{On Gorenstein log del Pezzo surfaces }
Qiang Ye
In this paper, we first present the complete list of the singularity types of the Picard number one Gorenstein log del Pezzo surface and the number of the isomorphism classes with the given singularity type. Then we give out a method to find out all singularity types of Gorenstein log del Pezzo surface. As an application, we present the complete list of the Dynkin type of the Picard number two Gorenstein log del Pezzo surfaces. Finally we present the complete list of the singularity type of the relatively minimal Gorenstein log del Pezzo surface and the number of the isomorphism classes with the given singularity type.
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\begin{document}


\title{ {\bf   On Gorenstein log del Pezzo surfaces }}
\author{Qiang Ye} 
 \date{}
\input prepictex
\input pictex
\input postpictex
\maketitle
\begin{abstract} 
In this paper, we first present the complete list of the singularity types of the Picard number one Gorenstein log del Pezzo surface and the number of the isomorphism classes with the given singularity type. Then we give out a method to find out all singularity types of Gorenstein log del Pezzo surface. As an application, we present the complete list of the Dynkin type of the Picard number two Gorenstein log del Pezzo surfaces. Finally we present the complete list of the singularity type of the relatively minimal Gorenstein log del Pezzo surface and the number of the isomorphism classes with the given singularity type.
\end{abstract}




\section{Introduction}

Let $V$ be a normal projective surface defined over the complex numbers field {\bf C}. We say that $V$ is a {\it log del Pezzo surface} if $V$ has at worst quotient singularities and if the anticanonical divisor $-K_V$ of $V$ is an ample divisor. Then $V$ is {\it Gorenstein}, i.e., $K_V$ is a Cartier divisor, if and only if $V$ has at worst rational double singularities.  We note that a log del Pezzo surface is a rational surface (cf. [GZ]). The log del Pezzo surface has been studied from the various points of view. A general theory on the structure of such singular surfaces is developed in  [M1] , [Z1] and [Z2].
The topological properties of the smooth part of the log del Pezzo surfaces and log Fano varieties are studied in [Fujiki], [GZ], [GZ1], [T1] and [Z3-Z6].


Let $V$ be a Gorenstein log del Pezzo surface. Denote by $V^0:=V-({\rm Sing}V)$ the smooth part of $V$. Let $g: U \longrightarrow V$ be a minimal resolution of singularities, we denote by $D:=g^{-1}({\rm Sing}V)$ the exceptional 
divisor and $\sharp (D)$ the number of the irreducible components of $D$. 

In [Fu], the possible singularities of a Picard number one Gorenstein log del Pezzo surface are classified (see also [Brenton] ).

\begin{question} Let $V$ be a Picard number one Gorenstein log del Pezzo surface,
 is $V$ then determined uniquely by its singularities?
\end{question}

In general,  $V$ is not determined by its singularities. However, when $V^0$ is simply connected, this is almost the case. In [GPZ], [KM] and [MZ1],  they have shown that
for  a Picard number one Gorenstein log del Pezzo surface $V$  with simply connected smooth part,  the singularity type of $V$ is one of the following:
$$
A_1, \ A_1+A_2, \ A_4, \ D_5, \ E_6, \ E_7, \ E_8.
$$
Furthermore, $V$ is uniquely determined by its singularities except in the case of $V(E_8)$ where there are exactly two isomorphism classes.
\vskip 0.5cm

One of our main results is the following:
\vskip 0.5cm

\begin{theorem} Let $V$ be a Picard number one Gorenstein
log del Pezzo surface $(\not= {\bf P}^2)$, then the singularity types of $V$ and the number $m$ of the isomorphism classes of the Picard number one Gorenstein log del Pezzo surfaces with the given singularity type are listed in Table $1.1$.
\vskip 0.5cm
\begin{center}
Table $1.1$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c|c|c|c|}\hline
Dynkin type of $V$ & $m$ & Dynkin type of $V$  & $m$ & Dynkin type of $V$ & $m$
\\ \hline
$E_8$  & $2$ & $2A_4$& $1$ & $A_1+2A_3$ & $1$ 
\\ \hline
$A_1+E_7$  &$2$ &$A_1+A_2+A_5$   & $ 1$ & $E_6$& $1$  
\\ \hline
  $D_8$ & $1$& $2A_1+2A_3$   & $1$ & $A_1+A_5$& $1$       \\ \hline
$A_8$  &$1$ & $4A_2$ &  
$1$  & $3A_2$&$1$ 
\\ \hline
$A_2+E_6$  & $2$ & $E_7$   & $1$  & $D_5$& $1$ 
\\ \hline
$2A_1+D_6$  & $1$ & $A_1+D_6$  & $ 1$ &  $2A_1+A_3$ & $1$
\\ \hline
$2D_4$  & $\infty$& $A_7$  & $1$  & $A_4$ &$1$ 
\\ \hline
$A_3+D_5$  & $1$ &$A_2+A_5$  & $1$  & $A_1+A_2$ &$1$ 
\\ \hline

$A_1+A_7$  & $1$ &$3A_1+D_4$  & $1$  & $A_1$ &$1$ 
\\ \hline

\end{tabular} 
\end{center} 
All isomorphism classes of any given singularity type are realizable (see the proof).
\end{theorem}
\begin{corollary}  Let $V$ be a Picard number one Gorenstein
log del Pezzo surface. Then $V$ is  determined uniquely up to isomorphisms by the Dynkin type of $V$,  except for $V(E_8)$, $V(A_1+E_7)$, $V(A_2+E_6)$ and $V(2D_4)$.
\end{corollary}
\begin{remark} There are several results which are related to our Theorem $1.2$. 

$(1)$ \ In {\rm [Fu] }, the possible singularities of $V$ are classified (see also {\rm  [Brenton]} ).

$(2)$ \ In {\rm  [MZ1]}, a classification of Picard number one Gorenstein log del Pezzo surfaces is given.

In the proof,  we need the result about the extremal rational elliptic surfaces which is proved in {\rm [MP]} but we do not use the classification mentioned in {\rm [Brenton]},  {\rm [Fu]} and {\rm [MZ1]} above. 
\end{remark}
We also find out the singularity types of the Picard number two Gorenstein log del Pezzo surfaces.

\begin{theorem}  Let $V$ be a Picard number two Gorenstein log del Pezzo surface $(\not= {\bf P}^1 \times {\bf P}^1 \ and  \   \Sigma_1 )$. Then there exist exactly  $45$ singularity types which  are listed in  
Table $1.2$.
\begin{center}
Table $1.2$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c|c|c|c|}\hline
$E_7$ &  $A_7$ &   $2A_3+A_1$&  $A_4+A_2$ &  $A_4+A_1$ & $A_3$    \\ \hline
$E_6+A_1$ & $A_6+A_1$ & $3A_2+A_1$  & $2A_3$ & $A_3+2A_1$  & $A_2+A_1$\\ \hline                                                      
$D_7$  & $A_5+A_2$ & $E_6$ & $A_3+A_2+A_1$ & $2A_2+A_1$ &  $A_2$\\ \hline
$D_6+A_1$ & $A_5+2A_1$ &  $D_6$ & $A_3+3A_1$ & $D_4$ &  $2A_1$\\ \hline
$D_5+A_2$ & $A_4+A_3$ & $D_5+A_1$& $3A_2$ & $A_4$ & $A_1$\\ \hline
$D_5+2A_1$ & $A_4+A_2+A_1$ & $D_4+2A_1$ & $6A_1$ & $A_3+A_1$ &  \\ \hline
 $D_4+A_3$ & $A_3+ A_2+2A_1$& $A_6$ &$D_5$ & $A_2+2A_1$ &   \\ \hline
$D_4+3A_1$ & $A_3+4A_1$ & $A_5+A_1$ & $A_5$ & $4A_1$ &   \\ \hline
\end{tabular} 
\end{center} 
All the above singularity types are realizable (see the proof).
\end{theorem}
\vskip 0.5cm
The relatively minimal Gorenstein log del Pezzo surface (see Definition $2.1$) is introduced in [MZ2] where the Dynkin type of the Picard number two relatively minimal Gorenstein log del Pezzo surface is given but the number of the isomorphism classes with the given singularity is not provided. 
In this paper, we prove the following theorem. 
\vskip 0.5cm
\begin{theorem}  Let $V$ be a Picard number two relatively minimal Gorenstein
log del Pezzo surface $(\not= {\bf P}^1 \times {\bf P}^1 \  and \ \Sigma_1 )$. Then $V$ has one of the $10$ singularity types in Table $1.3$.
The number $m$ of the isomorphism classes of the Picard number two relatively minimal Gorenstein log del Pezzo surfaces with the given singularity type are also listed in Table $1.3$.
\begin{center}
Table $1.3$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c|c|c|c|}\hline
Dynkin type of $V$ & $m$ & Dynkin type of $V$  & $m$ & Dynkin type of $V$ & $m$
\\ \hline
$D_7$  & $\infty$ & $D_6$& $\infty$  & $D_4 $ & $1$ 
\\ \hline
$D_5+2A_1$  & $\infty$ &$ D_4+2A_1$   & $1$ &   $4A_1$ & $1$  
\\ \hline
$A_3+4A_1$  & $1$  & $ 2A_3$ & $\infty$  &    &   
\\ \hline
$A_3+D_4$  & $\infty$  & $ 6A_1$ & $1$  &    &   
\\ \hline

\end{tabular} 
\end{center} 
All isomorphism classes of the above singularity types are realized  in the course of  the proof.
\end{theorem}
\begin{corollary}  Let $W$ be a Gorenstein log del Pezzo surface.

$(1)$  Then there are disjoint curves $C_1$, $\cdots$, $C_k$ and a birational morphism 
$\sigma : W \longrightarrow W_{\rm min}$ such that $\sigma(C_i)$ are smooth points , $\sigma$ is an isomorphism outside $\cup C_i$ and $W_{\rm min}$ is isomorphism to one of the surfaces $V$ in Theorem $1.2$ or $1.6$.

$(2)$  The singularity type of $W$ is that of $V$ plus a few type $A_{n_i}$ $(1\leq i \leq k)$ singularities lying on $C_i$.

\end{corollary} 
 
\begin{remark} From {\rm [MZ2, Lemma 2]} or Lemma $2.3$, we know that Theorems $1.2$ and $1.6$ present the complete list of the singularity types of the relatively minimal Gorenstein log del Pezzo surfaces and the number of the isomorphism classes with the given singularity type.
\end{remark}

This paper is organized as follows. In section $2$, we introduce some terminology and preliminaries which will be used throughout the paper. In section $3$, we establish the connection between Gorenstein log del Pezzo surfaces and rational elliptic surfaces. We will also give out a method to classify all singularity types of the Gorenstein log del Pezzo surfaces. In sections $4$ and $5$, we prove Theorems $1.2$, $1.5$ and $1.6$.
\vskip 0.5cm
\hskip -0.6cm {\bf Acknowledgement.} I would like to thank Professor D.-Q. Zhang for many enlightening advice during the preparation of this paper.

\section{Preliminaries}

{\bf A. Relatively minimal Gorenstein log del Pezzo surfaces}
\vskip 0.2cm
\hskip -0.6cm 

Let $V$ be a Gorenstein log del Pezzo surface. Let $f : U \longrightarrow V$ be a minimal resolution of singularities and let $D:=f^{-1}({\rm Sing} V)$. Then there are no $(-n)$-curves on $U$ with $n \geq 2$ except for those $(-2)$-curves contained in $D$. The morphism $f$ is the contraction of all $(-2)$-curves on $U$. 

Let $\Sigma = \{ F_1, \cdots, F_t \}$ be a maximum set of disjoint union of $(-1)$-curves on $U$ such that the connected component of $\sum F_j +D$ containing $F_i$ is a linear chain $F_i+\Delta_i$ of $F_i$ and $n_i$ ($n_i \geq 0$) $(-2)$-curves with the dual graph:
$$
(-1)-(-2)-(-2)-\cdots -(-2).
$$
Let $U \longrightarrow U_{\rm min}$ be the smooth blow-down of all connected components of $\sum_{i=1}^{t} (F_i +\Delta_i )$. Let $D'$ be the image on $U_{\rm min}$ of $D$ and let $U_{\rm min} \longrightarrow V_{\rm min}$ be the contraction of $D'$. Then we have a commutative diagram:
\begin{center}
\begin{tabular}{rrr}

$ U$ & $ \longrightarrow $ & $U_{\rm min}$ \\
 $\downarrow$ & & $\downarrow$ \\
$ V$ & $\longrightarrow $& $  V_{\rm min}$
\end{tabular}
\end{center}

Here the map $V \longrightarrow V_{\rm min}$ contracts ${\bar F}_i :=f(F_i)$ to $t$ smooth points
and is isomorphism outside $\cup {\bar F}_i$. Note that $\rho (V_{\rm min})=\rho (V) - t$.

\begin{definition} The surface $V_{\rm min}$ is  called a 
relatively minimal model of $V$. $V$ is relatively minimal if $V=V_{\rm min}$. Note that $V_{\rm min}$ is not uniquely determined by $V$.
\end{definition}

From the above discussion, we have

\begin{lemma} Let $V$ be a Gorenstein log del Pezzo surface, $U$ the minimal resolution of $V$ and $D$ the exceptional divisor.
  
$(1)$ $V$ is not relatively minimal if and only if there is a $(-1)$-curve $E$ on $U$ such that the connected component of $E+D$ has dual graph:
$$
(-1) \ \ or \ \ (-1)-(-2)-(-2)-\cdots -(-2).
$$
$(2)$ If Sing$V_{\rm min}$ has Dynkin type $\sum a_iA_i+\sum d_iD_i + \sum e_iE_i$, then SingV has Dynkin type  $\sum a_iA_i+\sum d_iD_i + \sum e_iE_i + \sum_{i=1}^t A_{n_i}$.

\end{lemma}


  
Thanks to the virtue of Miyanishi-Zhang (cf. [25, Lemma 2]), we have the following Lemma.

\begin{lemma}  Let $V$ be a Gorenstein log del Pezzo surface. Then there exists a birational morphism $\eta : V \longrightarrow W$ such that the following assertions hold.

$(1)$ $W$ is a relatively minimal Gorenstein log del Pezzo surfaces of Picard number one or two.

$(2)$  Let $E_1$,...,$E_t$ be all exceptional curves of $\eta$. Then $w_i:=\eta (E_i) (1\leq i\leq t)$ are $t$ distinct smooth points of $W$ for each $i$. Moreover $V$ consists of the same singular points as on $W$ and additional points $x_i$ of the Dynkin type $A_{n_i}$ with $n_i \geq 1$, where  $1\leq i\leq t$.

\end{lemma} 
\vskip 0.5cm
{\bf B. Mordell-Weil groups of  elliptic surfaces}
\vskip 0.5cm
\hskip -0.6cm $(a)$ \ \  Definitions
\vskip 0.5cm
\hskip -0.6cm
Let $C$ be a smooth projective curve over an algebraically
closed field of characteristic 0, and $Y$ a (Jacobian)  {\em elliptic
surface} over $C$. By this we mean the following: $Y$ is a 
smooth projective surface with a relatively minimal elliptic
fibration
$$
  f : Y \longrightarrow C 
$$ 
such that

$(i)$\ $f$ has a global section $\cal O$, and

$(ii)$\ $f$ is not smooth, i.e., there is at 
least one singular fibre.
\vskip 0.5cm
\hskip -0.6cm For every (Jacobian) elliptic fibration $Y$ there 
is a group structure of sections $MW (Y)$ with the distinguished section 
$\cal O$  as zero which we call {\it Mordell-Weil group}. 
Up to a finite group, $MW (Y)$ is identified 
with the relative automorphism group of the fibration. The basic results on the Mordell-Weil groups are proved in [Sh].
\vskip 0.5cm
\hskip -0.6cm Due to a formula of Shioda-Tate we have  the basic 
inequalities
$$
0\leq {\rm rank}MW (Y) \leq \rho (Y)-2 
$$
where $\rho (Y)$ is the Picard number and the discrepancy
in the upper bound is related to the degree of reducibility
of the fibre.

\begin{definition}
An elliptic fibration $Y$ is called {\rm extremal} if $\rho (Y)=h^{1,1} (Y)$  
{\rm (maximal\  Picard \  number) } and
rank $MW (Y)=0$.
\end{definition}

The extremal elliptic surfaces have  been studied from various points of view. In [MP],  the complete classification of extremal rational elliptic surfaces is given. For extremal elliptic $K3$ surfaces, we refer to [ATZ], [MP1],
[MP2],  [SZ] and [Ye].
\begin{definition}\hskip 0.5cm 
$Y$ is called  a  {\rm Mordell-Weil rank $r$} rational elliptic surface if $Y$ is a rational surface and rank $MW (Y)=r$.
\end{definition}

In [M] and [P],  the complete classification of the  rational elliptic surfaces is given. In [OS], they describe the complete structure theorem of the Mordell-Weil groups of the rational elliptic surfaces. 
\vskip 0.5cm
\hskip -0.6cm The following result is proved in [OS, Corollary 2.3 and Theorem 2.5].

\begin{theorem} The Mordell-Weil rank one rational elliptic surface is generated by the section $P$ which is disjoint from the zero-section ${\cal O}$ with  $<P,P> \leq 2$,  where $<\cdot ,\cdot> $ is the height pairing on Mordel-Weil groups which is defined below.
\end{theorem}

\vskip 0.5cm
\hskip -0.6cm $(b)$ \ \  Height Pairing

\vskip 0.5cm
\hskip -0.6cm Let $f$: $Y \longrightarrow {\bf P}^1$ be a 
(relatively minimal) elliptic surface over ${\bf P}^1$ with
a distinguished section ${\cal O}$. The complete list of 
possible fibres has been given by Kodaira [K1]. It 
encompasses two infinite families $(I_n, I_n^*, n\geq 0)$
and six exceptional cases $(II, III, IV, II^*, III^*, IV^*)$.
And they can be considered as sublattices of the N$\acute{e}$ron-Severi group of $Y$.
\vskip 0.5cm
\hskip -0.6cm  Given  an ellipric surface $f$: $Y\longrightarrow {\bf P}^1$,
let $F_{\nu} = f^{-1} ({\nu})$ denote the fibre over $\nu \in
{\bf P}^1$, and let

$  Sing(f) =\{ \nu \in {\bf P}^1 | F_{\nu} \ {\rm is\  singular} \}$.

