---
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
============

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 the normalization map. Then

-   either $\Delta(\bar{X},\pi^*(K_X+\Lambda))=1$ and $(X,\Lambda)$ is as in Thm \[delta1,1\],

-   or $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$ and $(\bar{X},\bar{\Lambda}+\bar{D})$ is as in Thm \[delta1,0\].

If $X$ is reducible, write $X=\bigcup X_i$, where $X_i$ is an irreducible component. Then $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$ or $0$ for each component $X_i$, $X$ has a unique minimal connected component $U$ such that $\Delta(U,(K_X+\Lambda)|_{U})=1$, $X\setminus U$ is composed with several trees $T_j$ with $\Delta(T_j,(K_X+\Lambda)|_{T_j})=0$ and $X$ is glued by $U$ and $T_j$ along lines.

Moreover, $U$ is one of the followings:

-   $X=U$ is a string of log surfaces $X_i$ with $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$ and $|(K_{X}+\Lambda)|_{X_i}|$ composed with a pencil of elliptic curves. The connecting curves are all fibers. Moreover, in this case $|K_X+\Lambda|$ is composed with a pencil of elliptic curves.

-   $U$ is a string of surfaces glued along lines. The end surfaces $X_i$ of the string $U$ are non-normal with $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$.

-   $U$ is composed with a single irreducible log surface $X_k$ with $\Delta(X_k,(K_X+\Lambda)|_{X_k})=1$ and $(K_{X}+\Lambda)|_{X_k}$ very ample. Moreover, in this case $K_X+\Lambda$ is very ample.

-   $U=X_j\cup X_k $ where $X_j$, $X_k$ are two Gorenstein log surfaces with $\Delta(X_j,(K_X+\Lambda)|_{X_j})=\Delta(X_k,(K_X+\Lambda)|_{X_k})=0$. All the connecting curves of $X$ are lines except $X_j\cap X_k$.

-   $U$ is a cycle of 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}
-----------------

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
-------------------------

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 $\Gamma$ as a fiber. It has a unique 1-section $\Delta_0$ whose self-intersection is $-d$.

-   We use ’$\equiv$’ to denote linear equivalent relation of divisors.

-   If $D$ is a Cartier divisor on $X$, then we denote $\Phi_{|D|}\colon X\dashrightarrow \mathbb{P}:=|D|^*$ as the rational map defined by the linear system $|D|$.

-   A line $l$ on a variety $X$ with respect to $\mathcal{O}_X(1)$ is a rational curve such that $l\cdot \mathcal{O}_X(1)=1$.

Preliminaries {#prelim}
=============

Let $X$ be a demi-normal surface, i.e. $X$ satisfies $S_2$ and is at worst ordinary double at any generic point of codimension 1. Denote $\pi\colon  \bar X \to X$ as the normalisation map of $X$. The conductor ideal $ \mathrm{\mathcal{H}om}_{\mathcal{O}_X}(\pi_*\mathcal{O}_{\bar{X}}, \mathcal{O}_X)$ is an ideal sheaf both on $X$ and $\bar{X}$ and hence defines subschemes $D\subset X \text{ and } \bar D\subset \bar X$, both reduced and of pure codimension 1; we often refer to $D$ as the non-normal locus of $X$.

Let $\Lambda$ be a reduced curve on $X$ whose support does not contain any irreducible component of $D$. Then the strict transform $\bar \Lambda$ in the normalization is well defined. We have $\pi^*(K_X+\Lambda)=K_{\bar{X}}+\bar D+\bar \Lambda$ and $(K_X+\Lambda)^2 = (K_{\bar X}+\bar D+\bar \Lambda)^2$.

\[defin: slc\] We call a pair $(X, \Lambda)$ as above a *log surface*; $\Lambda$ is called the (reduced) boundary.

A log surface $(X,\Lambda)$ is said to have *semi-log-canonical (slc)* singularities if it satisfies the following conditions:

1.  $K_X + \Lambda$ is $\mathbb{Q}$-Cartier, that is, $m(K_X+\Lambda)$ is Cartier for some $m\in\mathbb{Z}^{>0}$; the minimal such $m$ is called the (global) index of $(X,\Lambda)$.

2.  The pair $(\bar X, \bar \Lambda+\bar D)$ has log-canonical singularities.

The pair $(X,\Lambda)$ is called stable log surface if in addition $K_X+\Lambda$ is ample. A stable surface is a stable log surface with empty boundary.

By abuse of notation we say $(X, \Lambda)$ is a Gorenstein stable log surface if the index is equal to one, i.e. $K_X+\Lambda$ is an ample Cartier divisor.

Gorenstein slc singularities and semi-resolutions
-------------------------------------------------

Normalizing a demi-normal surface looses all information on the gluing in codimension one. Often it is better to work on a simpler but still non-normal surface.

A surface $X$ is called semi-smooth if every singularity of $X$ is either double normal crossing or a pinch point [^1]. The normalization of a semi-smooth surface is smooth.

A morphism of demi-normal surfaces $f\colon Y\rightarrow X$ is called a semi-resolution if the following conditions are satisfied:

1.  $Y$ is semi-smooth;

2.  $f$ is an isomorphism over the semi-smooth open subscheme of $X$;

3.  $f$ maps the singular locus of $Y$ birationally onto the non-normal locus of $X$.

A semi-resolution $f\colon Y\rightarrow X$ is called minimal if no $(-1)$-curve is contracted by $f$, that is, there is no exceptional curve $E$ such that $E^2 =K_Y\cdot E = -1$.

Semi-resolutions always exist and one can also incorporate a boundary [@KollarSMMP 10.5].

\[rem: classification of sings\] Semi-log-canonical surface singularities have been classified in terms of their resolution graphs, at least for reduced boundary [@KSB88].

Let $x\in (X, \Lambda)$ be a Gorenstein slc singularity with minimal log semi-resolution $f\colon Y\to X$. Then it is one of the followings (see [@Kollar-Mori Ch. 4], [@KollarSMMP Sect. 3.3], [@kollar12] and [@LR13]):

Gorenstein lc singularities, $\Lambda=0$

:   In this case $x\in X$ is a canonical singularity, or a simple elliptic respectively cusp singularity. For the latter the resolution graph is a smooth elliptic curve, a nodal rational curve, or a cycle of smooth rational curves (see also [@Lau77] and [@Reid97 Ch. 4]).

Gorenstein lc singularities, $\Lambda\neq 0$

:   Since the boundary is reduced, $\Lambda$ has at most nodes. If $\Lambda$ is smooth so is $X$ because of the Gorenstein assumption.

    If $\Lambda$ has a node at $x$ then $x$ is a smooth point of $X$ or $(X, \Lambda)$ is a general hyperplane section of a cyclic quotient singularity. In the minimal log resolution the dual graph of the exceptional curves is $$\bullet   \ {-}\ c_1  \ -\ \cdots  \ - \ c_n  \ {-}
    \ \bullet \qquad (c_i\geq1),$$ where $c_i$ represents a smooth rational curve of self-intersection $-c_i$ and each $\bullet$ represents a (local) component of the strict transform of $\Lambda$. If $c_i=1$ for some $i$ then $n=1$ and $\Lambda$ is a normal crossing divisor in a smooth surface.

non-normal Gorenstein slc singularities, $\Lambda=0$

:   We describe the dual graph of the $f$-exceptional divisors over $x$: analytically locally $X$ consists of $k$ irreducible components, on each component we have a resolution graph as in the previous item, and these are glued together where the components intersect. In total we have a cycle of smooth rational curve.

non-normal Gorenstein slc singularities, $\Lambda\neq 0$

:   The difference to the previous case is that the local components are now glued in a chain and the ends of the chain intersect the strict transform of the boundary. In this case $X$ itself might not even be $\mathbb{Q}$-Gorenstein.

