Chen18]. Here we continue to classify non-normal Gorenstein stable log surfaces with $\Delta(X,K_X+\Lambda)=1$. The main result is as follows.

\[Main Theorem \] Let $(X,\Lambda)$ be a Gorenstein stable log surface with $\Delta(X,K_X+\Lambda)=1$ which is not normal.

If $X$ is irreducible, let $\pi\colon \bar{X}\to X$ be 