$  {\bf {\cal R}}= {\rm Red}(f) = \{ \nu \in {\bf P}^1 | F_{\nu}\  {\rm is \ 
reducible}\}. $

\hskip -0.6cm For each $\nu \in {\bf {\cal R}}$, let
$$
  F_{\nu} = f^{-1} (\nu ) = \Theta_{\nu ,0} + \sum_{i=1}^{m_{\nu}
-1} \mu_{\nu ,i} \Theta_{\nu ,i}  \ \ (\mu_{\nu ,0} =1 )
$$
where $ \Theta_{\nu ,i}$ ($ 0\leq i \leq m_{\nu} -1$) are the 
irreducible components of $F_{\nu}$, $m_{\nu}$ being their
number, such that $ \Theta_{\nu ,0}$ is the unique component
of $F_{\nu}$ meeting the zero section. 

\begin{lemma} {\rm (cf. [Sh] )}
For each $\nu \in {\bf P}^1$, the intersection matrix
$$
A_{\nu} = ( (\Theta_{\nu ,i} \cdot \Theta_{\nu ,j}))_{1\leq i,j\leq m_{\nu}-1}
$$
is negative-definite.
\end{lemma} 

\hskip -0.6cm Here we denote by $K=K({\bf P}^1)$ the function field,
$$
  E(K) = {\rm the \ group\ of \ sections \ of\ }\  f,
$$
and
$$
  NS(Y) = {\rm the \ group \ of \ divisors \ on}\  Y \ \ {\rm
 modulo \ algebraic \ equivalence.}
$$
In general, for a rational elliptic surface $Y$ with a section and with at least one singular fibre, there is a unique homomorphism $\varphi$ of $E(K)$ into the 
N$\acute{e}$ron-Severi group $NS(Y)\otimes {\bf Q}$ such that 
$\varphi (P) \equiv (P)$ {\rm mod} $T_{\bf Q}$ and $\varphi (P) \perp T$ where $T$ is the subgroup of $NS(Y)$ generrated by the zero section ${\cal O}$ and all the irreducible components of fibres (cf. [Sh, Lemma 8.1]).

\begin{definition} 
The pairing 
$$
  <P,Q> = - (\varphi (P) \cdot \varphi (Q))
$$
on the Mordell-Weil group $E(K)$ is called
the {\rm height pairing}, and the lattice
$$
   (E(K)/E(K)_{\it tor} , <,>)
$$
is called the {\rm Mordell-Weil Lattice} of the elliptic
curve $E/K$ or of the elliptic surface $f$: $Y \longrightarrow
{\bf P}^1$.
\end{definition}
\begin{theorem} {\rm (Explicit formula for the height pairing) 
[30, Theorem 8.6]} For any $P$,$Q \in E(K)$, 
we have
$$
<P ,Q> = \chi + (P{\cal O}) + (Q{\cal O}) - (PQ) - \sum_{\nu \in R} contr_{\nu}
(P,Q),
$$
$$
<P,P>=2\chi + 2(P{\cal O}) - \sum_{\nu \in R} contr_{\nu}
(P).
$$
\end{theorem}
\begin{remark} {\rm Here $\chi =\chi ({\cal O}_S )$, and $(P{\cal O})$ is the intersection number of the sections $(P)$
and $({\cal O})$, and similarly for $(Q{\cal O})$,$(PQ)$. The term  $contr_{\nu}
(P,Q)$ stands for the local contribution at $\nu \in R$, which is
 defined as follows: suppose that $(P)$ interests $\Theta_{\nu ,i}$
and $(Q)$ intersects $\Theta_{\nu ,j}$. Then we let
$$
contr_{\nu} (P,Q) = \left \{\begin{array}{ll} 
                (-A_{\nu}^{-1})_{i,j} ,& if \ i\geq 1,j\geq 1, \\
                               0                 ,& otherwise.
 \end{array}
\right.
$$
where the first one means the $(i,j)$-entry of the matrix $(-A_{\nu}^{-1})$.
Further we set
$$
 contr_{\nu} (P) =  contr_{\nu} (P,P).
$$

\hskip -0.5cm Arrange $\Theta_i = \Theta_{\nu ,i}$ $(i=0,1,\cdots ,m_{\nu} -1)$ so 
that the simple components are numbered as in the figure below.
\vskip 0.5cm
\centerline{\input mw0.tex}  
\vskip 0.5cm
\begin{center}
Figure $2.1$
\end{center}
 For the other types of reducible fibres, the numbering is irrelevant.
Assume that $(P)$ intersects $\Theta_{\nu ,i}$ and $(Q)$ intersect
$\Theta_{\nu ,j}$ with $i>1$,$j>1$. Then we have the following 
Table 2.1: the fourth row is for the case $i<j$ (interchange $P$, $Q$
if necessary).
\begin{center}
Table $2.1$
\end{center}
\vskip 0.5cm
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$T_{\nu}^-$ & $A_1$ & $E_7$ & $A_2$ & $E_6$ & $A_{b-1}$ &  $D_{b+4}$ 
\\ \hline 
type of $F_{\nu}$ & $III$ & $III^*$ & $IV$ & $IV^*$ & $I_b (b\geq 2 )$
& $I_b^* (b\geq 0)$ \\  \hline
$contr_{\nu} (P)$ &$\frac1 2$ & $\frac 3 2$ & $\frac2 3$ & $\frac4 3$&
$\frac{i(b-i)} b$ & $\left \{\begin{array}{ll} 
                 1 , & i=1 \\  
                  1+\frac b 4, & i>1
           \end{array}
\right.$ \\  \hline
$contr_{\nu} (P,Q) (i<j)$& $-$ & $-$ & $\frac1 3$ &$\frac2 3$ &
$\frac{i(b-j)} b$ &  $\left \{\begin{array}{ll} 
                         \frac1 2 , & i=1 \\  
                         \frac{(2+b)}4 ,& i>1
                  \end{array}
\right.  $ \\ \hline
\end{tabular}
\end{center}
}
\end{remark}

\vskip 0.5cm
\hskip -0.6cm {\bf C. Weierstrass model of elliptic surfaces}
\vskip 0.5cm
\hskip -0.6cm Let $f: Y \longrightarrow C$ be a relatively minimal elliptic surface with the global section ${\cal O}$, and $K(Y)$ and $K(C)$ the function fields of $Y$ and of $C$ respectively. $f$ induces a homomorphism $f^* : K(C) \longrightarrow K(Y)$, and $K(Y)$ is a transcendential extension of $K(C)$ of transcendence degree one. $Y$ is birationaly equivalent to the subscheme $Y^*$ in $Proj({\cal O}\oplus {\cal O}(2L)\oplus {\cal O}(3L))$, which is given by the equation
$$
V^2W=4U^3-g_2UW^2-g_3W^3 ,
$$
where ${\cal O}$ is the structure sheaf of $C$, $L$ a line bundle on $C$ with $deg L= p_g -q+1$,  $g_2 \in H^0(C, {\cal O}(4L))$ and $g_3 \in H^0(C, {\cal O}(6L))$ are the sections with $\Delta =g_2^3-27g_3^2 
\not\equiv 0$.
\begin{theorem} {\rm (cf. [Kas] )} \  \  $Y^*$ is an algebraic surface with rational double points as the only singularities. $Y$ is the minimal resolution of $Y^*$. $Y^*$ is determined by $g_2$, $g_3$ up to ${\bf C}^*$-operation
$$
(g_2,g_3) \longrightarrow (\lambda^4 g_2 , \lambda^6 g_3),  \lambda \in {\bf C}^*={\bf C}-\{0\}.
$$
$g_2$ , $g_3$ satisfy

$(i)$ \ $\Delta =g_2^3 -g_3^2 \not\equiv 0$,

$(ii)$ {\rm min}$(3\nu_p (g_2), 2\nu_p (g_3)) <12$ for all $p \in C$,

where $v_p(g_2)$, $v_p(g_3)$ and $v_p(\Delta)$ are the orders of the zeros of $g_2$, $g_3$ and $\Delta$ at $p$. The singular fiber in $Y^*$ over $p$ consists of the minimal resolution of the rational double point and the rational curve, which is defined by the section ${\cal O}$. The type of rational double and thereby the type of the singular fibre determines $v_p(g_2)$, $v_p(g_3)$ and $v_p(\Delta)$. $Y^*$ is call the {\rm Weierstrass model} of the elliptic surface.
\vskip 0.2cm
\hskip -0.6cm The ${\cal J}$-invariant of the model is ${\cal J} = \frac{g_2^3}{\Delta}$.
\end{theorem}
For the value of $v_p(g_2)$, $v_p(g_3)$ and $v_p(\Delta)$ of the singular fibre, we refer to [Her].
\vskip 0.5cm
\hskip -0.6cm {\bf D. Classification of elliptic surfaces with at most $4$ singular fibers}
\vskip 0.5cm
In [Hir], U.Schmickler-Hirzebruch classifies all relatively minimal elliptic fibrations $Y$ over the complex projective line ${\bf P}^1$ which have at most three singular fibers. There are $36$ possible combinations of singular fibers (each of the $36$ potential representations yields different fiber types). Furthermore, U.Schmickler-Hirzebruch exhibits each of the $36$ combinations of singular fibers on an elliptic surface of the desired type and shows that except for the trivial case $(I_0,I_0,I_0)$ and the case $(I_0^*,I_0^*)$, the surface is determined by the singular fiber types. On the other hand, except for the trivial case, those surfaces corresponding to rational or $K3$ surfaces. There are exactly {\em eleven} elliptic $K3$ surfaces with $3$ singular fibers (cf. [SZ], [Ye]). Thus we get the following theorem.

\begin{theorem} Let $Y$ be a rational elliptic surface with at least one and at most three singular fibers. Then except for singular fiber type $(I_0^*,I_0^*)$, the surface is unique up to isomorphisms.
\end{theorem}

In [Her], S.Herfurtner studies the elliptic surfaces with four singular fibers and nonconstant ${\cal J}$-invariant over ${\bf P}^1$. He also distinguish two sets:
$$
T^+=\{I_n (n\geq 0), II,III, IV\} \ \ {\rm and} \ \ T^-=\{I_n^*(n\geq 0), IV^*,III^*,II^*\}.
$$ 
\begin{theorem} {\rm (cf. [Her] )} \ \ Let $Y\longrightarrow {\bf P}^1$ be a minimal elliptic surface with section,  nonconstant ${\cal J}$-invariant and four singular fibers, of which at most one is in $T^-$.

$(i)$ \ If one singular fiber is in $T^-$, three in $T^+$, the Weierstrass model depend on a parameter. Given four different base points, in the case $(I_1^*,I_1,I_1,II)$, there exists four, in the cases $(I_1^*,I_1,I_2,II)$ and $(I_1,I_1,II,IV^*)$, there exist two elliptic surfaces, each depending on the ${\cal J}$-invariant, and for all other fiber combinations there exists precisely one elliptic surface.

$(ii)$\ \ If all four singular fibers are in $T^+$, then the Weierstrass models are determined uniquely up to isomorphisms, except for one combination: $(I_1,I_6,II,III)$, where there are two nonisomorphic models.
\end{theorem} 
 


\vskip 0.5cm
\hskip -0.6cm {\bf E. Rational surfaces}
\vskip 0.5cm
\begin{lemma} Let $Y$ be a smooth rational surface. Then we have

$(1)$ Suppose that $L$ is a smooth rational curve with $L^2 \geq 0$. Then $|L|$ is base point free and $h^0(Y,L)=2+L^2$.

$(2)$ Suppose that $L$ is an irreducible curve with $p_a(L)=1$ and $L^2 \geq 1$. Then a general member of $|L|$ is smooth and $h^0(Y,L)=L^2+1$.
If $L^2 \geq 2$, then $Bs|L|=\emptyset$. If $L^2=1$, then $|L|$ has exactly one base point.

\end{lemma}

{\em Proof.} (1) is clear when $L^2=0$. So we assume $L^2 \geq 1$. 
Consider the exact sequence below
$$
0 \longrightarrow {\cal O}_Y \longrightarrow {\cal O}_Y (L) \longrightarrow {\cal O}_{{\bf P}^1}(L^2) \longrightarrow 0. 
$$
Then,  we have $h^0(Y,L) = 1 + h^0({\bf P}^1, {\cal O} (L^2)) = 1 + (L^2 +1)
=L^2 +2$. 

To see $|L|$ is base point free, we let $\sigma : L_1 \longrightarrow L$ be the blow-up at $q\in L$ with $E$ the exceptional curve and let 
$L'$ be the proper transform of $L$. Then  $L'^2=L^2-1$ and $h^0(L')=L'^2 +2=L^2 + 1= h^0(L)-1$. On the other hand, $|L'|=|\sigma^* (L) - E| = |L-q|$; thus $|L|$ is base point free for
$$
h^0(L-q)= h^0(L)-1.
$$

(2) Considering the similar exact sequence as in (1) and the equality
$$
\chi ({\cal O}_L (L)) = h^0(L, {\cal O}_L (L))- h^1(L, {\cal O}_L (L)) =
\chi({\cal O}_L) + deg {\cal O}_L (L),
$$
we get
\begin{eqnarray*}
h^0 (Y,L)&=&1+ h^0(L, {\cal O}_L (L)) = 1+ h^0(L, \omega_L (-L))+ 1-p_a(L) + L^2  \\
&=& 1+L^2 + h^0 (L, \omega_L (-L)) = 1+L^2.
\end{eqnarray*} 

Here $\omega_L$ is the dualizing sheaf and we have applied Serre  duality for $L$. From this precise formula for $h^0(Y, L)$ and the similar discussion as in (1), we see that $Bs|L|=\emptyset$ if $L^2 \geq 2$. For the last assertion, see [DZ].

\section{Reduction to the Rational Elliptic Surfaces }
\vskip 0.5cm
Before starting the results, we explain our terminology.
\vskip 0.5cm
$(a)$ \ \  Let $V$ be a projective, normal surface with only rational double points as singularities. As usual, ratinal double points are indicated by their Dynkin types $A_n$, $D_n$, $E_6$, $E_7$ and $E_8$. When we say $V$ a surface of type $2A_1+D_6$ for example, it means that $V$ has $3$ singular points, one of which is of type $D_6$ and the other two are of type $A_1$. We indicate this by writing $V(2A_1+D_6)$.
\vskip 0.5cm
$(b)$ \ For  a log del Pezzo surface $V(2A_1+D_6)$ for example,  we denote by $U(2A_1 +D_6)$ the minimal resolution of  $V(2A_1 +D_6)$.
\vskip 0.3cm

For readers' convenience, we give a short proof of a result in [5] using Kawamata-Viehweg Vanishing Theorem which was not available at that time.

\begin{lemma}  Let $U$ be the minimal resolution of a Gorenstein log del Pezzo surface $V$. Then $|-K_U|$ has a reduced irreducible member. 
\end{lemma}

{\em Proof.}\ Note that $K_U$ is the pull back of $K_V$ and $K_U^2=K_V^2$ (cf. [1, 2]). Since $-K_U$, being the pull back of $-K_V$, is nef and big,  by the Kawamata-Viehweg Vanishing theorem,
we have $H^1(U, -K_U)= (0)$ (cf. [KMM]). Hence by the Riemann-Roch theorem, we get
$\dim |-K_U| = K_U^2$. Suppose that a member $A$ of $|-K_U|$ contains an
arithmetic genus $\geq 1$ irreducible component $A_0$. The Riemann-Roch theorem implies that $ |A_0 + K_U| \not= \emptyset$. From this and $0=A+K_U=(A_0 +K_U) +(A-A_0)$, we deduce that $A_0=A$ with $p_a (A_0)=1$ 
and Lemma 3.1 is true.

  Thus we may assume that every member of $|-K_U|$ is a union of smooth rational curves. The Stein factorization and the fact that $q(U)=0$ imply that a general member of $|-K_U|$ is of the form $M_1 + \cdots + M_k +F$, where $F$ is the fixed part of the linear system, $M_i \cong {\bf P}^1$, and $M_i \sim M_j$.  

  Now suppose that $K_U^2 =1$. If $F=0$, then $k=1$ and $M_1^2 =1$ and Lemma 3.1 is true. Since $-K_U$ is nef and big, it is $1$-connected by a result of C.P.Ramanujam (cf. [Reid]). Hence $1=K_U^2 \geq (kM_1 + F).kM_1 \geq 1+ k^2 M_1^2$. Thus $M_1^2 =0$, $k=1$, $M_1.F=1$ and $K_U.F=0$. Now intersecting the relation $-K_U \sim M_1 +F$ with the smooth rational curve $M_1$ of self-intersection $0$, one gets a contraction. So Lemma 3.1 is true when $K_U^2=1$.