Normal Gorenstein stable log surfaces with small $\Delta$-genus {#zero-Delta-genus}
===============================================================

Let $X$ be a variety and $\mathcal{L}$ be an ample line bundle on it. Fujita introduced several invariants for such polarized varieties. One important of them is the $\Delta$-genus $\Delta(X,\mathcal{L}):=\mathcal{L}^{\dim X}-h^0(X,\mathcal{L})+\dim X$.

\[gepg-2\] Let $(X,\Lambda)$ be a normal irreducible Gorenstein stable log surface. Then $\Delta(X,K_X+\Lambda)\ge 0$.

Moreover if ’=’ holds, then $(X,\Lambda)$ is one of the followings:

-   $X$ is $\mathbb{P}^{2}$, $\mathcal{O}_X(K_X+\Lambda) =\mathcal{O}_{\mathbb{P}^{2}}(1)$ and $\Lambda\in|\mathcal{O}_{\mathbb{P}^{2}}(4)|$;

-   $X$ is $\mathbb{P}^{2}$, $\mathcal{O}_X(K_X+\Lambda)=\mathcal{O}_{\mathbb{P}^{2}}(2)$ and $\Lambda\in|\mathcal{O}_{\mathbb{P}^{2}}(5)|$;

-   $X$ is $\Sigma_d$, $K_X+\Lambda\equiv \Delta_0+\frac{N+d-1}{2}\Gamma$ and $\Lambda\in|3\Delta_0+\frac{N+3d+3}{2}\Gamma|$; ($N-d-3\ge0$ is an even number);

-   $X$ is a singular quadric $C_2$ in $\mathbb{P}^{3}$, $\mathcal{O}_X(K_X+\Lambda)=\mathcal{O}_{C_2}(1)=\mathcal{O}_{\mathbb{P}^{2}}(1)|_{C_2}$ and $\Lambda\in|\mathcal{O}_{C_2}(3)|$;

-   $X$ is a cone $C_{N-1}\hookrightarrow \mathbb{P}^{N}$, $\mathcal{O}_X(K_X+\Lambda)=\mathcal{O}_{\mathbb{P}^{N}}(1)|_{C_{N-1}}$ and the proper transformation $\bar{\Lambda}$ in the minimal resolution $\Sigma_{N-1}$ is linearly equivalent to $2\Delta_0+2N\Gamma$. ($N>3$)

\[delta-genus-one\] Let $(X,\Lambda)$ be a normal Gorenstein stable log surface with $(K_X+\Lambda)^2= p_g(X,\Lambda)-1$.

Then $(X,\Lambda)$ is one of the followings:

-   $X$ is a double cover of $\mathbb{P}^2$. $\Phi_{|K_X+\Lambda|}\colon X\to \mathbb{P}^2$ is the double covering map. The branch curve $B\in |\mathcal{O}_{\mathbb{P}^2}(2k)|$ is a reduced curve which admits curve singularities of lc double-covering type, and $\Lambda\in |{\Phi_{|K_X+\Lambda|}}^*\mathcal{O}_{\mathbb{P}^2}(4-k)|$. ($p_g(X,\Lambda)=3$, and $k=2,3,4$)

-   $X$ 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}
=========================================

Let $(X,\Lambda)$ be a Gorenstein stable log surface which is non-normal. $D$ is the non-normal locus of $X$.

Let $C\subset D$ be a subcurve of the non-normal locus of a Gorenstein stable log surface $(X,\Lambda)$ and $\nu\colon:\tilde{X}\to X$ be the partial normalization of $X$ along $C$. Denote $\tilde{C}$ and $\tilde{\Lambda}$ as the proper transformation of $C$ and $\Lambda$.

Then $(\tilde{X},\tilde{\Lambda}+\tilde{C})$ is a Gorenstein stable log surface.

We first notice that $\nu^*(K_X+\Lambda)=K_{\tilde{X}}+\tilde{\Lambda}+\tilde{C}$ is ample and Cartier. Second it is easy to verify that $(\tilde{X},\tilde{\Lambda}+\tilde{C})$ still has Gorenstein slc singularities only by the classification of Gorenstein slc singularities. Therefore $(\tilde{X},\tilde{\Lambda}+\tilde{C})$ is a Gorenstein stable log surface as well.

The map $\nu|_{\tilde C}\colon \tilde{C} \to C$ is generically a double cover and thus induces a rational involution $\tau$ on $\tilde{C}$.

\[prop: descend section\] If $(X, \Lambda)$ is a non-normal Gorenstein stable log surface and $\nu$, $C$, $\tilde{C}$, $\tau$ are defined as above, then $\nu^*H^0(X, K_X+\Lambda)\subset H^0(\tilde{X}, \nu^*(K_X+\Lambda)))$ is the subspace of those sections $s$ whose restriction to $\tilde{C}$ is $\tau$-anti-invariant.

\[separateNonnormalCurve\] It is easy to see that $\nu^*H^0(X, K_X+\Lambda)$ can not separate $\tilde{C}$. Actually, $\Phi_{\nu^*H^0(X, K_X+\Lambda)}|_{\tilde{C}}\circ \tau=\Phi_{\nu^*H^0(X, K_X+\Lambda)}|_{\tilde{C}}$ on $\tilde{C}$.

\[fibersection\] Let $(X,\Lambda)$ be a reducible Gorenstein stable log surface such that $X=X_1\cup X_2$ and $C:=X_1\cap X_2$ is the connecting curve. Write $(K_X+\Lambda)|_{X_i}:=(\pi^*(K_X+\Lambda))|_{X_i}=K_{X_i}+C+\Lambda|_{X_i}$.

Let $\mathcal{R}_{X_i\to C}\colon H^0(X_i,(K_X+\Lambda)|_{X_i})\to H^0(C,(K_X+\Lambda)|_C)$ be the restriction map. Then we have the following fiber product diagram of vector spaces $$\xymatrix{
    H^0(X, K_X+\Lambda) \ar[r]\ar[d] & H^0(X_1,(K_X+\Lambda)|_{X_1})\ar[d]^{\mathcal{R}_{X_1\to C}}\\
H^0(X_2,(K_X+\Lambda)|_{X_2})\ar[r]^{-\mathcal{R}_{X_2\to C}}& H^0(C, (K_X+\Lambda)|_C)
   }.$$

Moreover, denote $r_{X_i\to C}((K_X+\Lambda)|_{X_i}):=\dim\mathcal{R}_{X_i\to C}(H^0(X_i,(K_X+\Lambda)|_{X_i}))$. We have $$\begin{aligned}
h^0(X, K_X+\Lambda)\le& h^0(X_1,(K_X+\Lambda)|_{X_1})+h^0(X_2,(K_X+\Lambda)|_{X_2})\\
&-\max\{r_{X_1\to C}((K_X+\Lambda)|_{X_1}),r_{X_2\to C}((K_X+\Lambda)|_{X_2})\}.\end{aligned}$$

irreducible Non-normal Gorenstein stable surfaces with $\Delta(X,K_X+\Lambda)=1$ {#nnorm}
================================================================================

Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface and $\bar{X}$, $\pi$, $D$, $\bar{D}$, $\tau$ defined as before. Since $H^0(X,K_X+\Lambda)\cong \pi^*H^0(X,K_X+\Lambda) \subset H^0(\bar X,\pi^*(K_X+\Lambda))$, we have $\Delta(X,K_X+\Lambda)\ge \Delta(\bar{X},\pi^*(K_X+\Lambda))$, and ’=’ holds if and only if $\pi^*H^0(X,K_X+\Lambda) = H^0(\bar X,\pi^*(K_X+\Lambda))$.

\[nonnorm&irred\] Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface.