For $K_U^2 \geq 2$, let $U_1 \longrightarrow U$ be the blow-up of a point on $M_1-(M_2 + \cdots + M_k +F)$. then $-K_{U_1}$ is linearly equivalent to the proper transform of $M_1+\cdots +M_k+F$ and hence $nef$ and $big$. If $U_1 \longrightarrow V_1$ is the contraction of all $(-2)$-curves then $V_1$ is a Gorenstein log del Pezzo surface. For $U_1$, we may argue as in the case of $U$ and have reduced the proof of Lemma 3.1 to the case $K_U^2=1$, which has been dealt with in the previous paragraph. This completes the proof of Lemma 3.1.

\begin{lemma} Let $U$ be the minimal resolution of a Picard number one Gorenstein log del Pezzo surface $V$ with $K_V^2=d$ where $2\leq d \leq 7$. Then we can construct another surface $U_{d-1}$ which is the minimal resolution of a Picard number one Gorenstein log del Pezzo surface $V'$ with $K^2_{V'}=d-1$.
\end{lemma}

{\em Proof.}  \ From Lemma 3.1,  we may choose an irreducible member of $|-K_U|$, say $A$ and let $E_d$ be a $(-1)$-curve on $U$. Such a $(-1)$-curve exists because $K_V^2 \leq 7$. Then $A$ meets $E_d$ at a point $q$. We may choose $A$ general such that $E_d$ is the only $(-1)$-curve through $q$.  Let $U_{d-1} \longrightarrow U$ be the blow-up of $q$ with $E_{d-1}$ the except curve. Then $-K_{U_{d-1}}$ is linearly equivalent to the proper transform $A_{d-1}$ of $A$. Since $K_{U_{d-1}}^2 >0$, the divisor $-K_{U_{d-1}}$ is nef and big. And the curves having $0$ intersection with $A_{d-1}$ are precisely the inverse image 
of the $(-2)$-curves on $U$ (contractible to singular points on $V$)
and the proper transform $E'_d$ of $E_d$. It follows that the contraction of all the $(-2)$-curves on $U_{d-1}$ gives a Picard number one Gorenstein log del Pezzo surface $V'$.

\begin{lemma}  Let $V$ be a Picard number one Gorenstein log del Pezzo surface with $K_V^2=1$ and $U$ the minimal resolution of $V$ with $D$ the exceptional divisor. Then there exists an extremal rational elliptic surface $Y$ such that $g : Y \longrightarrow U$ is the
blow-down of one  $(-1)$-curve.
\end{lemma}

{\em Proof.}\hskip 0.5cm Since $K_U $ is the pull back of $K_V$, we have $K_U^2=K_V^2=1$, $K_U^2 =10-\rho (U)$, $\rho (U)=\rho (V)+ \sharp (D)$ and $\sharp (D) =8$. By Lemma 2.14, we know $Bs|-K_U|= \{p\}$.  

Let $f: Y \longrightarrow U$ be the blow-up at $p$, then with Lemma 2.14, 
we have

(1)   $Y$ is a rational surface.

(2)   $K_Y^2 =0$ and $|-K_Y|$ induces an elliptic fibration of $Y$.

(3)   $\rho (Y) = 10 - K_Y^2 =10$.

(4)  By the formula of Shioda (cf. [S])
$$
  10=\rho (Y)= \sharp (D) + {\rm rank \ MW(Y)} +2,
$$
we get  ${\rm rank \ MW(Y)}=0$.
\vskip 0.3cm
Thus $Y$ is an extremal rational elliptic surface. This proves the lemma.
\vskip 0.5cm
Combining Lemmas 3.2 and 3.3, we get the following Theorem 3.4.

\begin{theorem}  Let $V$ be a Picard number one Gorenstein log del Pezzo surface with $K_V^2 \leq 7$ and $U \longrightarrow V$  the minimal resolution . Then there exists an extremal rational elliptic surface $Y$ such that $f: Y \longrightarrow U$ is a succession of blow downs of some $(-1)$-curves. 
\end{theorem}

\vskip 0.3cm
\hskip -0.6cm  We may generalize the above Theorem $3.4$ to the Picard number $r$ ($r\geq 2$) Gorenstein log del Pezzo surfaces.

 \begin{theorem}
 Let $V$ be a Picard number $r$ ($r\geq 2$) singular Gorenstein log del Pezzo surface and $U \longrightarrow V$ the minimal resolution. Then there exists a Mordell-Weil rank $(r-1)$ rational elliptic surface $Y$ such that $f: Y \longrightarrow U$ is a succession of blow downs of some $(-1)$-curves.
\end{theorem}
\vskip 0.3cm
The definition of the Mordell-Weil rank $r$ rational elliptic surface is given in the Secition $2$. 
\vskip 0.3cm

{\em Proof.} \ \ For simplicity, we only prove Theorem 3.5 for $r=2$. The general case can be
proved in the similar way.
\vskip 0.3cm
We let $U \longrightarrow V$  be the minimal resolution of a Picard number two Gorenstein log del Pezzo surface $V$ with $D$ the exceptional divisor and $d:=K_V^2\geq 2$. From Lemma 3.1, we know that $|-K_U|$ has a reduced irreducible member. 

Since  $10-K_V^2=\rho (V)+ \sharp (D)$,  $\rho (V)=2$ and  $\sharp (D) \geq 1$ beacuse $V$ is a singular surface, 
we get $ 10- K_V^2 \geq 3$ and $K_V^2 \leq 7$. On the other hand, since $K_U$ is the pull back of $K_V$, $K_U^2 =K_V^2 \leq 7$.  

Thus we may choose an irreducible member of $|-K_U|$ say $A$ and let $E_d$ be a $(-1)$-curve on $U$. Such a $(-1)$-curve exists because $K_V^2 \leq 7$. Then $A$ meets $E_d$ at a point $q$. 

Since $K_U^2 \geq 2$, we know $|-K_U|$ is base point free by Lemma 2.14. Furthermore since the smooth reduced irreducible member of $|-K_U|$ is dense in $|-K_U|$ and since there exists only finitely many $(-1)$-curves on $U$ (cf. the proof of [6, Lemma $13$]), we get the following Claim.

\begin{claim} We may choose $A \in |-K_U|$ such that there doesn't exist another $(-1)$ curve passing through the point $q$.
\end{claim}
Let $U_{d-1}$ be the blow-up of $q$ with $E_{d-1}$ the exceptional curve and $E_d'$ the proper transform of $E_d$. Then by the similar discussion as Lemma 3.2, we know that $U_{d-1}$ is the minimal resolution of a Gorenstein log del Pezzo surface $V'$ with $K^2_{V'}=d-1$. 

\begin{claim} 
 $\rho (V')=2$.
\end{claim}

{\em Proof of the Claim.}\ It is clearly that $\rho (V') \leq 2$. Thus to prove the Claim, we need to show that $\rho (V')\not= 1$. Suppose $\rho (V')=1$. Then from the above construction, we know that there must exist a curve $C$ on $U$ such that $C^2 \geq 0$ and $mult_p C \geq 2$ and the proper transform $C'$ of $C$ in $U_{d-1}$ is an $(-2)$-curve. Thus we have $C'.E_{d-1} = mult_p C=m \geq 2$, $E_d'.E_{d-1}=1$ where $C'$ and $E_d'$ are $(-2)$-curves.
  
  Now we pay attention to the set $I= \{E_d', E_{d-1} ,C'\}$ in $U_{d-1}$.  
Clearly we may do the same operation (blow-up) on $U_{d-1}$ as before and finally get an rational elliptic surface $Y$.  Since $C'$ and $E_d'$ are already $(-2)$-curves  we will keep the same symbols in $Y$ and let $E'$ be the proper transform of $E_{d-1}$. To prove the Claim, we need to consider two cases.

1.  $E'$ is an $(-1)$-curve.

In this case and from the elliptic fibration of $Y$, we see that $C'$ and
$E_d'$ will be in two different elliptic fibers and $E'$ is a section of $Y$ which is a contradiction since $C'.E'=C'.E_{d-1} \geq 2$ .

2. $E'$ is an $(-2)$-curve.

In this case we see that $E_d'$, $E'$ and $C'$ will be in one elliptic fiber. We also get a contracdiction since  $C'.E' \geq 2$ and we can't find an elliptic fiber contains $C'$ and $E'$.

Thus we prove the Claim. We note that the above proof  can easily be generalized to prove the general case.









\vskip 0.3cm
\hskip -0.6cm   Continue this process. Since $K_{U_{d-1}}^2=d-1 < K_U^2$, this process must stop at the $(d-1)$-th step. Thus,  we obtain a surface $U_1$ which is the minimal resolution of the Picard number two Gorenstein log del Pezzo surface $V_1$ with $K^2_{V_1}=1$. On $U_1$ we can not do the above process again. But since $K_{U_1}^2=1$, $|-K_{U_1}|$ has exactly one base point $q$ by Lemma 2.14. Blowing up $q$ will give rise to a Mordell-Weil rank one rational elliptic surface $Y$ as in Lemma 3.3. Thus Theorem 3.5 is proved.  

\begin{remark}
 From Theorems $3.4$ and   $3.5$, together with {\rm [OS]} and {\rm [P]}, we may classify all singularity types of the Gorenstein log del Pezzo surfaces.

\end{remark}

 

\begin{question} From the proofs of Theorems $3.4$ and $3.5$, it is natural for us to ask the converse question: If $U \longrightarrow V$ is the minimal resolution of a Picard number  one (resp. Picard number $r \geq 2$) Gorenstein log del Pezzo surface $V$ . Let $E$ be a $(-1)$-curve on $U$ and $\phi : U \longrightarrow U'$  the blow-down of $E$, then could $U'$ be a minimal resoultion of a Picard number one (resp. Picard number $r \geq 2$) Gorenstein log del Pezzo surface $V'$?  
\end{question}



\begin{lemma} Let the notation be the same as in Question $3.9$. Let $f: U \longrightarrow U'$ be the blow-down of a $(-1)$-curve $E$ and $D$ the exceptional divisor of the minimal resolution $U \longrightarrow V$. Let $U' \longrightarrow V'$ be the contraction of all $(-2)$-curves and let $D'$ be the exceptional divisor.  Then

$(1)$ \ \ $V'$ is a Gorenstein log del Pezzo surface.

$(2)$ \ \ If $D.E=1$, $\rho (V')= \rho (V)$. The converse is also true.
\end{lemma}

{\em Proof.} \ We fix a nonsingular member  $A \in |-K_U|$, the existence
of which is proven in [D] . We also let $A'=f_*(A)$.
Then $A' \in |-K_{U'}|$ and $(A'^2)\geq (A^2) \geq 1$. Since $-K_{U}$ is nef and big, so is $-K_{U'}$. To vertify $(1)$, we have only to show that $-K_{V'}$ is ample. In other words, for every curve $C$ on $U'$ with $(-K_{U'},C)=0$, we must show that $C \leq D'$. Since $-K_{U'}$ is nef and big, by Hodge Index Theorem, we get $C^2 <0$. On the other hand,
$$
2p_a(C)-2 =C^2+C.K_{U'}=C^2 <0.
$$
Thus we have $p_a(C)=0$ and so $C={\bf P}^1$, $C^2=-2$ and $C \leq D'$. 
$(1)$ is proved. 
\vskip 0.5cm
\hskip -0.6cm
 $(2)$ Since $E.D=1$, we have $K_{U'}^2=K_U^2+1$ and
$\sharp (D)= \sharp (D') +1$. Thus the first part in the assertion  $(2)$ follows from the relations $K_U^2=K_V^2=10-\rho (U)$, $K_{U'}^2=K_{V'}^2=10-\rho (U')$, $\rho (U) = \rho (V) + \sharp (D)$and $\rho (U')=\rho (V') + \sharp (D')$. The converse part is clear from the above discussion.
\vskip 0.5cm
\hskip -0.6cm
We will call a $(-1)$-curve  which satisfies the part $(2)$ of Lemma $3.10$ a {\it Nice Exceptional Curve} or simply an {\it NEC}.  

  

\section{Complete Classification of the Picard number one Gorenstein log del Pezzo Surfaces} 


We shall prove Theorem $1.2$ in the present section.
\vskip 0.5cm
\hskip -0.6cm Let $V$ be a Picard number one Gorenstein log del Pezzo surfaces  and $g$ : $U \longrightarrow V$  a minimal resolution of singularities. Then $K_U^2=K_V^2 >0$, $|-nK_U| \not= \phi$ for $n>0$ and $U$ is a rational surface. Hence $K_U^2 + \rho (U)=10$. This implies that $1\leq K_U^2 \leq 9$.
We observe:
\vskip 0.5cm
(i) If $K_U^2=9$, then $V$ is smooth and isomorphic to ${\bf P}^2$.
\vskip 0.5cm
(ii) If $K_U^2=8$, then $U$ is the Hirzebruch surface $\Sigma_2$ and $V$ is obtained by contracting the $(-2)$-curve to an $A_1$-singularity. In this case $V$ is isomorphic to the quadric cone $Q := \{ X^2+Y^2+Z^2=0 \}$ in ${\bf P}^3$.
\begin{lemma}  $K_U^2\not= 7$. 
\end{lemma}

{\em Proof.} Assume the contrary, i.e., $K_U^2=7$. In this case
$U$ is obtained from a Hirzebruch surface $\Sigma_n$ by one blow-up with $E$ the exceptional curve. Let $S$ be a curve on $\Sigma_n$ with self-intersection $-n$  and $L$ a fibre of the ${\bf P}^1$-fibration on $\Sigma_n$. We may assume that the blow-up point $p\in L$. We have $K_{\Sigma_n} \sim -2S-(2+n)L$. 

If $p\not\in S$, then $K_U\sim -2S-(2+n)L'-(1+n)E$, where $L'$ is the proper transform of $L$. Clearly, $U$ contains a $(-2)$-curve different from $S$, say $C$. Since $K_U .C=0$, the curve $C$ is disjoint from $S$, $L'$ and $E$. This is impossible. 

If $p\in S$, then $K_U \sim -2S'-(2+n)L'-(3+n)E$ where $S'$ is the proper transform of $S$. Again there is a $(-2)$-curve different from $S$. Thus we get a contradiction as above. This completes the proof of the lemma.

\vskip 0.5cm
\hskip -0.6cm In the rest of this section, we assume $1\leq K_V^2 \leq 6$.
\vskip 0.3cm
\hskip -0.6cm 
We start with $K_V^2 =1$.
\vskip 0.3cm
\hskip -0.6cm
From Lemma 3.3, we know that if $U$ is the minimal resolution of a Picard number one Gorenstein log del Pezzo surface $V$,  then every NEC on $U$ comes from either the $(-1)$-curve on an extremal rational elliptic  surface $Y$ or the 
$(-2)$-curve which intersects with the exceptional curve of the blow-down $Y \longrightarrow U$. The $(-1)$-curves on $Y$ are the sections of the elliptic fibration and form the Mordell-Weil group of  the elliptic fibration $Y \longrightarrow {\bf P}^1$.   

 From [MP], we get a complete list of the extremal rational elliptic surfaces $Y$ which are listed in Table $4.1$. For convenience, we label the surfaces. By contracting the zero section $\cal O$ in $MW(Y)$, we get a surface $U$ which is the minimal resolution of a Picard number one Gorenstein log del Pezzo surface $V$ with $K_V^2 = 1$.   By a surface $U_j^i$, we mean that it is the blow-down of some $(-1)$-curves from the No.$i$ extremal rational elliptic  surface and $K^2=j$. We let $n$ be the number of the NEC on $U_1$.
\vskip 0.5cm
\begin{center}
Table $4.1$
\end{center}
\begin{center}
\begin{tabular}{|c|c| c|c|c|c|c|c|}\hline
$\sharp$ & $Y$  &  $U_1$  & $n$& $\sharp$ & $Y$ & $U_1$ & $n$
\\ \hline
$1$& $II, II^*$  & $U^1_1(E_8)$& $1$& $9$&$I_1^*,I_4,I_1$  & $U^9_1(D_5+A_3)$ &$1$
\\ \hline
$2$& $III, III^*$  &$U^2_1(A_1+E_7)$  &$1$ & $10$&$ I_2^*,I_2,I_2$ & $U^{10}_1(D_6+2A_1)$ &$1$ 
\\ \hline
 $3$& $IV, IV^*$ & $U^3_1(A_2+E_6)$ &$1$  & $11$&$ I_9,I_1,I_1,I_1  $ & $U^{11}_1(A_8)$ &$2$      \\ \hline
$4$&$I_0^*,I_0^*$   & $U^4_1(2D_4)$ & $2$&$12$& $I_8,I_2,I_1,I_1$  &$U^{12}_1(A_7+A_1)$ &$1$
\\ \hline
$5$&$II^*,I_1,I_1$   & $U^5_1(E_8)$  &$1$ &$13$& $ I_5,I_5,I_1,I_1$  &$U^{13}_1(2A_4)$ &$0$
\\ \hline
$6$&$ III^*,I_2,I_1$   & $U^6_1(E_7+A_1)$ &$1$ &$14$& $ I_6,I_3,I_2,I_1$  & $U^{14}_1(A_5+A_2+A_1)$ &$0$
\\ \hline
$7$&$IV^*,I_3,I_1$  & $U^7_1(E_6+A_2)$  &$1$& $15$&$I_4,I_4,I_2,I_2$  &$U^{15}_1(2A_1+2A_3)$ &$0$
\\ \hline
$8$&$I_4^*,I_1,I_1$   &$U^8_1(D_8)$ & $2$ & $16$&$I_3,I_3,I_3,I_3$   &$U^{16}_1(4A_2)$ &$0$
\\ \hline
\end{tabular} 
\end{center} 

From the proof of Lemma 3.3, we know there is a one to one correspondence between the No.$i$ extremal rational elliptic surfaces $Y$ and $U^i_1$.  On the other hand, for every $i\not=4$, there is only one isomorphism class of the No.$i$ extremal rational elliptic surface (cf. [MP, Theorem $5.4$]). The extremal rational elliptic surfaces of No.$4$, i.e., with singular fibers $(I_0^*, I_0^*)$, are parameterized by $j\in {\bf C}$ [ibid.].  Thus we get the following lemma:

\begin{lemma} Let  $V$ be the Picard number one  Gorenstein log del Pezzo surface with $K^2_V=1$. Then the singularity type of $V$, the number $m$ of the isomorphism class with the given singularity type are listed in Table $4.2$.