Then $\Delta(X,K_X+\Lambda)\ge 1$.

We only need to show that the case $\Delta(X,K_X+\Lambda)=\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$ does not occur. First, $h^0(X,K_X+\Lambda)=h^0(\bar{X},\pi^*(K_X+\Lambda))$ implies every section in $H^0(\bar{X},\pi^*(K_X+\Lambda))$ is $\tau$-anti-invariant restricting to $\bar{D}$. Hence $H^0(\bar{X},\pi^*(K_X+\Lambda))$ can not separate points of $\bar{D}$ by Remark \[separateNonnormalCurve\]. However, $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$ implies $\pi^*(K_X+\Lambda)$ is very ample by Cor \[gepg-2\], therefore it will separate points of $\bar{D}$, a contradiction.

\[delta1,1\]

Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface as before. Assume $\Delta(X,K_X+\Lambda)=\Delta(\bar{X},\pi^*(K_X+\Lambda))=1$. Then

-   either $X$ is a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branched curve is $2C+B\in |\mathcal{O}_{\mathbb{P}^2}(2m+2k)|$, where $C,B$ are reduced curves of degree $m,2k$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(4-k-m)|$. $k=2,3$. $0<m\le 4-k$; ($p_g(X,\Lambda)=3$)

-   or $\Lambda=0$, $K_X^2=(K_{\bar{X}}+\bar D)^2=1$. $(\bar{X},\bar{D})$ is a normal Gorenstein stable log surface as in Thm \[delta-genus-one\] (6). $\bar D$ is a 2 section with $p_a(\bar D)=2$. $\tau$ is induced from the double covering $\bar D \to \mathbb{P}^1$. ($p_g(X)=2$)

We see that $\pi^*H^0(X,K_X+\Lambda)=H^0(\bar{X},\pi^*(K_X+\Lambda))$ and $(\bar{X},\bar D+\bar{\Lambda})$ is a stable log surface as in Thm \[delta-genus-one\].

First $\Phi_{|\pi^*(K_X+\Lambda)|}$ is not an embedding by Remark \[separateNonnormalCurve\]. Second $\Phi_{|\pi^*(K_X+\Lambda)|}$ is not composed with a pencil of genus 2 curves since $\bar D+\bar{\Lambda}\not= 0$.

If $\Phi_{|\pi^*(K_X+\Lambda)|}$ is composed with a pencil of elliptic curves, $\bar{D}+\bar{\Lambda}$ is either a 2-section or a sum of two elliptic fibers of the fibration map. For the former case, we see that $\bar{\Lambda}=0$ and $\bar{D}$ is a genus 2 curve by Remark \[separateNonnormalCurve\]. Moreover, $(K_{\bar X}+\bar D)^2=1$. $\tau$ is induced from the double map $\bar{D} \to \mathbb{P}^1$. For the latter case, we see that either $\bar{D}=F_1+F_2$ and $\bar{\Lambda}=0$, or $\bar{D}=F_1$ and $\bar{\Lambda}\not=0$, where $F_i$ is a fiber of the fibration map. Both cases can be excluded since the $F_1\cap F_2$ or $F_1\cap\bar{\Lambda}$ on $\bar{X}$ can not be glued into a Gorenstein slc singularity on $X$.

If $\bar{X}$ is a double cover of $\mathbb{P}^2$, then $\bar{D}+\bar{\Lambda}\in |\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}^*\mathcal{O}_{\mathbb{P}^2}(4-k)|$, $k=2,3$. Remark \[separateNonnormalCurve\] indicates $\bar{D}=\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}^{-1}(C)$, where $C$ is a curve on $\mathbb{P}^2$. Denote $m$ as the degree of $C$. Then $\bar{\Lambda}\in |\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}^*\mathcal{O}_{\mathbb{P}^2}(4-k-m)|$. We have the following commutative diagram: $$\xymatrix{
 \bar{D}\ar@{^{(}->}[r]\ar[d] & \bar{X}\ar[d]^{\pi}\ar[rr]^{\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}} && \mathbb{P}^2\\
 D\ar@{^{(}->}[r] & X \ar[rru]_{\Phi_{|K_{X}+\Lambda|}}
 }.$$

We see that $X$ is also a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branch curve is $2C+B\in |\mathcal{O}_{\mathbb{P}^2}(2m+2k)|$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(4-k-m)|$.

Next we consider those irreducible non-normal Gorenstein stable log surfaces $(X,\Lambda)$ with $\Delta(X,K_X+\Lambda)=1$ and $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$.

We first notice that $\pi^*H^0(X,K_X+\Lambda)\subset H^0(\bar{X},\pi^*(K_X+\Lambda))$ is of codimension one. Hence it has a base point $c\in \mathbb{P}(H^0(\bar{X},\pi^*(K_X+\Lambda)))$. We have the following commutative diagram: $$\xymatrix{
      \bar{X} \ar@{^{(}->}[rrr]^<(0.3){\Phi_{|\pi^*(K_X+\Lambda)|}}\ar[d]^{\pi} & & & \mathbb{P}(H^0(\bar{X},\pi^*(K_X+\Lambda)))\ar@{-->}[d]^{pr_c}\\
X\ar@{-->}[rrr]^<(0.3){\Phi_{|K_X+\Lambda|}}& & & \mathbb{P}(\pi^*H^0(X,K_X+\Lambda))
  }.$$

We regard $\Phi_{|\pi^*(K_X+\Lambda)|}$ as an inclusion. To describe $pr_c|_{\bar{X}}$, we use the following theorem (see [@Ber06 Thm 2.1]):

\[linearsectionthm\] Let $X$ be a non-degenerate irreducible subvariety of $\mathbb{P}^r$ and $L$ be a linear subspace of $\mathbb{P}^r$ of dimension $s\le r$ such that $L\cap X$ is a 0-dimensional scheme $\zeta$. Then $\mathrm{length}\, \zeta \le \Delta(X,\mathcal{O}(1))+s+1$.

\[lineIntersecting\] Let $X$ be a non-degenerate irreducible subvariety of $\mathbb{P}^r$ with $\Delta(X,\mathcal{O}(1))=0$ and $L$ is a line intersecting $X$ along a 0-dimensional scheme $\zeta$. Then $\mathrm{length}\, \zeta\le 2$.

Let $X$ be a non-degenerate irreducible subvariety of $\mathbb{P}^N$ with $\Delta(X,\mathcal{O}(1))=0$ and $c\in \mathbb{P}^N$ be a point outside $X$. Denote $pr_c\colon \mathbb{P}^N\dashrightarrow \mathbb{P}^{N-1}$ as the projection from $c$. Assume $pr_c|_X$ is birational and is not an isomorphism. Denote by $W$ the image of $pr_c|_X$. Then $W$ is non-normal and $pr_c|_X\colon X\to W$ is a normalization map. The non-normal locus $D$ of $W$ is a line in $\mathbb{P}^{N-1}$. The pre-image $\bar{D}\colon = pr_c|_X^{-1}(D)$ is described as follows:

-   if $X$ is $\mathbb{P}^2$ embedded in $\mathbb{P}^5$, then $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$.

-   if $X$ is a cone $C_{N-1}\hookrightarrow \mathbb{P}^N$, then $\bar{D}$ is a sum of two rulings.

-   if $X$ is $\Sigma_d\hookrightarrow \mathbb{P}^N$, either $\bar{D}\in |\Delta_0+\Gamma|$, $N=d+3$, or $\bar{D}=\Delta_0$, $N=d+5$.

$pr_c|_X\colon X\to W$ is a normalization map by Cor \[lineIntersecting\].

We show that $D$ is a line in $\mathbb{P}^{N-1}$ by contradiction hypothesis. Assume $D$ is not a line, then there are two points $p,q\in D$ such that the line $L_{p,q}$ passing $p,q$ is not contained in $D$. Let $H_{p,q}\subset \mathbb{P}^N$ be the pre-image of $L_{p,q}$ with respect to $pr_c$. Then $H_{p,q}\cap X$ has length great than $4$ which contradicts Thm \[linearsectionthm\]. Therefore $D$ is a line in $\mathbb{P}^{N-1}$.

Finally, we describe $\bar{D}$. For the case where $X$ is a cone $C_{N-1}$, $\bar{D}$ must be a sum of two rulings. This can be shown by considering the plane $H_{v, L}\subset \mathbb{P}^{N-1}$ containing $v,L$, where $v$ is the vertex of $C_{N-1}$ and $L$ is a secant line of $X$ passing through $c$.

For the case where $X$ is $\mathbb{P}^2$ embedded in $\mathbb{P}^5$, we have $\bar{D}\cdot \mathcal{O}_{X}(1)=2$ since $D$ is a line in $\mathbb{P}^{N-1}$. Hence $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$.

For the case where $X$ is $\Sigma_d\hookrightarrow \mathbb{P}^N$, similarly as the former case we have $\bar{D}\cdot \mathcal{O}_{X}(1)=\bar{D}\cdot(\Delta_0+\frac{N+d-1}{2}\Gamma)=2$. Hence we have either $\bar{D}\in |\Delta_0+\Gamma|$, $N=d+3$, or $\bar{D}=\Delta_0$, $N=d+5$.

We first consider the cases where $c\in \bar{X}$.

Case I) $\bar{X}$ is $\mathbb{P}^2$ and $\pi^*(K_X+\Lambda)=\mathcal{O}_{\mathbb{P}^2}(1)$. Then $\bar{\Lambda}+\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$ and $pr_c|_{\bar{X}}$ is a fibration over $\mathbb{P}^1$. By the classification of Gorenstein slc singularities, we see that $\bar{\Lambda}\cdot\bar{D}$ is even. Hence we have either $\bar{\Lambda},\bar{D}\in  |\mathcal{O}_{\mathbb{P}^2}(2)|$ or $\bar{D}\in  |\mathcal{O}_{\mathbb{P}^2}(4)|$ and $\bar{\Lambda}=0$. Examples will be given later. We note further that $c\not\in\bar{D}$, otherwise it can not be glued into a Gorenstein slc singularity. Case II) $\bar{X}$ is $C_{N-1}$ and $c$ is the vertex. We see that $\bar{\Lambda}$ is two rulings and $\bar{D}\in \mathcal{O}_{C_{N-1}}(2)$.

Case III) $\bar{X}$ is $C_{N-1}$ and $c$ is not the vertex. We show that this does not occur. By Cor \[lineIntersecting\] $pr_c|_{\bar{X}}$ is an isomorphism outside the ruling $l$ passing $c$. Hence $\bar{D}$ must be $l$. However this is impossible as the vertex of $C_{N-1}$ would not be glued into a Gorenstein slc singularity.

Case IV) $\bar{X}$ is $\Sigma_{d}$. We show that this does not occur as well. Similarly as in Case III), we see that $\bar{D}$ should be a ruling $\Gamma$ passing through $c$. Hence $\bar{D}\cdot \bar{\Lambda}=3$, which is impossible.

Case V) $\bar{X}$ is $\mathbb{P}^2$ embedding in $\mathbb{P}^5$. We show that this does not occur. Since by Cor \[lineIntersecting\], we see that $pr_c|_{\bar{X}}$ is an isomorphism outside $c$, which is impossible.

Next we consider the cases where $c\not\in \bar{X}$. We see $pr_c|_{\bar{X}}$ is a morphism of degree at most two by Cor \[lineIntersecting\] and it contracts no curve on $\bar{X}$.

Case VI) $pr_c|_{\bar{X}}$ is a double cover. In this case $\bar{X}$ is a quadric in $\mathbb{P}^3$. Denote $pr_c|_{\bar{X}}$ as $\delta$. The branch curve $B\in |\mathcal{O}_{\mathbb{P}^2}(2)|$. $K_{\bar{X}}+\bar{D}+\bar{\Lambda}=\mathcal{O}_{\bar{X}}(1)=\delta^*\mathcal{O}_{\mathbb{P}^2}(1)$. Hence $\bar{D}+\bar{\Lambda}\in |\delta^*\mathcal{O}_{\mathbb{P}^2}(3)|=|\mathcal{O}_{\bar{X}}(3)|$. $\bar{D}$ is the pre-image of a curve $C$ on $\mathbb{P}^2$ under $pr_c|_{\bar{X}}$. Denote $m$ as the degree of $C$. $m\le 3$. We have the following commutative diagram: $$\xymatrix{
 \bar{D}\ar@{^{(}->}[r] &\bar{X}\ar[d]^{\pi}\ar[drr]^{pr_c|_{\bar{X}}}\ar@{^{(}->}[rr]^{\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}} && \mathbb{P}^3 \ar@{-->}[d]^{pr_c}\\
  & X \ar[rr]_{\Phi_{|K_{X}+\Lambda|}} && \mathbb{P}^2
 }.$$ We see that $X$ is also a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branch curve ia $2C+B$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(3-m)|$.

If $pr_c|_{\bar{X}}$ is birational, it must be a normalisation map. Its image must be $X$ and $\pi=pr_c|_{\bar{X}}$.

Case VII) $\bar{X}$ is $\mathbb{P}^2$ embedding in $\mathbb{P}^5$. The non-normal locus $D$ is a line in $\mathbb{P}^5$ and $\bar{D}\in  |\mathcal{O}_{\mathbb{P}^2}(1)|$. $\bar{\Lambda}\in  |\mathcal{O}_{\mathbb{P}^2}(4)|$.

Case VIII) $\bar{X}$ is $C_{N-1}$, $N>3$. $\bar{D}$ is two rulings. $\bar{\Lambda}$ is linearly equivalent to $2H_{\infty}$ where $H_{\infty}$ is a hyperplane.

Case IX) $\bar{X}$ is $\Sigma_{d}$ embedded in $\mathbb{P}^N$ , $N>3$. Either $N=d+3$, $\bar{D}\in |\Delta_0+\Gamma|$ and $\bar{\Lambda}\in |2\Delta_0+(2d+2)\Gamma|$; or $N=d+5$, $\bar{D}=\Delta_0$ and $\bar{\Lambda}\in |2\Delta_0+(2d+4)\Gamma|$.