\vskip 0.5cm
\begin{center}
Table $4.2$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c|c|}\hline
 Dynkin type of $V$ & $m$ & Dynkin type of $V$  & $m$\\ \hline
$E_8$ & $2$  &  $A_8$ & $1$\\ \hline
$A_1+E_7$ & $2$  &  $A_1+A_7$& $1$  \\ \hline
$A_2+E_6$ & $2$ & $2A_4$ & $1$ \\ \hline
$2D_4$ & $\infty$ &$A_1+A_2+A_5$ & $1$  \\ \hline
$D_8$ & $1$ & $2A_1+2A_3$ & $1$ \\ \hline
$A_3+D_5$ & $1$ & $4A_2$ & $1$  \\ \hline
$2A_1+D_6$ & $1$  & & \\ \hline
\end{tabular} 
\end{center} 
\end{lemma}

Next we consider $K_V^2=2$.
\vskip 0.5cm

By blowing down an NEC on $U_1$ which is listed in Table $4.3$, we get a surface $U_2$ which is the minimal resolution of a Picard number one Gorenstein log del Pezzo surface $V_2$. They are listed in the following table where we let $n$ be number of the NECs on $U_2$.
\vskip 0.5cm
\begin{center}
Table $4.3$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c| c|c|c|c|} \hline
  $U_1$ & $U_2$ &$n$ & $U_1$ & $U_2$ &$n$ \\ \hline
  $U^1_1 (E_8)$ & $U^1_2(E_7)$ &$1$ &$U^8_1(D_8)$ & $U^8_2(A_1+D_6)$& $1$ \\ \hline
$U^2_1(A_1+E_7)$ & $U^2_2(A_1+D_6)$ &$1$ &$U^8_1(D_8)$ & $U^8_2(A_7)$ &$2$ \\ \hline
  $U^3_1(A_2+E_6)$ & $U^3_2(A_2+E_5)$ &$1$ &$U^9_1(A_3+D_5)$ & $U^9_2(A_1+2A_3)$& $0$\\ \hline
 $U^4_1(2D_4)$ & $U^4_2(3A_1+D_4)$& $0$ &$U^{10}_1(2A_1+D_6)$ & $U^{10}_2(3A_1+D_4)$ &$0$  \\ \hline
 $U^5_1(E_8)$ & $U^5_2(E_7)$ &$1$ &$U^{11}_1(A_8)$& $U^{11}_2(A_2+A_5)$& $1$ \\ \hline
 $U^6_1(A_1+E_7)$ & $U^6_2(A_1+D_6)$& $1$ &$U^{12}_1(A_1+A_7)$ & $U^{12}_2(A_1+2A_3)$ &$0$ \\ \hline
 $U^7_1(A_2+E_6)$ & $U^7_2(A_2+A_5)$& $1$  & & & \\ \hline
 

\end{tabular} 
\end{center} 

\begin{remark} \ $(i)$ \ In the following discussion, the Figure $i$ $(i=1,2,3,4)$ are listed in the configurations of  Appendix.

(ii) To see $U_2^{11}(A_2+A_5)$ also doesn't depend on the choice of the blow-down curve $E_i$($i=1,2$) on $U_1^{11} (A_8)$ (cf. Figure $2$),  we look at the No.$11$ extremal rational elliptic surface $Y$ (cf. Table $2$). A simple calculation shows that the Mordell-Weil group of $Y$ is ${\bf Z}/3{\bf Z}$ and there are three sections $E_0$,$E_1$ and $E_2$ (cf. Figure $2$). The section $E_1$ defines an automorphism $g$ of $Y$ such that $g(E_0)=E_1$, $g(E_1)=E_2$ and $g(E_2)=E_0$. For simplicity, we denote by $bd_i$ the blow-down of the $E_i$ $(i=0,1,2)$. Then
$$
bd_2 (U_1^{11}(A_8)) = bd_2 \circ bd_0 (Y) \cong bd_0 \circ bd_1 (Y) \cong bd_1 \circ bd_0 (Y) = bd_1 (U_1^{11} (A_8)).
$$
\end{remark}

\begin{claim} $U^1_2(E_7) \cong U^5_2(E_7)$.
\end{claim}
{\em Proof.} \ \ \ For simplicity,  we let 
$$
\begin{array}{llll}
U_1 =U_1^1(E_8), &   V_1 = V_1^1(E_8), &  U_2 =U_2^1(E_7), & V_2 = V_2^1(E_7),  \\
U'_1 = U_1^5(E_8), &  V'_1 =V_1^5(E_8), &  U'_2 = U_2^5(E_7), & V'_2 =V_2^5(E_7).
\end{array}
$$
By the construction of $U_2$, $U_2'$, there is a unique $(-1)$-curve $E_2$ on $U_2$. To prove $U_2 \cong U'_2$, we only need to show that $|-K_{U_2}|$ has a cuspidal member $A$ which is the image of the type $II$ fiber and also a nodal member $A'$; in fact if $U_1 \rightarrow U_2$ (resp. $U'_1 \rightarrow U_2$) is the blow-up of $q=A \cap E_2$ (resp. $q'=A' \cap E_2$) 
then $|-K_{U_2}|$ (resp.$|-K_{U'_2}|$) has a cuspidal (resp. nodal) member, whence 
$$U_2 \cong {\rm \ (the \ blow-down\ of \ the \ unique \ (-1)-curve  \ 
on \ U_1' \  )} = U'_2.
$$
 We let $Y$ (resp. $Y'$) be the extremal rational elliptic  surface with singular fiber type $II^*$, $II$ (resp. $II^*$,$I_1$,$I_1$). There is a   composition $Y \longrightarrow {\bf P}^2$ (resp. $Y' \longrightarrow {\bf P}^2$) of blow-down of the section $E$ and all components in the type $II^*$ fiber except for a multiplicity-$3$ component $C'_3$.

  Clearly the above birational morphism $Y \longrightarrow {\bf P}^2$ factors as $Y \longrightarrow U_2 \longrightarrow {\bf P}^2$, where $Y\longrightarrow U_2$ is the blow-down of $E$ and the multiplicity-1 component $C_1$ of $II^*$. 

  Thus the existence of $A$, $A'$ on $U_2$ is equivalent to the existence of a cuspidal cubic $B$ (the image of the type $II$ fibre) and a nodal cubic $B'$ on ${\bf P}^2$ with a common inflex (the image of the multiplicity-3 component $C'_3$ in the type $II^*$ fibre) defined by $Z=0$, with local intersection $I(B,B';[0,1,0])=7$ at the inflex point $[0,1,0]$. We may assume that $B=\{ Y^2Z=X^3\}$ and just let
$B'=\{ Y^2Z=X^3+XZ^2+\sqrt{-4/27} \}$. This completes the proof of the Claim.
    

\begin{claim}
$(1)$ $U^2_2(A_1+D_6)\cong U^6_2(A_1+D_6) \cong U^8_2(A_1+D_6)$  \ \ {\rm (cf. Figure $1$)}.

$(2)$ $U^3_2(A_2+A_5)\cong U^7_2(A_2+A_5) \cong U^{11}_2(A_2+A_5)$ \ \ {\rm (cf. Figure $2$)}.

$(3)$  $U^4_2(3A_1+D_4) \cong U^{10}_2(3A_1+D_4)$ \ \ {\rm (cf. Figure $3$)}.

$(4)$  $U_2^9(A_1+2A_3) \cong U_2^{12}(A_1+2A_3)$\ \ {\rm (cf. Figure $4$)}.
\end{claim}

{\em Proof.} 
(1)\ \ From the construction of $U^2_2(A_1+D_6)$,  there is an NEC $E_2$ 
on $U^2_2(A_1+D_6)$ and there is a smooth rational curve $L$ and $L^2=0$ such that $L\cap E_2 = \{p \}$. On the other hand, by Lemma 2.14, we know that $|L|$ is base point free and defines a ${\bf P}^1$-fibration. We choose a general member $L_1$ of $|L|$, and let  $\{q \}=L_1 \cap E_2$. It is easy to see that $U_1^2(A_1+E_7)$ is the blow-up of $U_2^2(A_1+D_6)$ at point $p$ and $U_1^6(A_1+E_7)$ is the blow-up of $U_2^2(A_1+D_6)$ at point $q$. Thus
$$
U_2^2(A_1+D_6) \cong ({\rm \ the \ blow-down \  of \ the \ uniqe \ NEC \ on\ } U_1^6(A_1+E_7)) = U^6_2(A_1+D_6)
$$
On the other hand, blowing up at point $r$ in $U_2^6(A_1+D_6)$ will produce
$U_1^8(D_8)$, and there is unique way from $U_1^8(D_8)$ to $U_2^8(A_1+D_6)$.
Thus
$$
U^2_2(A_1+D_6) \cong U^8_2(A_1+D_6).
$$
By the similar argument as (1), we may prove (2), (4).

\vskip 0.5cm
(3) \  Since $U_1^{10}(2A_1+D_6)$ is unique and there is only one $NEC$ on $U_1^{10}(2A_1+D_6)$,  we know $U_2^4(3A_1+D_4)$ is unique and
$$
U^4_2(3A_1+D_4) \cong U^{10}_2(3A_1+D_4).
$$
This proves the Claim.
\vskip 0.5cm
Since $U_i^j \cong U_{i'}^{j'}$ if and only if $V_i^j \cong V_{i'}^{j'}$, we get the following lemma:

\begin{lemma} Let  $V$ be the Picard number one  Gorenstein log del Pezzo surface with $K^2_V=2$. Then the singularity type of $V$ is one of the  following:
$$
E_7,  A_7,  A_1+D_6, A_1+2A_3, A_2+A_5,  3A_1+D_4.
$$
Furthermore, $V$ is uniquely determined by its singularity type.
\end{lemma}

Next we deal with $K_V^2=3$.
\vskip 0.5cm
By blowing down an NEC on $U_2$ which is listed in Table $4.4$, we get a surface $U_3$ which is the minimal resolution of a Picard number one Gorenstein log del Pezzo surface $V_3$. They are listed in the following table where we let $n$ be the number of the NEC on $U_3$.
\vskip 0.5cm
\begin{center}
Table $4.4$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c|c|c|c|} \hline
  $U_2$ & $U_3$ &$n$   &$U_2$ & $U_3$ &$n$ \\ \hline
  $U^1_2 (E_7)$ & $U^1_3(E_6)$ &$1$ & $U^3_2(A_2+A_5)$ & $U^3_3(3A_2)$ &$0$ \\ \hline
$U^6_2(A_1+D_6)$ & $U^6_3(A_1+A_5)$ &$1$   & $U^8_2(A_7)$ & $U^8_3(A_1+A_5)$& $1$ \\ \hline
\end{tabular} 
\end{center} 

\begin{remark} $(i)$ There are two NECs, $E_0'$ and $E_1'$ on $U_2^8(A_7)$ (cf. Figure $1$).  To see $U_3^8(A_1+A_5)$ doesn't depend on the choice of the blow-down of the
$E_i'$ on $U_2^8(A_7)$ where $i=1,2$, we look at the No.8 extremal rational elliptic surface $Y$ (cf. Table $2$). A direct calculation shows that the Mordell-Weil group of $Y$ is ${\bf Z}/2{\bf Z}$ and there are two sections $E_0$, $E_1$ (cf. Figure $1$). The section $E_1$ define an automorphism $g$ of $Y$ which interchanges $E_0$ and $E_1$. And this $g$ will induce an automorphism of $U_2^8(A_7)$ which interchanges $E_i'$ ($i=0,1$). Thus blowing down $E_0$ or $E_1$ will produce the same surface. 

$(ii)$ \  By the similar argument as in Claim $4.5$, we have
 $$
U^8_3(A_1+A_5) \cong U^6_3(A_1+A_5) \ \ {\rm (cf. Figure \ 1)}.
$$ 
\end{remark}
Thus we have the following lemma:
\begin{lemma} Let  $V$ be the Picard number one  Gorenstein log del Pezzo surface with $K^2_V=3$. Then the singularity type of $V$ is one of the  following:
$$
E_6,  3A_2, A_1+A_5.
$$
$V$ is uniquely determined by its singularity type.

\end{lemma}

Finally we deal with \ $ 4\leq K_V^2 \leq 6$.
\vskip 0.5cm 
By blowing down an NEC on $U_3$ which is listed in Table $4.5$, we get a surface $U_4$ which is the minimal resolution of a Picard number one Gorenstein log del Pezzo surface $V_4$. 
\vskip 0.5cm
\begin{center}
Table $4.5$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c|} \hline
  $U_3$ & $U_4$ &$n$   \\ \hline
  $U^1_3 (E_6)$ & $U^1_4(D_5)$ &$1$   \\ \hline
$U^8_3(A_1+A_5)$ & $U^8_4(2A_1+A_3)$ &$0$  \\ \hline
\end{tabular} 
\end{center} 


\begin{lemma} Let  $V$ be the Picard number one  Gorenstein log del Pezzo surface with $K^2_V=4$. Then the singularity type of $V$ is one of the following:
$$
D_5, 2A_1+A_3
$$
Furthermore,  $V$ is uniquely determined by its singularity type.
\end{lemma}

By blowing down the unique NEC on $U^1_4(D_5)$, we get $U_5^1(A_4)$ and so the Picard number one Gorenstein log del Pezzo surface $V(A_4)$.  Then blowing down the unique NEC on $U_5^1(A_4)$, we get $U(A_1+A_2)$ with $\sharp \{{\it NEC} \ {\rm on}\  U(A_1+A_2) \}=0$, and so the Picard number one Gorenstein log del Pezzo surface $V(A_1+A_2)$.
\vskip 0.5cm
Combining Lemmas $4.2$, $4.6$, $4.8$, $4.9$ and the above discussion, we prove  Theorem $1.2$.



\section{Complete classification of the Picard number two relatively minimal Gorenstein log del Pezzo Surfaces}

We shall prove Theorems $1.5$ and $1.6$ in the present section. 
\vskip 0.3cm
\hskip -0.6cm
Let $V$ be a Picard number two Gorenstein log del Pezzo surface and $g$ : $U \longrightarrow V$  a minimal resolution of singularities. We also denote by $D:=g^{-1}({\rm Sing}V)$ the exceptional divisor.
\vskip 0.3cm
\hskip -0.6cm
We start with $K_V^2=1$.
\vskip 0.3cm
\hskip -0.6cm
 From Theorem $3.5$,we know that any rational surface $U$ which is the minimal resolution of a Picard number two Gorenstein log del Pezzo surface $V$ with $K_V^2=1$  comes from the Mordell-Weil {\em rank one} rational elliptic surface $Y$. Meanwhile, every $NEC$ on $U$ comes from either the $(-1)$-curve on  
$Y$ or the $(-2)$-curve which intersects the NEC  of the blow-down $Y \longrightarrow U$. The $(-1)$-curves on $Y$ are the sections of the elliptic fibration and form the Mordell-Weil group of  the elliptic fibration $Y \longrightarrow {\bf P}^1$.   
\vskip 0.3cm
\hskip -0.6cm
   From [M], [OS] and [P], we get a complete list of the Mordell-Weil {\em rank one} rational elliptic surfaces $Y$ and its Mordell-Weil group.  By contracting the zero section ${\cal O}$ in $MW(Y)$, we get a surface $U_1$ which is the minimal resolution of a Picard number two Gorenstein log del Pezzo surfaces $V$ with $K_V^2 = 1$.  For convenience, we label the surfaces which are listed in Table $1$ of  Appendix. By a surface $U_j^i$, we mean that it is the smooth blow-down of some $(-1)$-curves from the No.$i$ Mordell-Weil rank one rational elliptic  surface and $K^2=j$. Thus we get the following lemma.

\begin{lemma} Let  $V$ be the Picard number two  Gorenstein log del Pezzo surface with $K^2_V=1$. Then there exist $18$ singularity types which are listed in Table $5.1$.

\vskip 0.3cm
\begin{center}
Table $5.1$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c|c|c|}\hline
$E_7$ & $D_5+A_2$  &  $A_7$ & $A_4+A_3$  & $2A_3+A_1$ \\ \hline
$E_6+A_1$ & $D_5+2A_1$  &  $A_1+A_6$& $A_4+A_2+A_1$  & $3A_2+A_1$\\ \hline
$D_7$ &  $D_4+A_3$ & $A_5+A_2$ & $A_3+A_2+2A_1$ &\\ \hline
$D_6+A_1$ & $ D_4+3A_1$ &$A_5+2A_1$ & $A_3+4A_1$ & \\ \hline
\end{tabular} 
\end{center} 
\end{lemma}
In order to find out the list of all Picard number two {\em relatively minimal}  Gorenstein log del Pezzo surfaces, we need to exclude the surfaces on which there is a $(-1)$-curve $E$ and a connected component $R$ of $D$ ($R$ may be zero) such that $E+R$ is a linear chain and $E+R$ is a connected component of $E+D$ (cf. Table $1$ of  Appendix).
In fact we have the following lemma:

\begin{lemma} None of $U_1^i$ ($7\leq i\leq 38$) is  the minimal resolution of a Picard number two relatively minimal Gorenstein log del Pezzo surface.
\end{lemma}

{\em Proof.} We verify the above assertions by checking each case separately. To exhibit our arguments, we treat only  No.$12$ and No.$24$,  i.e., 
$U_1^{12}(2A_3+A_1)$ and $U_1^{24}(3A_2+A_1)$ respectively. The other cases can be treated  similarly. We also write out the linear chain exists in the corresponging  surface in  Table $1$ of the Appedenix.
\vskip 0.3cm
\hskip -0.6cm
(1) $U_1^{12}(2A_3+A_1)$. 
\vskip 0.3cm
\hskip -0.6cm
The Mordell-Weil group of the No.$12$ Mordell-Weil rank one rational elliptic surface $Y$ (with fibre type $2I_4,I_2,2I_1$)  is $A_1^*\oplus {\bf Z}/4{\bf Z}$ where $A_1^*$ is the rank one lattice generated by an element $e$ such that $<e,e>=1/2$.  We let $P$ be the generator of the lattice $A_1^*$ and $Q$ the generator of the torsion part of the Mordell-Weil group,i.e., ${\bf Z}/4{\bf Z}$.  A direct calculation shows that there is one element in the set $\{ P_2, P_2+Q, P_2+2Q, P_2+3Q \}$ where $P_2:=2P$, say $P_2$, passing through the $0$-th components of the singular fibres $2I_4, I_2$
and $(P_2{\cal O} )=0$. Since $U_1^{12}(2A_3+A_1)$ is the blow-down of the zero section ${\cal O}$ of $Y$, the image of $P_2$ will be the $(-1)$-curve disjoint from $D=2A_3+A_1$. Thus $U_1^{12}(2A_3+A_1)$ is not the minimal resolution of a Picard number two relatively minimal Gorenstein log del Pezzo surface.
\vskip 0.3cm
\hskip -0.6cm      
$(2)$ $U_1^{24}(3A_2+A_1)$.
\vskip 0.3cm
\hskip -0.6cm
To start with, we let $P$ be the generator of the rank $1$ lattice $<1/6>$. Firstly we need to find out all the $(-1)$-curves on $U_1^{24}(3A_2+A_1)$. Since we blow down the zero section ${\cal O}$ on the No.24 Mordell-Weil rank one rational elliptic surface with the fibre type $(3I_3, I_2, I_1)$, the section $P_m :=mP$ ($m\in {\bf Z}$) which intersects
 ${\cal O}$ will not be a  $(-1)$-curves on $U_1^{24}(3A_2+A_1)$ again. Thus we get the following claim:

\begin{claim} A $(-1)$-curve on $U_1$ is either the image of the zero component of the singular fibre or the image of the section disjoint from the ${\cal O}$. 
\end{claim}
Suppose $(P{\cal O})=0$. Then from $m^2/6=<P_m,P_m>=2\chi + 2(P_m{\cal O}) - \sum_{\nu \in R} contr_{\nu} 
(P_m) = 2 -\sum_{\nu \in R} contr_{\nu}(P_m) \leq 2$,
we get $m^2 \leq 12 $ and $|m|\leq 3$. Conversely,  a simple calculation shows that $(P_3{\cal O})=0$ and $P_3$ only passes through the $1$-st component of $I_2$. Thus $P_3+A_1$ will form a linear chain required. 
\vskip 0.3cm
 
By a similar discussion as in Lemma 4.2, we know there is a one to one correspondence between No.$i$ Mordell-Weil rank one rational elliptic surfaces and $U^i_1$, where $1\leq i \leq 6$. Thus finding the number of the isomorphism classes of the relatively minimal Dynkin type on $K_V^2=1$ is equivalent to finding the number of the isomorphism classes of the rational elliptic surfaces with the corresponding singular fibre type.

\begin{remark} \ \ $(1)$ \ \ From {\rm [Hir]} or Theorem $2.12$ , we know 
the Nos.$1$ and $4$ Mordell-Weil rank one rational elliptic surfaces are unique up to isomorphism,i.e., are determined by the singular fibre types. 

$(2)$  \ \ From {\rm [Her]} or Theorem $2.13$, we know that the number of the isomorphism classes of each of the Nos. $2$, $5$ and $6$ Mordell-Weil rank one rational elliptic surfaces is  infinity. In fact, the isomorphism classes depend on a single parameter.
 
\end{remark}

\begin{lemma} The No.$3$ Mordell-Weil rank one rational elliptic surface $S$ is uniquely determined by its singular fibre 
type $(I_4, I_2,I_2,I_2, I_2)$, up to isomorphisms.
\end{lemma}

{\em Proof.} \ \ To see the uniqueness of $S$, we need to calculate the polynomial $g_2$, $g_3$ and $\Delta$ in the homogeneous coordinates $(X,Y)$ of ${\bf P}^1$ where $g_2$, $g_3$ are the sections which determine the {\it Weierstrass model} of $S$ (cf. Theorem $2.11$). For simplicity,
we let $G_2=g_2$ and $G_3=3\sqrt{3}g_3$. Then $\Delta=G_2^3-G_3^2$.

The orders $\nu_p$ of zeros for the fibre type $(I_4, I_2,I_2,I_2, I_2)$ (cf. [Her]) have to be:
\begin{center}
Table $5.2$
\end{center}     
\begin{center}
\begin{tabular}{ c||c | c||c}\hline
$p$   & $\nu_p(G_2)$  &  $\nu_p(G_3)$ &  $\nu_p (\Delta)$ \\ \hline
$p_1$ & $0$  &  $0$& $4$   \\
$p_2$ &  $0$ & $0$ & $2$ \\ 
$p_3$ & $ 0$ &$0$ & $2$ \\ 
$p_4$ &  $0$ & $0$ & $2$ \\ 
$p_5$ &  $0$ & $0$ & $2$ \\ \hline
{\rm sum} & $0$ & $0$ & $12$ \\ \hline 
\end{tabular} 
\end{center} 

Therefore 
%\setcounter{equation}{0}
%\renewcommand{\theequation}{\alph{equation}}    %(*)
\begin{equation}  
-{\bar \Delta}^2 = \Delta =G_2^3-G_3^2    
\end{equation}

with ${\bar \Delta}$, $G_3 \in H^0({\bf P}^1, {\cal O}(6L))$ where $deg L=p_g-q+1=1$ since $S$ is a rational surface. Thus $(1)$ is equivalent to
$$
G_2^3 = G_3^2 -  {\bar \Delta}^2 = (G_3-{\bar \Delta} )(G_3 + {\bar \Delta}).
$$
It follows from Table $5.2$ that $G_2$, $G_3$ and ${\bar \Delta}$ are relatively 
prime. We let
$$
H_1^3=G_3-{\bar \Delta} \ \ {\rm and} \ \ H_2^3=G_3+ {\bar \Delta}
$$
It is easy to see that  $H_1$, $H_2 \in H^0({\bf P}^1, {\cal O}(2L))$ and they are relatively prime. Thus
$$
-{\bar \Delta} = \frac{1}{2} (H_1^3-H_2^3)=\frac{1}{2}(H_1-H_2)(\eta^2 H_2-\eta H_1)(\eta H_2-\eta^2 H_1)
$$
where $\eta = e^{\frac{2\pi i}{3}}$. By $(1)$, ${\bar \Delta}$ has one double zero at $p_1$ and four single zeros at $p_i$ where $i=2$,$3$,$4$,$5$. Let
\begin{eqnarray*}
J_1 &:=&  H_1-H_2 \\
J_2 &:=&   \eta^2 H_2-\eta H_1 \\
J_3 &:=& \eta H_2-\eta^2 H_1 
\end{eqnarray*}
 with $J_i \in H^0({\bf P}_1, {\cal O}(2L))$, $i=1$, $2$, $3$. Similarly, $J_1$, $J_2$ and $J_3$ are relatively prime.

It can be assumed that double point of ${\bar \Delta}$ is that of  $J_2$ at $p_1=0$, two of the four single zeros of ${\bar \Delta}$ are that of  $J_3$ at $p_2=\infty$ and $p_3=1$. Therefore we may write $J_2=X^2$ and $J_3=Y(X-Y)$. It follows
\begin{eqnarray*}
J_1 &=& X^2+XY-Y^2 \\
H_1 &=&   \frac{1}{1-\eta}(X^2- \eta XY +\eta Y^2) \\
H_2 &=& \frac{1}{1-\eta} (\eta X^2-XY+Y^2) \\
{\bar \Delta} &=& -\frac{1}{2}(Y^2-XY-X^2)\cdot X^2\cdot Y(X-Y)
\end{eqnarray*}
and
\begin{eqnarray*}
G_2 &=& H_1\cdot H_2 \\
G_3 &=&  H_1^3+  {\bar \Delta}  \\
 \Delta &=& -\frac{1}{4}(Y^2-XY-X^2)^2\cdot X^4\cdot Y^2(X-Y)^2
\end{eqnarray*}
This proves the lemma.
\vskip 0.3cm
Thus we get the following lemma (cf. the proof of Lemma $3.3$):

\begin{lemma} Let  $V$ be a Picard number two relatively minimal  Gorenstein log del Pezzo surface with $K^2_V=1$. Then the singularity type of $V$ and the number $m$ of the isomorphism classes of the given singularity type are listed in Table $5.3$.
\vskip 0.5cm
\begin{center}
Table $5.3$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c|c|}\hline
 Dynkin type of $V$ & $m$ & Dynkin type of $V$  & $m$\\ \hline
$D_7$ & $\infty $  &  $D_5+2A_1$ & $\infty $\\ \hline
$A_3+4A_1$ & $1$  &  $D_4+A_3$& $\infty $  \\ \hline
\end{tabular} 
\end{center} 
\end{lemma}
\vskip 0.3cm
Next we deal with $K_V^2=2$.
\vskip 0.3cm
By blowing down an NEC on $U_1$, we get a surface $U_2$ which is the minimal resolution of the Picard number two Gorenstein log del Pezzo surface $V$ with $K_V^2=2$.

\begin{theorem} Let  $V$ be the Picard number two  Gorenstein log del Pezzo surface with $K^2_V=2$. 

\hskip -0.6cm $(1)$ There exist $12$ singularity types which are listed in the following:
$E_6$, $D_4+2A_1$, $A_4+A_2$, $A_3+3A_1$, $D_6$, $A_6$, $2A_3$, $3A_2$, $D_5+A_1$, $A_5+A_1$, $A_3+A_2+A_1$, $6A_1$.

\hskip -0.6cm 
$(2)$ If $V$ is a relatively minimal surface, then the singularity type of $V$  and the number $m$ of the isomorphism classes with the given singularity type are listed in Table $5.4$.
\vskip 0.5cm
\begin{center}
Table $5.4$
\end{center}
\begin{center}
\begin{tabular}{| c|c|c|c|}\hline
 Dynkin type of $V$ & $m$ & Dynkin type of $V$  & $m$\\ \hline
$D_6$ & $\infty$  &  $D_4+2A_1$ & $1 $\\ \hline
$6A_1$ & $1$  &  $2A_3$& $\infty $  \\ \hline
\end{tabular} 
\end{center} 
\end{theorem}
{\em Proof.} \ The singularity types and the relatively minimal singularity types can be found by a direct calculation together with the definition of the relatively minimal surfaces (cf. Definition $2.1$). For the details, we refer to Table $3$ of  Appendix. 
\vskip 0.5cm
\hskip -0.6cm
Now we shall deal with the number of isomorphism classes with the given relatively minimal singularity type separately. Firstly, we prove the following lemma.



\begin{lemma} The number of the isomorphism classes of the Picard number two Gorenstein log del Pezzo surface with the singularity type $D_6$ is infinity.
\end{lemma}
{\em Proof.} \ We let $U$ be the minimal resolution of $V$. By Table $3$ of  Appendix, 
there are four possibilities for $U$: start from No.$1$ ($2$, $7$ or $8$) Mordell-Weil rank one rational elliptic surface.  Since each of these surfaces has at least one type $I_1$ fibre, we have
\begin{claim} There exists a reduced irreducible nodal member $A \in |-K_U |$.
\end{claim}
From the configuration of the negative curves in $U$ which is listed below, we know there are three $(-1)$-curves: $l_1$, $l_2$ and $l_3$.
\vskip 0.5cm
\centerline{\input ex61.tex}
\vskip 0.5cm
\begin{center}
Figure $5.1$
\end{center}
\begin{claim} There exists an automorphism on  $U$ which 
interchanges $l_1$ and $l_2$.
\end{claim}
{\em Proof of the claim.} \ \ Let $W \longrightarrow U$ be the blow up of 
$p=A \cap l_3$. It is easy to see that $W$ is a surface which is the minimal resolution of the Picard number two Gorenstein log del Pezzo surface with the singularity type $E_7$. Thus $W$ is either $U_1^7(E_7)$ or $U_1^8(E_7)$ and the proper transform of $l_i$ (i=1,2) intersects the $1$st component of $E_7$. From Table $2$ of  Appendix, we know that $l_i$ (i=1,2) comes from the Mordell-Weil section $P_1$ or $P_{-1}$ of the No.$7$ or $8$ Mordell-Weil rank one rational elliptic surface where $P_1$ is the generator of the rank one Mordell-Weil lattice $A_1^*$. In any case, we may define an automorphism $\Phi$ on No.7 (No.8) Mordell-Weil rank one rational elliptic surface which is the involution on a general fibre. Then
$\Phi (P_1)= P_{-1}$ and  $\Phi$ will induce an automorphism on $U$ which interchanges $l_1$ and $l_2$. This proves the claim.
\vskip 0.3cm
\hskip -0.6cm Let $U \longrightarrow {\bf P}^2$ be the composition of blowing down of $l_3+G_2+G_6+G_3$, $l_1+G_1+G_4$ to $p$, $q$ respectively where $G_i$ 
$(i=1, \cdots , 6)$ is the component of $D_6$ (cf. Figure $5.1$). 
\vskip 0.5cm
\centerline{\input ex62.tex}
\vskip 0.5cm
\begin{center}
Figure $5.2$
\end{center}
From the above construction and Claim $5.10$, we know that $U$ is uniquely determined by a pair $M=(S,C,p)$ where $S={\bf P}^2$, $C$ is a nodal cubic  curve on $S$ and $p$ is a point of $C$ which is neither a singular point of $C$ nor an inflex of $C$. An automorphism of $U$ will induce an isomorphism on $M=(S,C,p)$ by making use of Claim $5.10$ if necessary. In order to prove the lemma, we have only to show that there exist infinitly many isomorphism classes of such pairs. 
\vskip 0.3cm
\hskip -0.6cm
Before stating the next result, we employ the following notation.

$S$ = ${\bf P}^2$ with coordinates $X$,$Y$ and $Z$.

 $C$ = $\{[X:Y:Z] | Y^2 Z=X^2(X+Z) \}$. 


$S'$ = ${\bf P}^2$ with coordinates $X'$,$Y'$ and $Z'$. 