We summarize the above results in the following theorem:

\[delta1,0\] Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface as before. Assume $\Delta(X,K_X+\Lambda)=1$ and $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$. Then there are the following possibilities:

-   $\bar{X}$ is $\mathbb{P}^2$. $\bar{\Lambda}\in |\mathcal{O}_{\mathbb{P}^2}(2)|$ and $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(2)|$. Moreover, $c\not\in \bar{D}$. ($p_g(X,\Lambda)=2$)

-   $\bar{X}$ is $\mathbb{P}^2$. $\bar{\Lambda}=0$ and $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. Moreover, $c\not\in \bar{D}$. ($p_g(X,\Lambda)=2$)

-   $\bar{X}$ is $C_{N-1}$. $\bar{\Lambda}$ is two rulings and $\bar{D}\in \mathcal{O}_{C_{N-1}}(2)$. $c$ is the vertex of $C_{N-1}$. ($p_g(X,\Lambda)=N$)

-   $\bar{X}$ is a quadric in $\mathbb{P}^3$. $\bar{D}\in |\mathcal{O}_{\bar{X}}(m)|$ and $\bar{\Lambda} \in |\mathcal{O}_{\bar{X}}(3-m)|$. $X$ is a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branch curve is $2C+B$, where $C$,$B$ are reduced curves of degree $m$, $2$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(3-m)|$. $0<m \le 3$. ($p_g(X,\Lambda)=3$)

-   $X$ is the projection image of a Veronese embedding of $\mathbb{P}^2$. $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$ and $\bar{\Lambda}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. ($p_g(X,\Lambda)=5$)

-   $X$ is the projection image of $C_{N-1}\subset\mathbb{P}^N$, $N>3$. $\bar{D}$ is a sum of two lines passing the vertex, and $\bar{\Lambda}\in \mathcal{O}_{C_{N-1}}(2)$. ($p_g(X,\Lambda)=N$)

-   $X$ is the projection image of $\Sigma_d$ embedded in $\mathbb{P}^N$, $N>3$. $D\subset X$ is a line. We have either $N=d+3$, $\bar{D}\in |\Delta_0+\Gamma|$ and $\bar{\Lambda}\in |2\Delta_0+(2d+2)\Gamma|$, or $N=d+5$, $\bar{D}=\Delta_0$ and $\bar{\Lambda}\in |2\Delta_0+(2d+4)\Gamma|$. ($p_g(X,\Lambda)=N$)

Let $\bar{D}$ be a smooth quadric in $\mathbb{P}^2$ defined by $x^2+y^2+z^2=0$. $\tau$ acts on $\bar{D}$ by $x\mapsto -x, y \mapsto -y, z\mapsto z$. Let $\bar{\Lambda}=L_{x=0}+L_{y=0}$, where $L_{x=0}$, $L_{y=0}$ are two lines. Gluing $\mathbb{P}^2$ along $\bar{D}$ by $\tau$ we get a non-normal Gorenstein stable log surface $(X,\Lambda)$ with $(K_X+\Lambda)^2=p_g(X,\Lambda)-1=1$.

Let $\bar{D}$ be a smooth quartic in $\mathbb{P}^2$ defined by $x^4+y^4+z^4=0$. $\tau$ acts on $\bar{D}$ by $x\mapsto -x, y \mapsto -y, z\mapsto z$. Gluing $\mathbb{P}^2$ along $\bar{D}$ by $\tau$ we get a non-normal Gorenstein stable surface $X$ with $K_X^2=p_g-1=1$.

\[irrnonnormalstabledelta1\] Let $X$ be an irreducible non-normal Gorenstein stable surface with $\Delta(X,K_X)=1$. Then $X$ is one of the followings:

-   $X$ is a double cover of $\mathbb{P}^2$. The branch curve is $2C+B$, where $C$,$B$ are reduced curves of degree $4-k$, $2k$. $k=2,3$. ($p_g(X)=3$)

-   $X$ is obtained from a log surface $(\bar{X},\bar{D})$ by gluing the 2-section $\bar{D}$. $(\bar{X},\bar{D})$ is a normal Gorenstein stable log surface as in Thm \[delta-genus-one\] (6). ($p_g(X)=2$)

-   $\bar{X}$ is $\mathbb{P}^2$. $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. ($p_g(X)=2$)

-   $\bar{X}$ is a quadric in $\mathbb{P}^3$. $\bar{D}\in |\mathcal{O}_{\bar{X}}(3)|$. ($p_g(X)=3$)

reducible Gorenstein stable log surfaces with $\Delta(X,K_X+\Lambda)=1$ {#reducible-stable}
=======================================================================

In this section we consider reducible Gorenstein stable log surfaces. They are glued by irreducible ones along some connecting curves.

\[restsecions\] Let $X$ be a connected $S_2$ scheme of pure dimension, $\mathcal{L}$ be an invertible sheaf such that $\dim \mathrm{Bs} |\mathcal{L}|<\dim X-1$ and $C$ be a subscheme of codimension 1. Then $$\begin{aligned}
r_{X\to C}(\mathcal{L})=\dim <\Phi_{|\mathcal{L}|}(C)>+1,\end{aligned}$$ where $<\Phi_{|\mathcal{L}|}(C)>$ is the projective subspace of $|\mathcal{L}|^*$ spanned by $\Phi_{|\mathcal{L}|}(C)$.

Denote $\mathbb{P}:=|\mathcal{L}|^*$. We have the following commutative diagram: $$\xymatrix{
      H^0(\mathbb{P},\mathcal{O}_{\mathbb{P}}(1))\ar[rr]^<(0.2){\mathcal{R}_{\mathbb{P}\to \Phi_{|\mathcal{L}|}(C)}}\ar[d]^{\cong}_{\Phi_{|\mathcal{L}|}^*} & & H^0(\Phi_{|\mathcal{L}|}(C),\mathcal{O}_{\mathbb{P}}(1)|_{\Phi_{|\mathcal{L}|}(C)})\ar@{^{(}->}[d]_{\Phi_{|\mathcal{L}|}^*}  \\
    H^0(X,\mathcal{L})\ar[rr]^{\mathcal{R}_{X\to C}}& & H^0(C,\mathcal{L}|_C)
   }.$$ Therefore $r_{X\to C}(\mathcal{L})=r_{\mathbb{P}\to \Phi_{|\mathcal{L}|}(C)}(\mathcal{O}_{\mathbb{P}}(1))=\dim <\Phi_{|\mathcal{L}|}(C)>+1$.

Still we call a curve $C$ on a demi-normal scheme $X$ a *[line]{} if 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: $$\xymatrix{
  \bar{C}\ar@{^{(}->}[r]\ar[d] & \bar{X}\ar@{^{(}->}[rr]^{\Phi_{|\pi^*(K_X+\Lambda)|}}\ar@{-->}[rrd]^{\Phi'}\ar[d]_{\pi}& & \mathbb{P}\ar@{-->}[d]^{pr_c}\\
  C\ar@{^{(}->}[r] &X\ar@{-->}[rr]_{\Phi_{|K_X+\Lambda|}} &&\mathbb{P}',
  }$$ where $\Phi'$ is a map defined by the partial linear system $\pi^*H^0(X,K_X+\Lambda)$. We regard $\Phi_{|\pi^*(K_X+\Lambda)|}$ as an inclusion. By Lemma \[restsecions\], $r_{X\to C}(K_X+\Lambda)\ge1$. If ’=’ holds, $\Phi_{|K_X+\Lambda|}(C)$ is a point. Hence $pr_c(\bar{C})$ is a point, which implies $\bar{C}$ is a line passing $c$. Hence $C$ is a line passing through $\pi(c)$, which is the base point of $|K_X+\Lambda|$.

\[2comps\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface. Assume $X=X_1\cup X_2$, where $X_1$ is connected and $X_2$ is irreducible. Denote $C:=X_1\cap X_2$ as the connecting curve of $X_1$ and $X_2$.

Then:

-   $\Delta(X,K_X+\Lambda)\ge \Delta(X_1,(K_X+\Lambda)|_{X_1})$.

-   if ’=’ holds and $\Delta(X_2,(K_X+\Lambda)|_{X_2})\le 1$, then

    -   either $X_2$ is non-normal and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=1$. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=1$. $C$ is a line on $X_2$ passing through the base point of $|(K_X+\Lambda)|_{X_2}|$. Moreover, $r_{X_1\to C}((K_X+\Lambda)|_{X_1})\le 1$.