$C'$ = $\{[X':Y':Z'] | Y'^2 Z'=X'^2(X'+Z') \}$.
  
\begin{claim} If $\Phi : S \longrightarrow S'$ is an isomorphism such that $\Phi (C)=C'$, then either $\Phi ([X:Y:Z]) =[X':Y':Z']$ or $\Phi ([X:Y:Z])=[X':-Y':Z']$.
\end{claim}
{\em Proof of the claim.}  Clearly, $\Phi$ is linear map. We set: 
$$
\left(\begin{array}{r}
   X' \\
   Y' \\
   Z'
   \end{array}
   \right)  =
\Phi \left(\begin{array}{r}
   X \\
   Y \\
   Z 
   \end{array}
   \right) =
 \left( \begin{array}{rrr}
   a_{11} & a_{12} & a_{13} \\
   a_{21} & a_{22} &a_{23} \\
   a_{31} & a_{32} & a_{33}
   \end{array}
   \right) \cdot
 \left( \begin{array}{r}
   X \\
   Y  \\
   Z
    \end{array}
   \right) = A \cdot
 \left( \begin{array}{r}
   X \\
   Y \\
   Z
   \end{array}
   \right)
$$
Then $det A \not=0$. Since $\Phi (C)=C'$, we have
$$
\Phi ([0:0:1])=[0:0:1]   \ \ {\rm and} \ \ \Phi ([1:0:-1])=[1:0:-1].
$$
From the above equalities, we may set
$$
a_{33}=1, a_{11}=1- a_{31}, a_{13}=a_{23}=a_{21}=0.
$$
For simplicity, we denote $a_{12}=a$, $a_{22}=b$, $a_{32}=c$ and $a_{11}=d$. Thus we have
$$
X'=dX+aY, \ \ Y'=bY, \ \ Z'=(1-d)X+cY+Z.
$$
Substituting above equalities to $Y'^2Z'=X'^2(X'+Z')$, we get
%\setcounter{equation}{0}
%\renewcommand{\theequation}{\alph{equation}}    %(1)
\begin{equation}  
(bY)^2\cdot ((1-d)X+cY+Z)=(dX+aY)^2\cdot (X+aY+cY+Z).  
\end{equation}
(2) is hold for all $[X:Y:Z] \in C$. On the other hand, if $[X:Y:Z]\in C$, then
$[X:-Y:Z] \in C$, thus we also have
%\setcounter{equation}{0}
%\renewcommand{\theequation}{\alph{equation}}    %(2)
\begin{equation}  
(bY)^2\cdot ((1-d)X-cY+Z)=(dX-aY)^2\cdot (X-aY-cY+Z). 
\end{equation}
From $(2)+(3)$, we may get
\begin{eqnarray*}
b^2Y^2[(1-d)X+Z] &=&(X+Z)(d^2X^2+a^2Y^2)+2ad(a+c)Y^2X \\
&=& d^2Y^2Z+a^2Y^2(X+Z)+2ad(a+c)XY^2.
\end{eqnarray*}
So for all $[X:Y:Z]\in C-\{ [0:0:1], [1:0:-1] \}$, we must have
$$
b^2[(1-d)X+Z]=d^2Z+a^2(X+Z)+2ad(a+c)X
$$
Since we can take $Z=1$ and arbitrarily many $X$,  we get
%\setcounter{equation}{0}
%\renewcommand{\theequation}{\alph{equation}}    %(3)
\begin{equation}  
b^2=d^2+a^2  
\end{equation}

%\setcounter{equation}{0}
%\renewcommand{\theequation}{\alph{equation}}    %(4)
\begin{equation}  
b^2(1-d)=a^2+2ad(a+c)
\end{equation}
By $(2)-(3)$, we get
$$
cb^2Y^3=2adXY(X+Z)+(a+c)d^2YX^2+a^2(a+c)Y^3.
$$
Since above equality holds for all $[X:Y:Z] \in C-\{ [0:0:1], [1:0:-1] \} $, we get
$$
cb^2Y^2=2adX(X+Z)+(a+c)d^2X^2+a^2(a+c)Y^2.  
$$
We let $g(X,Y,Z)=cb^2Y^2-[2adX(X+Z)+(a+c)d^2X^2+a^2(a+c)Y^2]$. Then we know $C$ is included in the closure of the zero set of $g$. On the other hand, $C$ is an irreducible cubic. Thus we know
$g = 0$ and so
%\setcounter{equation}{0}
%\renewcommand{\theequation}{\alph{equation}}    %(5)
\begin{equation}  
2ad=0,
\end{equation}
%\setcounter{equation}{0}
%\renewcommand{\theequation}{\alph{equation}}    %(6)
\begin{equation}  
 b^2c=a^2(a+c).
\end{equation}
By $(6)$, we have $a=0$ or $d=0$. But $det A \not=0$, we see that $a$ and $d$ can't both equal to $0$. If $d=0$, then by $(4)$ and $(7)$, we get $a=0$, a contradiction. Thus $a=0$. Then by $(5)$ and $det A \not =0$ which implies $b\not=0$, we have $d=1$ and so by $(4)$, $b=\pm 1$.
This proves the claim. 
\vskip 0.3cm
\hskip -0.6cm
Let $M=(S, C, p_1 )$ and $M_1=(S, C, p_2)$ where $S$ and $C$ are defined above and $p_1=[X_1:Y_1:Z_1]$ and $p_2=[X_2:Y_2:Z_2]$. From the above claim, we see that $M_1$ and
$M_2$ are not isomorphic if $[X_1:Y_1:Z_1]\not=[X_2:\pm Y_2:Z_2]$. Moving $p \in C $, we may get infinitely many non-isomorphism classes of the pair $(S,C,p)$ and so the infinite many non-ismorphism classes of $U$. This proves the lemma.
\vskip 0.5cm
\hskip -0.6cm
Next we consider the singularity type $6A_1$.
\vskip 0.5cm
\hskip -0.6cm
From Table $3$ of the Appedenix, there are at most two possibilities for  
$U(D_6)$: Start with either $U_1^3(A_3+4A_1)$ or $U_1^{17}(D_4+3A_1)$ and blow down an NEC.  
Since the configuration of $U_2^3(6A_1)$ and $U_2^{17}(6A_1)$ are the same, we only discuss one of these, say, $U_2^3(6A_1)$. By Tables $1$ and $2$ in the Appedenix, we find that blowing up a point on $Q_1$ or $Q_2$ will give rise to the surface $U_1$ which is the minimal resolution of a Picard number two log del Pezzo surface with singularity type $A_3+4A_1$. By Lemma 5.6, we know $U_1=U_1^3(A_3+4A_1)$. On the other hand, there is a unique way to blow down an NEC on $U_1^3(A_3+4A_1)$ to $U_2^{3}(6A_1)$, modular an automorphism switching the two NECs (as in Claim $5.10$). Thus the uniqueness of the surface with the singularity type $6A_1$ is due to the uniqueness of the Picard number two log del Pezzo surface with the singularity type $A_3+4A_1$. 
\vskip 0.5cm
\hskip -0.6cm
Now we deal with the singularity type $D_4+2A_1$.
\vskip 0.5cm
\hskip -0.6cm
 We let $V(D_4+2A_1)$ be the  Picard number two Gorenstein log del Pezzo surface with the singularity type $D_4+2A_1$, $U(D_4+2A_1)$ the minimal resolution of $V(D_4+2A_1)$ and $D$ the exceptional divisor. The configuration of $D$ is in the left part of the following figure. Blowing up $p$, $q$ of $D$ will give rise to the No.$9$ extremal rational elliptic surface $Y$ with the singular fibre type $(I_1^*, I_4, I_1)$ which is unique up to isomorphisms (cf. [MP, Theorem 5.4]). On the other hand, a direct calculation shows that the Mordell-Weil group of $Y$ is ${\bf Z}/4{\bf Z}$ with the generator $Q$ and $Y \longrightarrow U(D_4+2A_1)$ is the blow-down of ${\cal O}$ and $2Q$ on $Y$.
\vskip 0.5cm 
 \centerline{\input ex53.tex}
\vskip 0.5cm
\begin{center}
Figure $5.3$
\end{center}
\vskip 0.5cm
\hskip -0.6cm 
Finally, we look at the singularity type $2A_3$. 
\vskip 0.5cm
\hskip -0.6cm
 We let $V(2A_3)$ be the  Picard number two Gorenstein log del Pezzo surface with the singularity type $2A_3$, $U(2A_3)$ the minimal resolution of $V(2A_3)$ and $D$ the exceptional divisor. The configuration of $D$ is in the left part of the following figure.
\vskip 0.5cm
  \centerline{\input ex33.tex}
\vskip 0.5cm
\begin{center}
Figure $5.4$
\end{center} 
We denote
\begin{center}
\begin{tabular}{ccc}
A  &= & {\rm  \ the \ set \ of \  the \ isomorphism \ classes \ of } $U(2A_3)$. \\
B  &= &  {\rm the \  set \ of \ the \ extremal \ rational \ elliptic \ surfaces } \\
   & & {\rm  with \ the \ singular \ fibre \ type } 
$(I_0^*,I_0^*)$.

\end{tabular}
\end{center}


 We define an surjective map $\Phi : A \longrightarrow B$ where $\Phi$ is the blow-up $p$ and $q$ of $D$. Thus the infinity of the number of the isomorphism classes of $U(2A_3)$ is due to the infinity of the  number of the isomorphism classes of the extremal rational elliptic surface with the singular fibre type $(I_0^*, I_0^*)$ (cf. [MP, Theorem $5.4$]).




We  may do the same operation as above to get a surface  which is the minimal resolution of the Picard number two Gorenstein log del Pezzo surface 
$V$ with $K_V^2\geq 3$. For details, we refer to the Table $3$ of the Appedenix.
\begin{lemma} Let  $V$ be the Picard number two  Gorenstein log del Pezzo surface with $K^2_V\geq 3$. Then the singularity type of $V$ is one of the following  listed in Table $5.5$.
\begin{center}
Table $5.5$
\end{center}
\begin{center}
\begin{tabular}{| c||c|}\hline
 $K_V^2$  & Dynkin type of $V$ \\ \hline

$3$ &   $D_5$, $A_5$, $A_4+A_1$, $A_3+2A_1$, $2A_2+A_1$    \\ \hline
$4$ &   $D_4$, $A_4$, $A_3+A_1$, $A_2+2A_1$, $4A_1$      \\ \hline
$5$ &  $A_3$, $A_2+A_1$   \\ \hline

$6$  &  $A_2$, $2A_1$   \\ \hline

$7$ &  $A_1$  \\ \hline

\end{tabular} 
\end{center} 
\end{lemma}
\begin{remark}  Let $V$ be a Picard number two Gorenstein log del Pezzo surface and $U$ the minimal resolution of $V$.

$(1)$ \ \ From a direct calulation, we find that the configurations of the negative curves with the same singularity type are the same except for the type $A_5+A_1$ which has two different configurations where one has two NECs (cf. $U_2^9(A_5+A_1)$) and the other has three NECs (cf. $U_2^{10}(A_5+A_1)$ or $U_2^{11}(A_5+A_1)$).


$(2)$ The configurations of the negative curves of $U$ with $K_U^2 \geq 3$ are listed in the Figure $6$ of  Appendix. From these configurations, it is easy to see  that  there are only two relatively minimal Dynkin types in the above table which are $4A_1$ and $D_4$.
\end{remark}

\begin{lemma} The Picard number two relatively minimal  Gorenstein log del Pezzo surface with the singularity type $4A_1$ is unique.
\end{lemma}
{\em Proof.} We let $U$ be the minimal resolution of $V(4A_1)$ and $D$ the exceptional divisor. Then the configuation of $D$ is in the left part  of the  following figure. 
\vskip 0.5cm
\centerline{\input ex41.tex}
 \vskip 0.5cm
\begin{center}
Figure $5.5$
\end{center}


Let $\sigma_1 : U \longrightarrow W$ be the contraction of $l_1$ and $l_2$ and
$\sigma_2 : W \longrightarrow {\bf P}^1 \times {\bf P}^1$ the contraction of $E_1$ and $E_2$. By changing the coordinates of ${\bf P}^1 \times {\bf P}^1$, we may assume $\sigma_2 \circ \sigma_1 (E_1)=p_1=(\infty ,\infty )$ and 
$\sigma_2 \circ \sigma_1 (E_2)=p_2=(0, 0)$. There is unique way from $({\bf P}^1\times {\bf P}^1, p_1,p_2)$ to $U$. Thus we prove the lemma.
\vskip 0.3cm 

\begin{lemma} The Picard number two relatively minimal Gorenstein log del Pezzo surface with the singularity type $D_4$ is unique.
\end{lemma} 
{\em Proof.} \ From Figure $6$ in  Appendix, we know that there is unique way from the surface $U(D_5)$ which is the minimal resolution of the Picard number two Gorenstein log del Pezzo surface with the singularity type $D_5$ to  $U(D_4)$. 
%\vskip 0.5cm
% \centerline{\input ex46.tex}
%\vskip 0.5cm

Let $U(A_1+D_6) \longrightarrow U(D_5)$ be  the blow up at the point $p$ (see Figure $6$ of  Appendix). Then  $U(A_1+D_6)$ is the minimal resolution of a Picard number one Gorenstein log del Pezzo surface with the singularity type $A_1+D_6$.
On the other hand,  $U(A_1+D_6)$ is unique by Lemma $4.6$ and there is a unique way from $U(A_1+D_6)$ to $U(D_5)$. Thus we know $U(D_5)$ is unique up to isomorphisms. This proves the lemma. 
\vskip 0.5cm
\hskip -0.6cm
Combining Lemma $5.1$, Theorem $5.7$ and Lemma $5.12$, we prove Theorem $1.5$.
\vskip 0.3cm
\hskip -0.6cm Combining Lemmas $5.6$, Theorem $5.7$, Lemmas $5.14$ and $5.15$, we prove Theorem $1.6$.


\appendix
\section*{Appendix. \ \ List of Configurations and Tables}
\vskip 0.5cm
{\bf Configurations which is used in the proof of Theorem $1.2$}
\vskip 0.3cm
\hskip -0.6cm In the following configurations, we exclude the fibres of type $I_1$ or $II$ which does not contribute to the calculation of the Mordell-Weil group of the corresponding fibration [Sh]. For simplicity, we only list the $(-1)$-curves which will be used in the proof of Theorem $1.2$. 

  \vskip 0.5cm
\centerline{\input ex0.tex}
   \vskip 0.5cm
   \centerline{\input ex2.tex}
   \vskip 0.5cm
   \centerline{\input ex6.tex}
   \vskip 0.5cm
   \centerline{\input ex8.tex}
   \vskip 0.5cm
   \begin{center}
   Figure $1$
   \end{center}
   \vskip 0.5cm
   \centerline{\input ex11.tex}
   \vskip 0.5cm
   \centerline{\input ex3.tex}
   \vskip 0.5cm
   \centerline{\input ex7.tex}
   \vskip 0.5cm
   \begin{center}
    Figure $2$
   \end{center}
   \vskip 0.5cm
   \centerline{\input ex10.tex}
   \vskip 0.5cm
   \centerline{\input ex4.tex}
   \vskip 0.5cm
   \begin{center}
    Figure $3$
   \end{center}
  \centerline{\input ex9.tex}
  \vskip 0.5cm
  \centerline{\input ex12.tex}
  \vskip 0.5cm
  \begin{center}
   Figure $4$
  \end{center}
{\bf Configurations of the Singularity types}
\vskip 0.3cm
\hskip -0.6cm $(a)$ The Picard number one Gorenstein log del Pezzo surfaces.
\vskip 0.3cm
\hskip -0.6cm
The following configurations are the figures of the negative curves on $U$ 
which is the minimal resolution of the Picard number one Gorenstein log del Pezzo surface and $2\leq K_U^2 \leq 7$.

\vskip 0.5cm 
\centerline{\input ex67.tex}
\vskip 0.5cm 
\centerline{\input ex68.tex}
\vskip 0.5cm
\centerline{\input ex69.tex}
\vskip 0.5cm 
\centerline{\input ex71.tex}
\begin{center}
Figure $5$
\end{center}
\vskip 0.3cm
$(b)$ The Picard number two Gorenstein log del Pezzo surface.
\vskip 0.3cm
\hskip -0.6cm
The following configurations are the figures of the negative curves on $U$ 
which is the minimal resolution of the Picard number two Gorenstein log del Pezzo surface and $K_U^2 \geq 3$.

\vskip 0.5cm 
\centerline{\input ex51.tex}
\vskip 0.5cm 
\centerline{\input ex52.tex}
\vskip 0.5cm
\centerline{\input ex63.tex}
\vskip 0.5cm 
\centerline{\input ex66.tex}
\begin{center}
Figure $6$
\end{center}