    -   $X_2$ is normal and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=0$, $|(K_X+\Lambda)|_{X_2}|$ is very ample. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=2$. $C$ is a line on $X_2$. Moreover, $r_{X_1\to C}((K_X+\Lambda)|_{X_1})\le 2$;

    -   or $X_2$ is normal and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=1$, $|(K_X+\Lambda)|_{X_2}|$ is composed with a pencil of elliptic curves. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=1$. $C$ is a fiber on $X_2$. Moreover, $r_{X_1\to C}((K_X+\Lambda)|_{X_1})\le 1$.

$(K_X+\Lambda)^2=(K_X+\Lambda)|_{X_1}^2+(K_X+\Lambda)|_{X_2}^2$ together with Cor \[fibersection\] gives $$\label{deltaIneq}
\begin{split}
\Delta(X,K_X+\Lambda) & \ge \Delta(X_1,(K_X+\Lambda)|_{X_1})+\Delta(X_2,(K_X+\Lambda)|_{X_2})\\
&+\max\{r_{X_1\to C}((K_X+\Lambda)|_{X_1}),r_{X_2\to C}((K_X+\Lambda)|_{X_2})\}-2.
\end{split}$$ 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 $\mathrm{im}\, \mathcal{R}_{X_1\to X_1\cap X_2}\subset \mathrm{im} \, \mathcal{R}_{X_2\to X_1\cap X_2}$, then $\mathcal{R}_{X\to X_1}$ is surjective.

-   If $(K_X+\Lambda)|_{X_2}$ is very ample and $X_1\cap X_2$ is a line on $X_2$, then $\mathcal{R}_{X\to X_1}$ is surjective and $\Phi_{|K_X+\Lambda)|}$ separates $X_1$ and $X_2$.

-   If each $(K_X+\Lambda)|_{X_i}$ is very ample and $X_1\cap X_2$ is a line on each $X_i$, then $K_X+\Lambda$ is very ample.

\(i) and (ii) is obvious by chasing the diagram.

For (iii), $X_1\cap X_2$ is a line implies $\mathcal{R}_{X_2\to X_1\cap X_2}$ is surjective. Then any section in $H^0(X_1,(K_X+\Lambda)|_{X_1})$ can be extended into a section in $H^0(X,K_X+\Lambda)$. Therefore $\mathcal{R}_{X\to X_1}$ is surjective. Next $\ker \mathcal{R}_{X_2\to X_1\cap X_2}$ is nontrivial and its base part is the line $X_1\cap X_2$. Then $\ker \mathcal{R}_{X\to X_1}$ is nontrivial and its base part is $X_1$. Therefore $\Phi_{|K_X+\Lambda)|}$ separates $X_1$ and $X_2$.

For (iv) we see that $\mathcal{R}_{X\to X_i}$ is surjective and the kernel is nontrivial by (ii). Then we have plenty of sections to separate points and tangents, which implies $K_X+\Lambda$ is very ample.

\[equaldelta\] Let $(X,\Lambda)$ be a log surface as in Lemma \[2comps\]. We assume further $X_1$ is irreducible and $\Delta(X,K_X+\Lambda)= \Delta(X_i,(K_X+\Lambda)|_{X_i})\le 1$ for $i=1,2$. Then

-   if $\Delta(X,K_X+\Lambda)=0$, then $X_1\cap X_2$ is a line on each $X_i$. Moreover, $K_X+\Lambda$ is very ample.

-   if $\Delta(X,K_X+\Lambda)=1$, then

    -   either $X_1$ and $X_2$ are both normal. Each $|(K_X+\Lambda)|_{X_i}|$ is composed with a pencil. $X_1\cap X_2$ is a fiber on each $X_i$. $|K_X+\Lambda|$ is composed with a pencil as well.

    -   or $X_1$ and $X_2$ are both non-normal. The base points of $|(K_X+\Lambda)|_{X_i}|$ coincide into the unique base point of $|K_X+\Lambda|$. $X_1\cap X_2$ is a line passing through the base point of $|(K_X+\Lambda)|_{X_i}|$ on each $X_i$.

\(i) follows from Lemma \[2comps\] and Lemma \[globalsection\].

For (ii), applying Lemma \[2comps\] we see that either $X_i$ is non-normal or $(K_X+\Lambda)|_{X_i}$ is composed with a pencil. It is easy to see that either $X_1$, $X_2$ are both non-normal or $(K_X+\Lambda)|_{X_1}$, $(K_X+\Lambda)|_{X_2}$ are both composed with a pencil of elliptic curve, since the geometric genus of $X_1\cap X_2$ on $X_1$ or $X_2$ should coincide.

If $(K_X+\Lambda)|_{X_1}$, $(K_X+\Lambda)|_{X_2}$ are both composed with a pencil of elliptic curve, then by Lemma \[2comps\] each $(K_X+\Lambda)|_{X_i}$ is composed with a pencil and $X_1\cap X_2$ is a fiber on each $X_i$. Therefore $|K_X+\Lambda|$ is composed with a pencil as well. If $X_1$ and $X_2$ are both non-normal, then by Lemma \[restdim\], $X_1\cap X_2$ is a line passing through the base point of $|(K_X+\Lambda)|_{X_i}|$ on each $X_i$. These two base points must coincide, otherwise no nontrivial section in $H^0(X_i,(K_X+\Lambda)|_{X_i})$ can be glued into a section of $H^0(X,K_X+\Lambda)$.

\[decrease1\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface which has two irreducible components $X_1$, $X_2$. Assume further $\Delta(X_1,(K_X+\Lambda)|_{X_1})=\Delta(X_2,(K_X+\Lambda)|_{X_2})=0$ and $\Delta(X,K_X+\Lambda)=1$. Then $$r_{X_1\to X_1\cap X_2}((K_X+\Lambda)|_{X_1})=r_{X_2\to X_1\cap X_2}((K_X+\Lambda)|_{X_2})=3.$$

By (\[deltaIneq\]), we have $\max\{r_{X_1\to X_1\cap X_2}((K_X+\Lambda)|_{X_1}),r_{X_2\to X_1\cap X_2}((K_X+\Lambda)|_{X_2}\}\le 3$. Moreover $r_{X_i\to X_1\cap X_2}((K_X+\Lambda)|_{X_i}\ge 2$ by Lemma \[restdim\]. Thus there must be one $r_{X_i\to X_1\cap X_2}((K_X+\Lambda)|_{X_i}=3$.

Now we suppose $r_{X_1}(X_1\cap X_2)=2$ and $r_{X_2}(X_1\cap X_2)=3$ for a contradiction. Then $X_1\cap X_2$ is a line on $X_1$ and not a line on $X_2$. Thus nonzero elements of $\mathrm{im}\mathrm{Res}_{X_i|X_1\cap X_2}$ have different degrees. Hence only those sections in $H^0(X_i,(K_X+\Lambda)|_{X_i})$ vanishing on $X_1\cap X_2$ can be glued together. Therefore $$\begin{aligned}
  p_g(X,\Lambda)&=\dim\ker \mathcal{R}_{X_1\to X_1\cap X_2}+\dim \ker \mathcal{R}_{X_2\to X_1\cap X_2}\\
  &\le p_g(X_1,(K_X+\Lambda)|_{X_1})-2+p_g(X_2,(K_X+\Lambda)|_{X_2})-2\\
  &\le (K_X+\Lambda)^2,
  \end{aligned}$$ a contradiction. This completes the proof.

\[diffdelta\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface. Assume $X=X_1\cup X_2$, where $X_i$ is irreducible. Assume further $\Delta(X,K_X+\Lambda)= \Delta(X_1,(K_X+\Lambda)|_{X_1})=1$ and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=0$.