\vskip 0.5cm
\hskip -0.6cm {\bf  Table $1$. \ Mordell-Weil Rank one Rational Elliptic Surfaces}
\vskip 0.3cm
In the following table, $Y \longrightarrow U_1 (D)$ is the blow-down of the zero section ${\cal O}$ and $U_1(D) \longrightarrow V$ is the minimal resolution of the singularities of $V$ with $D$ the exceptional divisor; this map is nothing but the contraction of all $(-2)$-curves. The symble $\langle m \rangle$ stands for a rank one lattice ${\bf Z}P$ with the generator $P$ such that $\langle P,P \rangle =m$. We also denote $P_n:=nP$ where $n \in {\bf Z}$. The $A_1^*$ indicates the lattice generated by the element $P$ such that $<P,P>=1/2$. In the last column, $E$ is a $(-1)$-curve and $R$ is a connected component of $D$ such that $E+R$ is a linear chain on $U_1(D)$. We can't find such a linear chain for the first $6$ cases.
\begin{center}
\begin{tabular}{| c|c|c|c|c|}\hline
 $\sharp$  &  Fibre type on $Y$ &  MW(Y) &  $U_1 (D)$   &  $E+R$ on $U_1(D)$   \\  \hline  
$1$ &  $I_3^*, II, I_1$    &  $<1/4>$   &  $U^1_1(D_7)$  &   \\ \hline
$2$ &$I_3^*, 3I_1$      &  $<1/4>$   &   $U_1^2( D_7)$    &   \\ \hline
$3$ & $I_4, 4I_2$     &  $<1/4>\oplus ({\bf Z}/2{\bf Z})^2$   &  $U_1^3( A_3+4A_1)$    &     \\ \hline
$4$ & $I_1^*, III, I_2$      &  $<1/4>\oplus ({\bf Z}/2{\bf Z}) $   &  $U_1^4( D_5+2A_1)$   &     \\ \hline
$5$ &  $I_1^*, 2I_2, I_1$     &  $ <1/4>\oplus ({\bf Z}/2{\bf Z})  $   & $U_1^5( D_5+2A_1)$     &   \\ \hline
$6$ & $I_0^*, I_4, 2I_1$     &  $  <1/4>\oplus ({\bf Z}/2{\bf Z})  $   & $U_1^6( D_4+A_3)$     &   \\ \hline 
$7$ & $III^*, II, I_1$     &  $A_1^*$   &  $U_1^7(E_7)$   &  $P_2$    \\ \hline
 $8$ &  $III^*, 3I_1$    &  $A_1^*$   &  $U_1^8(E_7)$     &   $P_2$ \\ \hline
$9$ & $I_8, 4I_1$     &  $A_1^*\oplus ({\bf Z}/2{\bf Z}) $   &   $U_1^9(A_7)$   &   $P_2$  \\ \hline
$10$ & $I_8, II, 2I_1$      &  $<1/8>$   &  $U_1^{10}(A_7)$     &   $P_3+A_7$  \\ \hline
$11$ &  $I_8, 4I_1$     &  $<1/8>$   &  $U_1^{11}(A_7)$    &   $P_3+A_7$  \\ \hline
$12$ & $2I_4, I_2, 2I_1$     &  $ A_1^* \oplus ({\bf Z}/4{\bf Z})  $   &  $U_1^{12}(2A_3+A_1)$    &   $P_2$ \\ \hline
$13$ &$I_2^*, III, I_1$      &  $ A_1^* \oplus {\bf Z}/2{\bf Z}$   &  $U_1^{13}(D_6+A_1)$    &  $P_2$ \\ \hline
$14$ & $I^*_2, I_2, 2I_1$      &  $A_1^* \oplus {\bf Z}/2{\bf Z} $   & $U_1^{14}(D_6+A_1)$       & $P_2$  \\ \hline
$15$ &  $I_6, IV, 2I_1$    &  $ A_1^* \oplus {\bf Z}/3{\bf Z}$   &  $U_1^{15}(A_5+A_2)$     &  $P_2$ \\ \hline
$16$ &  $I_6, I_3, 3I_1$     &  $A_1^* \oplus {\bf Z}/3{\bf Z}$   &  $U_1^{16}(A_5+A_2)$     & $P_2$  \\ \hline
$17$ &   $I_0^*, 3I_2$     &  $ A_1^* \oplus ({\bf Z}/2{\bf Z})^2$   & $U_1^{17}(D_4+3A_1)$ & $P_2$ \\ \hline
$18$ &  $IV^*, III, I_1$    &  $<1/6>$   &  $U_1^{18}(E_6+A_1)$    & $P_3+A_1$ \\ \hline
$19$ & $IV^*, I_2, II$     &  $<1/6>$   &  $U_1^{19}(E_6+A_1)$    &  $P_3+A_1$ \\ \hline
$20$ & $IV^*, I_2, 2I_1$     &  $<1/6>$   &  $U_1^{20}(E_6+A_1)$    & $P_3+A_1$  \\ \hline
$21$ & $I_4, I_3, III, I_2$     &  $<1/12> \oplus {\bf Z}/2{\bf Z}$   &  $U_1^{21}(A_3+A_2+2A_1)$ &  $P_4+A_2$  \\ \hline
$22$ & $I_4, I_3, 2I_2, I_1$     &  $ <1/12> \oplus {\bf Z}/2{\bf Z} $   &  $U_1^{22}(A_3+A_2+2A_1)$    & $P_4+A_2$ \\ \hline
$23$ & $IV, 2I_3, I_2$     &  $ <1/6> \oplus {\bf Z}/3{\bf Z}$   & 
$U_1^{23}(3A_2+A_1)$      & $P_3+A_1$  \\ \hline
$24$ &  $3I_3, I_2, I_1$    &  $ <1/6> \oplus {\bf Z}/3{\bf Z}$   &  $U_1^{24}(3A_2+A_1)$    & $P_3+A_1$ \\ \hline
$25$ &$I_7, III, 2I_1$      &  $ <1/14>$   &  $U_1^{25}(A_6+A_1)$  &  $P_4+A_6$ \\ \hline
$26$ &  $I_7, I_2, II, I_1$    &  $<1/14>$   &  $U_1^{26}(A_6+A_1)$  & $P_4+A_6$  \\ \hline
$27$ &  $I_7, I_2, 3I_1$    &  $<1/14>$   & $U_1^{27}(A_6+A_1)$       &  $P_4+A_6$ \\ \hline
$28$ & $I_1^*, IV, I_1$     &  $<1/12>$   & $U_1^{28}(D_5+A_2)$    &   $P_4+A_2$ \\ \hline
 $29$ & $I_1^*, I_3, II$      &  $<1/12>$   &  $U_1^{29}(D_5+A_2)$      &  $P_4+A_2$    \\ \hline
$30$ & $I_1^*, I_3, 2I_1$      &  $<1/12>$   & $U_1^{30}(D_5+A_2)$   & $P_4+A_2$    \\ \hline
$31$ &  $I_6, III, I_2, I_1$     &  $<1/6> \oplus {\bf Z}/2{\bf Z}$   &  $U_1^{31}( A_5+2A_1)$  & $P_3+A_1$ \\ \hline
$32$ &  $I_6, 2I_2, 2I_1$     &  $ <1/6> \oplus {\bf Z}/2{\bf Z}$   &   $ U_1^{32}(A_5+2A_1)$    & $P_3+A_1$     \\ \hline
$33$ & $I_5, I_4, II, I_1$     &  $ <1/20>$   &  $U_1^{33}(A_4+A_3)$       & $ P_5+A_3$  \\ \hline
$34$ &  $I_5, I_4, 3I_1$     &  $<1/20>$   &  $U_1^{34}(A_4+A_3)$    &  $ P_6+A_4$ \\ \hline
$35$ &   $I_5, IV, I_2, I_1$    &  $<1/30>$   &   $U_1^{35}(A_4+A_2+A_1)$   & $ P_6+A_4$   \\ \hline
$36$ &  $I_5, I_3, III, I_1$     &  $<1/30>$   &  $U_1^{36}(A_4+A_2+A_1)$    &$ P_6+A_4$   \\ \hline
$37$ &  $I_5, I_3, I_2, II$     &  $<1/30>$   &  $U_1^{37}(A_4+A_2+A_1)$      & $ P_6+A_4$  \\ \hline
$38$ &   $I_5, I_3, I_2, 2I_1$    &  $<1/30>$   &  $U_1^{38}(A_4+A_2+A_1)$      & $ P_6+A_4$   \\ \hline 
 
\end{tabular} 
\end{center}
 
\vskip 0.6cm
\hskip -0.6cm {\bf Table $2$. \  Nice Exceptinal Curves on $U_1$}
\vskip 0.5cm
\hskip -0.6cm
In the following table, we employ the following notation and convention:
\vskip 0.3cm
\hskip -0.6cm
We let $G_i$, $H_i$, $J_i$, $L_i$ and $M_i$ be the $i$-th
component in the corresponding fibres $F_1$, $F_2$, $F_3$ $F_4$ and $F_5$
respectively. The numbering of the singular fibre is defined in 
the following diagrams.
For the three types of singular fibres, the $0$-th component in a fibre intersects the zero section. For the other types, the numbering is natural. The components of the singular fibre are numbered as in [Ye]. 
%\vskip 0.3cm
%\centerline {\input mw1.tex} 
%\vskip 0.3cm
%\begin{center}
%Figure $7$
%\end{center}
 For a lattice $L=<m>\oplus {\bf Z}/k{\bf Z}$, we let $P_1$ to be the generator of $<m>$ and $P_n=nP_1$ where $n\in {\bf Z}$. If $Q\in {\bf Z}/k{\bf Z}$,
i.e., the torsion part of $L$, then we denote $P_{n,Q}=P_n+Q$.  $(R_1 R_2)$ is the intersection number of the sections $R_1$ and $R_2$.
 \vskip 0.5cm
\begin{center}
\begin{tabular}{| c|c|c|c|c|}\hline
 $\sharp$ &  Fibre type  & Sections disjoint from & $NECs$ & sections which intersects  \\
  & on $Y$    &  ${\cal O}$ on $Y$  &    on $U_1$    &   NEC  on $U_1$  \\ \hline
$1$ &  $I_3^*, II, I_1$  &  $ P_1=(2)$, $P_{-1}=(3) $  & $P_{\pm1}$, $P_{\pm 2}$ & $(P_{\pm 1} P_{\mp 2})=1$   \\ 

$2$ &$I_3^*, 3I_1$     &   $ P_{\pm 2}=(1)$  & $G_0$ & $(P_2 P_{-2})=1$    \\ \hline

$3$ & $I_4, 4I_2$       &  $Q_1=(2,1,1,0,0)$,& $P_{\pm 2}$&  $(P_2 P_{-2})=1 $ \\
    & &  $Q_2=(2,0,0,1,1)$, &  & $(P_{\pm 2} P_{{\mp 2},Q_1})=2 $     \\
    &      & $Q_3=(0,1,1,1,1)$,& & $ (P_{\pm 2} P_{\mp 1})=1 $ \\
&      &  $P_1=(1,0,1,1,0)$, &  &$ (P_{\pm 2} P_{{\mp 1},Q_2})=1$ \\

   &   & $P_{1,Q_1}=(3,1,0,1,0)$,& & $ (P_{\pm 2} P_{\mp 1 ,Q_1})=1 $  \\
&   &  $P_{1,Q_2}=(3,0,1,0,1)$, & & $(P_{\pm 2} P_{\mp 1 ,Q_2})=1 $  \\

 & &          $P_{1,Q_3}=(1,1,0,0,1)$,& & $(P_{\pm 2}  Q_3)=1$  \\
& &          $P_{-1}=(1,0,1,0,1)$, &   &\\


& &     $P_{{-1},Q_1}=(3,1,0,0,1)$,  &  &\\ 
& &      $P_{{-1},Q_2}=(3,0,1,1,0)$, &  &\\ 


& &          $P_{-1,Q_3}=(1,1,0,1,0)$,   &   & \\
& &             $P_{\pm 2}=(2,0,0,0,0)$, &  & \\

& &          $P_{\pm 2 ,Q_1}=(0,1,1,0,0)$,&  &  \\ 
& &        $P_{\pm 2 ,Q_2}=(0,0,0,1,1)$. & & \\ \hline

\end{tabular} 
\end{center} 

\begin{center}
\begin{tabular}{| c|c|c|c|c|}\hline

$4$ & $I_1^*, III, I_2$  & $Q=(1,1,1)$, $P_1=(2,1,0)$,& $P_{\pm 2}$ & $(P_{\pm 2} P_{\mp 1})=1$      \\
                           
 $5$ &  $I_1^*, 2I_2, I_1$    &     $P_{1,Q}=(3,0,1)$, $P_{-1}=(2,0,1)$,& $G_0$ &   $(P_{\pm 2} P_{\mp 1 ,Q})=1$    \\
 & &   $P_{-1 ,Q}=(3,1,0)$,  $P_{\pm 2}=(1,0,0)$,& &
 $(P_{\pm 2} P_{\mp 2,Q})=2$ \\
& & $P_{\pm 2,Q}=(0,1,1)$. &     & $(P_2  P_{-2})=1$ \\ 
& & & & $(G_0 P_{\pm 2,Q})=1$ \\
\hline
$6$ & $I_0^*, I_4, 2I_1$ &    $Q=(1,2)$, $P_1=(2,1)$  & $P_{\pm 2}$, &$(P_{\pm 2} P_{\mp 1})=1$   \\
     &   &         $P_{1,Q}=(3,3)$, $P_{-1}=(3,1)$ & $P_{\pm 2,Q}$  &$(P_{\pm 2} P_{\mp 1,Q})=1$    \\
    &  &    $P_{-1,Q}=(2,3)$, $P_{\pm 2}=(1,0)$ & $G_0$ & $(P_2P_{-2})=1$    \\                                   &  &  $P_{\pm 2,Q}=(0,2)$ & & $(P_{\pm 2,Q} P_{\mp 1})=1$   \\
 & &                        & & $(P_{\pm 2} P_{\mp 2,Q})=2$   \\
 & &                        & & $(P_{2,Q} P_{-2,Q})=1$    \\ 
& & & & $(G_0  P_{\pm 2,Q})=1$ \\
\hline

$7$ & $III^*, II, I_1$      &  $P_{\pm 1}=(1)$, $P_{\mp 2}=(0)$ & $P_{\pm 1}$ &    \\  
$8$ &  $III^*, 3I_1$    &              & $G_0$    & \\ \hline
$9$ & $I_8, 4I_1$      &   $Q=(4)$, $P_{\pm 1}=(2)$ & $Q$, $P_{\pm 1}$  & $(P_{\pm 1} P_{\mp 1,Q})=1$ \\    
    & &  $P_{\pm 1,Q}=(6)$, $P_{\pm 2}=(0)$  & $P_{\pm 1,Q}$  & $(P_{\pm 2}Q)=1$   \\ \hline

$10$ & $I_8, II, 2I_1$  &   $P_1=(3)$, $P_2=(6)$ & $P_{\pm 1}$    &$(P_{\pm 1} P_{\mp 3})=1$ \\
  $11$ &  $I_8, 4I_1$ & $P_3=(1)$, $P_{-1}=(5)$&$P_{\pm 2}$  &$(P_2 P_{-2})=1$   \\
       & &  $P_{-2}=(2)$, $P_{-3}=(7)$ & $P_{\pm 3}$    &$(P_{\pm 3} P_{\mp 2})=1$    \\ \hline


$12$ & $2I_4, I_2, 2I_1$  & $Q_1=(1,1,1)$, $Q_2=(2,2,0)$  & $\sharp \{ NEC \}$ & \\
    &    &  $Q_3=(3,3,1)$, $P_{\pm 1}=(2,0,1)$ & $=\phi$  &        \\
      &   &  $P_{\pm 1,Q_1}=(3,1,0)$, $P_{\pm 1,Q_2}=(0,2,1)$ &   &    \\
      &  &  $P_{\pm 1,Q_3}=(1,3,0)$, $P_{\pm 2}=(0,0,0)$  &  &   \\ \hline

$13$ &$I_2^*, III, I_1$ &  $Q=(2,1)$, $P_{\pm 1}=(3,0)$&  $P_{\pm 1}$  &    $(P_{\pm 1} P_{\mp 1,Q})=1$ \\
 $14$ & $I^*_2, I_2, 2I_1$       &  $P_{\pm 1, Q}=(1,1)$, $P_{\pm 2}=(0,0)$    & $G_0$  & \\ \hline

$15$ &  $I_6, IV, 2I_1$ & $Q_1=(2,1)$, $Q_2=(4,2)$ & $P_{\pm 1}$   & $(P_{\pm 1} P_{\mp 1,Q_1})=1$ \\
  $16$ &  $I_6, I_3, 3I_1$   &        $P_{\pm 1}=(3,0)$, $P_{\pm 1,Q_1}=(5,1)$ &  & $(P_{\pm 1} P_{\mp 1,Q_2})=1$ \\
     & &             $P_{\pm 1,Q_2}=(1,2)$, $P_{\pm 2}=(0,0)$   &    & \\ \hline



$17$ &   $I_0^*, 3I_2$ & $Q_1=(1,1,1,0)$, $Q_2=(2,0,1,1)$ &   $G_0$ & $(G_0 P_{\pm1})=1$\\
    &       & $Q_3=(3,1,0,1)$, $P_{\pm 1}=(0,1,1,1)$ & &$(G_0 P_{\pm 2})=1$   \\
     &     &$P_{\pm 1, Q_1}=(1,0,0,1)$, $P_{\pm 1,Q_2}=(2,1,0,0)$ & &  \\
   &      & $P_{\pm 1,Q_3}=(3,0,1,0)$, $P_{\pm 2}=(0,0,0,0)$ & &   \\ \hline

$18$ &  $IV^*, III, I_1$ & $P_1=(1,1)$, $P_2=(2,0)$ & $P_{\pm 2}$ & $(P_{\pm 3} P_{\mp 1})=1$ \\         
  $19$ & $IV^*, I_2, II$  & $P_{\pm 3}=(0,1)$, $P_{-1}=(2,1)$ & $P_{\pm 3}$    &$(P_{2} P_{-2})=1$ \\    
  $20$ & $IV^*, I_2, 2I_1$ &   $P_{-2}=(1,0)$      & $G_0$      & $(P_{\pm 3} P_{\mp 2})=2$ \\    
     & & &   & $(P_{3} P_{-3})=2$     \\ 
& & & & $(G_0 P_{\pm 3})=1$ \\
\hline

$21$ & $I_4, I_3, III, I_2$ & $Q=(2,0,1,1)$, $P_1=(1,1,1,0)$& $P_{\pm 4}$  & $(P_{\pm 4} P_{\mp 1})=1$ \\    
  $22$ & $I_4, I_3, 2I_2, I_1$        & $P_{1,Q}=(3,1,0,1)$, $P_2=(2,2,0,0)$  &    &$(P_{\pm 4} P_{\mp 1,Q})=1$ \\    
 &     & $P_{2,Q}=(0,2,1,1)$, $P_3=(3,0,1,0)$  &  & $(P_{\pm 4} P_{\mp 2})=1$ \\    
 & & $P_{3,Q}=(1,0,0,1)$, $P_4=(0,1,0,0)$  &  & $(P_{\pm 4} P_{\mp 2,Q})=1$ \\    
 & & $P_{-1}=(1,2,0,1)$, $P_{-1,Q}=(3,2,1,0)$  &  & $(P_{\pm 4} P_{\mp 3})=2$ \\    
 &  & $P_{-2}=(2,1,0,0)$, $P_{-2,Q}=(0,1,1,1)$  &  & $(P_{\pm 4} P_{\mp 3,Q})=2$ \\    
 & & $P_{-3}=(3,0,0,1)$, $P_{-3,Q}=(1,0,1,0)$  &  &   $(P_{4} P_{-4})=2$ \\    
  &   & $P_{-4}=(0,2,0,0)$  &  &  \\ \hline