Then

-   either each $(K_X+\Lambda)|_{X_i}$ is very ample and on each $X_i$ the curve $X_1\cap X_2$ is a line on each $X_i$. $K_X+\Lambda$ is very ample;

-   or $X_1$ is non-normal. $X_1\cap X_2$ is a line on each $X_i$.

First by Lemma \[2comps\], $X_1\cap X_2$ will be a line on $X_2$ and $r_{X_1\to X_1\cap X_2}((K_X+\Lambda)|_{X_1})=1$, or $2$.

For the case $X_1$ is normal, we claim that $r_{X_1\to X_1\cap X_2}((K_X+\Lambda)|_{X_1})\not =1$. Otherwise, by Lemma \[restdim\], $|(K_X+\Lambda)|_{X_1}|$ is composed with a pencil of elliptic curves and the connecting curve $X_1\cap X_2$ will be an elliptic fiber on $X_1$. While on $X_1$ the connecting curve $X_1\cap X_2$ has geometric genus 0. This is impossible.

Therefore, $r_{X_1\to X_1\cap X_2}((K_X+\Lambda)|_{X_1})=2$. $(K_X+\Lambda)|_{X_1}$ is not composed with a pencil and the log canonical image of $X_1\cap X_2$ is a line. Next we show that $X_1$ is not a double cover of $\mathbb{P}^2$. Otherwise, on $X_1$, $X_1\cap X_2$ is a double covering curve of $\mathbb{P}^1$. Then nonzero sections in the image of $\mathcal{R}_{X_i\to X_1\cap X_2}$ have different degrees. Hence only those sections of $H^0(X_i, (K_X+\Lambda)|_{X_i})$ vanishing on $X_1\cap X_2$ can be glued together. Therefore, $$\begin{aligned}
   p_g(X,\Lambda)&=\dim\ker \mathcal{R}_{X_1\to X_1\cap X_2}+\dim \ker \mathcal{R}_{X_2\to X_1\cap X_2}\\
   &=p_g(X_1,(K_X+\Lambda)|_{X_1})+p_g(X_2,(K_X+\Lambda)|_{X_2})-4\\
   &< (K_X+\Lambda)^2,
  \end{aligned}$$ a contradiction. Therefore $(K_X+\Lambda)|_{X_1}$ is very ample. $X_1\cap X_2$ is a line on $X_1$. The other statements follow.

For the non-normal case, we take a partial normalization $\nu\colon \tilde{X}\to X$ along the non-normal locus on $X_1$. $\tilde{X}=\tilde{X_1}\cup X_2$, where $\tilde{X_1}$ is the normalization of $X_1$. It is easy to check that $\Delta(\tilde{X},\nu^*(K_X+\Lambda))=0$. Then we see that $X_1\cap X_2$ is a line on each $X_i$. Hence we complete the proof.

Let $X_1$ be a $\mathbb{P}^2$ blown up at $k$ distinct points as in Thm \[delta-genus-one\] (3). Each $E_i$ is a line w.r.t $-K_{X_1}$. Choosing a general $\Lambda_1=E_1+...+E_k+B\in |-2K_{X_1}|$, then $\Lambda_1$ is nodal and $E_i$ intersects $B$ at 3 distinct points. Let $(X_2,\Lambda_2)$ be a log surface as in 　Thm \[gepg-2\](iii) such that $\Lambda_2=\Gamma_1+...+\Gamma_s+D$, where $\Gamma_i$ is a ruling and it intersect $D$ at 3 distinct points. Then we can glue $X_1$, $X_2$ along $E_i$, $\Gamma_j$ to obtain a log surface as in Thm \[diffdelta\] (i).

The following theorem and corollary is first obtained in [@LR13].

\[thm: log noether\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface. Then $$\Delta(X,K_X+\Lambda)\ge 0.$$

This follows directly by Cor \[gepg-2\], Lemma \[nonnorm&irred\] and Lemma \[2comps\].

\[cor: nonnormal equality\] Let $(X, \Lambda)$ be a Gorenstein stable log surface such that $\Delta(X,K_X+\Lambda)=0$. Write $X=\bigcup X_i$, where $X_i$ is an irreducible component.

Then

1.  $\Delta(X_i, (K_X+\Lambda)|_{X_i})=0$.

2.  $K_X+\Lambda$ is very ample。 It defines an embedding $\phi\colon X\hookrightarrow \mathbb{P}=|K_X+\Lambda|^*$;

3.  if $X_{i}\cap X_{j}\not = \emptyset $, then $X_{i}\cap X_{j}$ is a line on both $X_i$ and $X_j$;

4.  $X$ is a tree of $X_i$ glued along lines.

In particular, $\Lambda\neq 0$.

Finally we are able to classify Gorenstein stable log surface with $\Delta(X,K_X+\Lambda)=1$.

\[nonnormdelta-1\] Let $(X,\Lambda)$ be a reducible Gorenstein stable log surface with $\Delta(X,K_X+\Lambda)=1$. Write $X=\bigcup X_i$, where $X_i$ is an irreducible component.

Then $X$ has a unique minimal connected component $U$ such that $\Delta(U,(K_X+\Lambda)|_{U})=1$, $X\setminus U$ is composed with several trees $T_j$ with $\Delta(T_j,(K_X+\Lambda)|_{T_j})=0$ and $X$ is glued by $U$ and $T_j$ along lines.

$U$ is one of the followings:

-   $X=U$ is a string of log surfaces $X_i$ with $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$ and $|(K_{X}+\Lambda)|_{X_i}|$ composed with a pencil of elliptic curves. The connecting curves are all fibers. Moreover, in this case $|K_X+\Lambda|$ is composed with a pencil of elliptic curves.

-   $U$ is a string of surfaces glued along lines whose end surfaces $X_i$ of the string $U$ are non-normal with $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$.

    Moreover, if $U$ is reducible, $|K_X+\Lambda|$ has a base point $c\in X$ and all the connecting curves of $U$ pass through $c$.

-   $U$ is composed with a single irreducible log surface $X_k$ with $\Delta(X_k,(K_X+\Lambda)|_{X_k})=1$ and $|(K_{X}+\Lambda)|_{X_k}|$ very ample. Moreover, in this case $K_X+\Lambda$ is very ample.

-   $U=X_j\cup X_k $ where $X_j$, $X_k$ are two Gorenstein log surfaces with $\Delta(X_j,(K_X+\Lambda)|_{X_j})=\Delta(X_k,(K_X+\Lambda)|_{X_k})=0$. All the connecting curves of $X$ are lines except $X_j\cap X_k$.

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

First by Cor \[genus-like\] we see each component $X_i$ has $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$ or $0$. Second, once we obtain a connected component $U\subset X$ such that $\Delta(U,(K_X+\Lambda)|_{U})=1$ and $\Delta(X_i,(K_X+\Lambda)|_{X_i})=0$ for each $X_i\subset X\setminus U$, we see that $X\setminus U$ will be composed with several trees of log surfaces $T_j$ with $\Delta(T_j,(K_X+\Lambda)|_{T_j})=0$ by Lemma \[2comps\] and induction hypothesis. Therefore it remains to describe $U$.

We distinguish between two cases whether there is an irreducible component $X_i$ such that $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$.

Case 1. There is a component $X_i$ such that $\Delta(X_i,(K_X+\Lambda)|_{X_i})=1$. Then by Cor \[equaldelta\] and Cor \[diffdelta\], either $(K_X+\Lambda)|_{X_i}$ is composed with a pencil of elliptic curves, $X_i$ is non-normal, or $(K_X+\Lambda)|_{X_i}$ is very ample.

If $(K_X+\Lambda)|_{X_i}$ is composed with a pencil of elliptic curves, then by Cor \[equaldelta\] and Cor \[diffdelta\], for each irreducible components $X_j$ connected to $X_i$, $(K_X+\Lambda)|_{X_j}$ is composed with a pencil of elliptic curves as well, $X_i\cap X_j$ is a fiber and $\Delta(X_j,(K_X+\Lambda)|_{X_j})=1$. Therefore, inductively, every $(K_X+\Lambda)|_{X_k}$ is composed with a pencil of elliptic curves. Each $X_i$ is connected to at most two other components, as there are at most two fibers as connecting curves which pass through the base point of $|(K_X+\Lambda)|_{X_i}|$ on $X_i$. Thus $X=U$ is a string of such surfaces glued along fibers.

If $(K_X+\Lambda)|_{X_i}$ is very ample. Then by Cor \[equaldelta\] and Cor \[diffdelta\], every other irreducible component $X_j$ connected to $X_i$ has $\Delta(X_j,(K_X+\Lambda)|_{X_j})=0$ and the connecting curves of them are lines. It is easy to check $U=X_i$. Moreover, $K_X+\Lambda$ is very ample by Lemma \[globalsection\].

If $X_i$ is non-normal, then by Lemma \[2comps\] other component $X_j$ is either non-normal or has $\Delta(X_j,(K_X+\Lambda)|_{X_j})=0$ and $X$ is a tree of such surfaces. Let $\nu\colon \tilde{X}\to X$ be a partial normalization along the non-normal curves on $X_j$. It is easy to check that $\Delta(\bar{X}, \nu^*(K_X+\Lambda))=0$. $\nu^*H^0(X,K_X+\Lambda)\subset H^0(\bar{X}, \nu^*(K_X+\Lambda))$ has a base point $c\in \mathbb{P}:=|\nu^*(K_X+\Lambda)|^*$. We see that $c$ is also the base point of $|(K_X+\Lambda)|_{X_j}|$ for any non-normal $X_j$. If there is only one non-normal surface $X_i$, then $U=X_i$. If there are two non-normal surfaces $X_i$, $X_j$, then $U$ is the unique string of surfaces contained in $X$ with $X_i$, $X_j$ as the ends. If there are more than two non-normal surfaces, we claim that they lies on a unique minimal string of surfaces, which is $U$. Otherwise there are three non-normal surfaces contained in a fork of surfaces $Y\subset X$. By Lemma \[2comps\], we see that the base point $c$ lies on each $X_k\subset Y$. However, in the central surface of $Y$, there are three connecting curve passing through $c$, which is impossible. Hence the claim is true. Case 2. Each irreducible component $X_i$ has $\Delta(X_i,(K_X+\Lambda)|_{X_i})=0$.

If there are two irreducible components $X_j$, $X_{k}$ such that $\Delta(X_j\cup X_k,(K_X+\Lambda)|_{X_j\cup X_k})=1$, then by Lemma \[decrease1\], $X_j\cap X_k$ is neither a line on $X_j$ nor $X_k$, and $r_{X_j\to X_j\cap X_k}((K_X+\Lambda)|_{X_j})=r_{X_k\to X_j\cap X_k}((K_X+\Lambda)|_{X_k})=3$. It is easy to see that $X$ is a tree of log surfaces. Ungluing $X$ along $X_j\cap X_{k}$, there will be two connected tree of surfaces $V_1\supset X_j$, $V_2\supset X_{k}$. We claim $\Delta(V_1,(K_X+\Lambda)|_{V_1})=\Delta(V_2,(K_X+\Lambda)|_{V_2})=0$. Otherwise assume $\Delta(V_1,(K_X+\Lambda)|_{V_1})=1$, then $\Delta(V_1\cup X_{k},(K_X+\Lambda)|_{V_1\cup X_{k}})=1$ implies $ V_1\cap X_{k}=X_j\cup X_{k}$ is a line on $X_k$, a contradiction. Hence $U=X_j\cap X_k$ and $X$ is a tree of log surfaces whose connecting curves are lines except $X_j\cap X_k$.

Finally we consider the case that all the connecting curve are lines. It is easy to see that $X$ is not a tree, otherwise $\Delta(X,K_X+\Lambda)=0$. Hence there is a minimal cycle of surfaces $U$ of log surfaces such that $\Delta(U,(K_X+\Lambda)|_U)=1$.

Let $X$ be a connected reducible Gorenstein stable surface with $K_X^2= p_g-1$. Then $X$ is a union of two $\mathbb{P}^2$ glued along a quartic curve. $X$ is a double cover of $\mathbb{P}^2$ and it can be deformed into a smooth one. ($K_X^2=2$)

If $X\setminus U$ is not empty, then there is a leaf log surface $X_i$ such that $\Delta(X_i,K_X|_{X_i})=0$. While $K_X|_{X_i}=K_{X_i}+\bar{D_i}$ and the connecting curve $\bar{D_i}$ is a line, which is impossible. Hence $X=U$. A similar discussion tells us that $X$ is not a string of $X_i$ such that $|K_X|_{X_i}|$ is composed with a pencil. Therefore $X$ is either a string of surfaces containing non-normal ones or a union of two irreducible components whose connecting curve is not a line. We show that the first case does not occur. Otherwise $X=U$ is a string of surfaces glued along lines passing through $c\in X$ where $c$ is the base point of $|K_X|$. However, in this case $c$ would not be a Gorenstein slc singularity, a contradiction.

Let $X=X_1\cup X_2$ with $\Delta(X_i, K_X|_{X_i})=0$ and $r_{X_i\to X_1\cap X_2}(K_X|_{X_i})=3$. Then $X_i$ is $\mathbb{P}^2$, $\Sigma_d$ or $C_{n-1}$. While in these cases $\ker \mathcal{R}_{X_i\to X_1\cap X_2}=0$ (otherwise $K_X|_{X_i}=K_{X_i}+X_1\cap X_2\ge X_1\cap X_2$ as divisors, which is impossible), thus $h^0(X_i,K_X|_{X_i})=r_{X_i\to X_1\cap X_2}(K_X|_{X_i})=3$. Hence $X_i$ is $\mathbb{P}^2$, and the connecting curve $X_1\cap X_2$ is a curve of degree 4.

Let $X$ be a connected Gorenstein stable surface with $K_X^2= p_g-1$.

If $X$ is irreducible, then it is one of the followings:

-   $X$ is a double cover of $\mathbb{P}^2$. The branch curve is $2C+B$, where $C$,$B$ are reduced curves of degree $4-k$, $2k$. $k=2,3$. ($p_g(X)=3$)

-   $X$ is obtained from a log surface $(\bar{X},\bar{D})$ by gluing the 2-section $\bar{D}$. $(\bar{X},\bar{D})$ is a normal Gorenstein stable log surface as in Thm \[delta-genus-one\] (6). ($p_g(X)=2$)

-   $\bar{X}$ is $\mathbb{P}^2$. $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. ($p_g(X)=2$)

-   $\bar{X}$ is a quadric in $\mathbb{P}^3$. $\bar{D}\in |\mathcal{O}_{\bar{X}}(3)|$. ($p_g(X)=3$)

If it is reducible, then $X$ is a union of two $\mathbb{P}^2$ glued along a quartic curve. ($p_g(X)=3$)

We have confirmed the conjecture in [@LR13] that there are no Gorenstein stable surfaces with $K_X^2= p_g-1\ge 3$.

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[^1]: A local model for the pinch point in $\mathbb{A}^3$ is given by the equation $x^2+yz^2=0$.