\end{tabular} 
\end{center} 



\begin{center}
\begin{tabular}{| c|c|c|c|c|}\hline

 $23$ & $IV, 2I_3, I_2$ &  $Q_1=(1,1,1,0)$, $Q_2=(2,2,2,0)$ &$P_{\pm 3}$  & $(P_{\pm 3} P_{\mp 1})=1$ \\      
 $24$ &  $3I_3, I_2, I_1$    &  $P_1=(1,2,0,1)$, $P_{1,Q_1}=(2,0,1,1)$ &     &  $(P_{\pm 3} P_{\mp 1,Q_1})=1$ \\ 
&  &  $P_{1,Q_2}=(0,1,2,1)$, $P_2=(1,0,2,0)$ &   & $(P_{\pm 3} P_{\mp 1,Q_2})=1$ \\ 
&  &  $P_{2,Q_1}=(2,1,0,0)$, $P_{2,Q_2}=(0,2,1,0)$ &    &$(P_{\pm 3} P_{\mp 2})=2$ \\ 
&  &  $P_{-1}=(1,0,2,1)$, $P_{-1,Q_1}=(2,1,0,1)$ &   & $(P_{\pm 3} P_{\mp 2,Q_1})=2$ \\ 
&  &  $P_{-1,Q_2}=(0,2,1,1)$, $P_{-2}=(1,2,0,0)$ &   &$(P_{\pm 3} P_{\mp 2,Q_2})=2$ \\ 
&  &  $P_{-2,Q_1}=(2,0,1,0)$, $P_{-2,Q_2}=(0,1,2,0)$ &     &$(P_{3} P_{-3})=2$ \\
&    &  $P_{\pm 3}=(0,0,0,1)$ &    & \\ \hline




$25$ &$I_7, III, 2I_1$ &  $P_1=(2,1)$, $P_2=(4,0)$&$P_{\pm 2}$ &$(P_{\pm 3} P_{\mp 2})=1$ \\  
  $26$ &  $I_7, I_2, II, I_1$  &  $P_3=(6,1)$, $P_4=(1,0)$& $P_{\pm 4}$ & $(P_{\pm 4} P_{\mp 1})=1$ \\  
  $27$ &  $I_7, I_2, 3I_1$  &  $P_{-1}=(5,1)$, $P_{-2}=(3,0)$&  & $(P_{\pm 4} P_{\mp 2})=1$ \\  
&  &  $P_{-3}=(1,1)$, $P_{-4}=(6,0)$& & $(P_{\pm 4} P_{\mp 3})=1$ \\
&  &   &   &  $(P_4  P_{-4})=2$ \\ \hline



$28$ & $I_1^*, IV, I_1$ &    $P_1=(2,1)$, $P_2=(1,2)$ &  $P_{\pm 3}$  &$(P_{\pm 3} P_{\mp 2})=1$ \\  
  $29$ & $I_1^*, I_3, II$    &  $P_3=(3,0)$, $P_4=(0,1)$  & $P_{\pm 4}$    &  $(P_{\pm 4} P_{\mp 2})=1$ \\ 
  $30$ & $I_1^*, I_3, 2I_1$    &  $P_{-1}=(3,2)$, $P_{-2}=(1,1)$  & $G_0$ & $(P_{\pm 4} P_{\mp 3})=2$ \\ 
   &  &  $P_{-3}=(2,0)$, $P_{-4}=(0,2)$  &   &$(P_{-4} P_{4})=2$ \\  
   &    &         &   & $(P_{-3} P_{3})=1$    \\ 
& & && $(G_0  P_{\pm 4})=1$ \\
\hline





$31$ &  $I_6, III, I_2, I_1$ & $Q=(3,1,0)$, $P_1=(2,0,1)$ & $P_{\pm 2}$ & $(P_{\pm 3} P_{\mp 1})=1$ \\  
  $32$ &  $I_6, 2I_2, 2I_1$        & $P_{1,Q}=(5,1,1)$, $P_2=(4,0,0)$& $P_{\pm 3}$      &  $(P_{2} P_{-2})=1$ \\  
     &   & $P_{2,Q}=(1,1,0)$, $P_{\pm 3}=(0,0,1)$ &  & $(P_{\pm 2} P_{\mp 2,Q})=1$ \\  
&         & $P_{-1}=(4,0,1)$, $P_{-1}+Q=(1,1,1)$ &  & $(P_{\pm 3} P_{\mp 2})=2$ \\  
&        & $P_{-2}=(2,0,0)$, $P_{-2}+Q=(5,1,0)$ &  & $(P_{\pm 3} P_{\mp 2,Q})=2$ \\  
&        &     &  & $(P_{3} P_{-3})=2$    \\ \hline


$33$ & $I_5, I_4, II, I_1$      & $P_1=(2,1)$, $P_2=(4,2)$ & $P_{\pm 4}$ &$(P_{\pm 4} P_{\mp 3})=1$ \\  
$34$ &  $I_5, I_4, 3I_1$        &  $P_3=(1,3)$, $P_4=(3,0)$ & $P_{\pm 5}$ & $(P_{\pm 5} P_{\mp 3})=1$ \\  
&   & $P_5=(0,1)$, $P_{-1}=(3,3)$ & &  $(P_{4} P_{-4})=1$ \\  
  &   & $P_{-2}=(1,2)$, $P_{-3}=(4,1)$ &  & $(P_{\pm 5} P_{\mp 4})=2$ \\  
 &    & $P_{-4}=(2,0)$, $P_{-5}=(0,3)$ &  &  $(P_{5} P_{-5})=2$ \\ \hline



$35$ &   $I_5, IV, I_2, I_1$  &  $P_1=(1,1,1)$, $P_2=(2,2,0)$ &$P_{\pm 6}$   & $(P_{\pm 6} P_{\mp 1})=1$ \\     
$36$ &  $I_5, I_3, III, I_1$   &  $P_3=(3,0,1)$, $P_4=(4,1,0)$ &   &$(P_{\pm 6} P_{\mp 2})=1$ \\    
  $37$ &  $I_5, I_3, I_2, II$ &  $P_5=(0,2,1)$, $P_6=(1,0,0)$ &  &$(P_{\pm 6} P_{\mp 3})=1$ \\  
  $38$ &   $I_5, I_3, I_2, 2I_1$ &  $P_{-1}=(4,2,1)$, $P_{-2}=(3,1,0)$ &    &  $(P_{\pm 6} P_{\mp 4})=1$ \\  
&  &  $P_{-3}=(2,0,1)$, $P_{-4}=(1,2,0)$ &  &$(P_{\pm 6} P_{\mp 5})=2$ \\  
&  &  $P_{-5}=(0,1,1)$, $P_{-6}=(4,0,0)$ &   &$(P_{6} P_{-6})=2$    \\ \hline


\end{tabular} 
\end{center} 
 
\vskip 1cm
\hskip -0.6cm {\bf Table $3$. \  The Singularity types of the Picard number two Gorenstein log del Pezzo surfaces}
$$
$$
To find out the singularity types of the Picard number two Gorenstein log del Pezzo surfaces,
we start blowing down the zero-section ${\cal O}$ of  the Mordell-Weil rank one rational elliptic surface (cf. Theorem $3.5$) and then blow down an NEC (if exists) on the resulting surface,
and continue this process. The following table indicates the all singularity types found by this method. 
\vskip 0.5cm
\hskip -0.6cm
In this table, the numbers $i$ in the first column indicates that we start from the  No.$i$ Mordell-Weil rank one rational elliptic surface (cf. Table $1$ of  Appendix). Since we pay attention to the singularity types, if there are more than one surfaces with the same Mordell-Weil group which may produce the same singularity type, then we only pick up one of those surfaces. For example, blowing down the zero section of No.1 and No.2 Mordell-Weil rank one rational elliptic surfaces will give rise to the  same singularity type $D_7$ and the same configuration of the negative curves on the surface; thus we may only choose one surface, i.e., the No.$1$ Mordell-Weil rank one rational elliptic surface. In the last column of the table, the number $m$ in `` Exceptional divisor of $U_1 \longrightarrow U_m$'' is the largest number $m$ such that $U_m$ is the minimal resolution of the singularities of the Picard number two Gorenstein log del Pezzo surface. For the notation of $P_i$, $G_j$, $H_k$ and $Q$, we refer to the Table $2$ of  Appendix. The suffix $n$ of $U_n$ singifies that $K_{U_n}^2=n$. 
\vskip 0.5cm
\hskip -0.6cm Let's give an example, say, the type $D_7$, to explain how to read this table. There are five NECs on $U_1(D_7)$: $P_{\pm 1}$, $P_{\pm 2}$ and $G_0$ (cf. Table $2$ of Appendix). Thus there are five possibilities to do operation (blowing down) on $U_1(D_7)$. But by the symmetry, we only need to blow down one of the three NECs: $P_1$, $P_2$ or $G_0$. Thus blowing down $P_1$ (resp. $P_2$, or $G_0$) of $U_1(D_7)$ will give rise to $U_2(A_6)$ (resp. $U_2(D_6)$ and $U_2(A_1+D_5)$) and the first component of ``Exceptional divisor of $U_1$ to $U_m$'' in the last column is $P_1$ (resp. $P_2$, or $G_0$). When we reach $U_2(A_6)$, we may do the similiar operation as on $U_1(D_7)$ and get $U_3(A_5)$ and $U_3(A_4+A_1)$. 
$$
$$
\begin{center}
\begin{tabular}{| c|c|c|c|c|c|c|c|c|}\hline
$No.$ & $U_1$  &  $U_2$ & $U_3$ & $U_4$ & $U_5$ & $U_6$ & $U_7$ & Exceptional divisor   \\
 & & & & & & & & of $U_1 \longrightarrow U_m$
 \\ \hline 
$1$   &  $D_7$ & $A_6$ & $A_5$ & $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_1+P_2+G_2+$  \\
 $2$ &  &    &     &         &       &   & & $G_1+G_4+G_5$ \\
 \cline{5-9}
 &   &  &  &  $A_4$& $A_2+A_1$ &  $2A_1$  & $A_1$ & $P_1+P_2+G_1$ \\
 &    &   &    &       &   & &  & $G_2+G_4+G_5$  \\ \cline{6-9}
&  &  &  &     & $A_3$ & $A_2$   & $A_1$ &$P_1+P_2+G_1$ \\
&        &       &     &         &       &   & & $G_4+G_5+G_2$ \\ \cline{7-9}
&  & &     &    &   & $2A_1$ & $A_1$ & $P_1+P_2+G_1$ \\

&  &  &  &    &       &   & & $G_4+G_2+G_5$ \\ \cline{4-9} 

   & &   & $A_4+A_1$ & $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ &  
  $P_1+G_0+P_2+$ \\
& & & & & & & & $G_1+G_4+G_5$ 
\\ \cline{5-9}

 & &   &   &$A_4$ & $A_2+A_1$& $2A_1$ & $A_1$ & $P_1+P_0+P_2+$ \\
& & & & & & & & $G_2+G_4+G_5$ \\ \cline{6-9}

& & & &  & $A_3$  & $A_2$ & $A_1$ 
& $P_1+G_2+P_{-1}+$ \\ 
&  & & & & & & & $G_7+G_6+G_5$
\\  \cline{7-9}

& & & & & &$2A_1$ &$A_1$ & $P_1+G_0+P_2+$ \\
& & & & & & & & $G_4+G_2+G_5$  \\ \cline{3-9}
   
     
&     & $D_6$ & $D_5$ & $D_4$ & $A_3$ & $2A_1$ & $A_1$ & $P_2+G_1+G_4+$\\ 
& & & & & & & & $G_5+G_6+P_1$ \\ \cline{7-9}

&    & &              & &             & $A_2$ & $A_1$ & $P_2+G_1+G_4+$  \\ 
& & & & & & & & $G_5+P_1+G_6$ \\ \cline{5-9}

&    & &              & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_2+G_1+P_1+$ \\ 
& & & & & & & & $G_2+G_4+G_5$ \\ \cline{6-9}


&    & &              &       & $A_3$ & $A_2$ & $A_1$ & $P_2+G_1+P_1+$ \\ 
& & & & & & & & $G_4+G_5+G_6$  \\ \cline{7-9}


&     & &             &       &       & $2A_1$ & $A_1$ & $P_2+G_1+P_1+$  \\ 

& & & & & & & & $G_4+G_2+G_6$   \\ \cline{3-9}


&  & $A_1+D_5$ & $D_5$ & $D_4$ & $A_3$ & $A_2$ & $A_1$ & $G_0+P_2+G_4+$ \\

& & & & & & & & $G_5+P_1+G_6$\\ \cline{7-9}       

&  & &                 & &             & $2A_1$ & $A_1$ & $G_0+P_2+G_4+$ \\

& & & & & & & & $G_5+G_6+P_1$\\ \cline{5-9}

&  & &                 & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$&  $G_0+P_2+P_1+$ \\

& & & & & & & & $G_2+G_4+G_5$\\ \cline{6-9}

&  & &                 &       & $A_3$ & $A_2$ & $A_1$ & $G_0+P_2+P_1+$ \\ 

& & & & & & & & $G_4+G_5+G_6$\\ \cline{7-9}

&  & &                 & &             & $2A_1$ & $A_1$ & $G_0+P_2+P_1+$ \\

& & & & & & & & $G_4+G_2+G_5$  \\ \cline{1-9}
\end{tabular} 
\end{center} 

\begin{center}
\begin{tabular}{| c|c|c|c|c|c|c|c|c|}\hline



&     &  & $A_5$ & $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_2+P_1+G_2+$\\ 
& & & & & & & & $G_1+G_4+G_5$ \\ \cline{5-9}

&     &  &   & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_2+P_1+G_1+$\\ 
& & & & & & & & $G_2+G_4+G_5$ \\ \cline{6-9}

&     &  &  &  & $A_3$ & $A_2$ & $A_1$ & $P_2+P_1+G_1+$\\ 
& & & & & & & & $G_4+G_5+G_2$ \\ \cline{6-9}

&     &  &  &  &  & $2A_1$ & $A_1$ & $P_2+P_1+G_1+$\\ 
& & & & & & & & $G_4+G_2+G_5$ \\ \cline{4-9}



&  &           & $A_1+A_4$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1  $ & $G_0+P_1+G_2+$ \\

& & & & & & & & $P_2+G_4+G_5$ \\ \cline{5-9}

&  & &  & $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ &  $G_0+P_1+P_{-1}+$ \\
& & & & & & & & $G_2+G_7+G_6$ 
\\ \cline{5-9}
 &  &           & & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1  $ & $G_0+P_1+P_2+$ \\
& & & & & & & & $G_2+G_4+G_5$ \\ \cline{6-9}
&  &      &  & & $A_3$ & $A_2$ & $A_1  $ & $G_0+P_1+P_2+$ \\
& & & & & & & & $G_4+G_5+G_2$ \\ \cline{6-9}

&  &    & & &  & $2A_1$ & $A_1  $ & $G_0+P_1+P_2+$ \\

& & & & & & & & $G_4+G_2+G_5$ \\ \cline{1-9}





$3$  & $A_3+4A_1$ & $6A_1$ &  & & & &   & $P_2$ 
\\ \hline

$4$ & $D_5+2A_1$ & $D_4+2A_1$ & $A_3+2A_1$ & $4A_1$ &  & &  & $P_2+G_1+G_4$
\\ \cline{3-9}


$5$   & & $A_3+3A_1$ & $A_3+2A_1$ & $4A_1$ &  & &  &  $G_0+P_2+G_4$
\\ \hline
$6$ & $D_4+A_3$ & $2A_3$ & $A_3+2A_1$ & $4A_1$ & & &  & $P_2+G_1+P_{2, Q}$ \\  \cline{3-9}
    &           & $D_4+2A_1$ & $A_3+2A_1$ & $4A_1$ &  & &  & $P_{2, Q}+P_2+G_1$ \\ \cline{3-9}
     &           & $A_3+3A_1$ & $A_3+2A_1$ & $4A_1$ &  & &  & $G_0+P_2+P_{2, Q}$\\ \hline
$7$  & $E_7$ & $E_6$ & $D_5$  & $A_4$ & $A_2+A_1$& $2A_1$ & $A_1$ & $P_1+G_1+G_7+$ \\
$8$& & & & & & & & $G_6+G_0+G_5$ \\ \cline{6-9}

  & & &                 &     &$A_3$& $2A_1$&$A_1$ & $P_1+G_1+G_7+$  \\
& & & & & & & & $G_0+G_6+G_5$ \\ \cline{7-9}

     & &&                  &&            & $A_2$&$A_1$ & $P_1+G_1+G_7+$  \\ 
& & & & & & & & $G_0+G_5+G_6$ \\ \cline{5-9}

     & &&                  &$D_4$&$A_3$& $2A_1$&$A_1$ & $P_1+G_1+G_0+$  \\

& & & & & & & & $G_5+G_4+G_7$ \\ \cline{7-9}
     & &&                  &&          & $A_2$&$A_1$ & $P_1+G_1+G_0+$  \\

& & & & & & & & $G_5+G_7+G_4$ \\ \cline{3-9}




&   &$D_6$&$D_5$ &$A_4$&$A_2+A_1$& $2A_1$&$A_1$ & $G_0+P_1+G_5+$  \\
 & & & & & & & & $G_4+G_1+G_7$ \\ \cline{6-9}

&   &&              &     &$A_3$& $2A_1$&$A_1$ & $G_0+P_1+G_5+$  \\
& & & & & & & & $G_1+G_4+G_7$ \\ \cline{7-9}

& &&                &&          & $A_2$&$A_1$ & $G_0+P_1+G_5+$  \\
& & & & & & & & $G_1+G_7+G_4$ \\ \cline{5-9}

& &&                &$D_4$    &$A_3$& $2A_1$&$A_1$ & $G_0+P_1+G_1+$  \\

& & & & & & & & $G_5+G_4+G_7$ \\ \cline{7-9}

& &&                &&             & $A_2$&$A_1$ & $G_0+P_1+G_1+$  \\
& & & & & & & & $G_5+G_7+G_4$ \\ \cline{1-9}

\end{tabular} 
\end{center} 

\begin{center}
\begin{tabular}{| c|c|c|c|c|c|c|c|c|}\hline



&    &  &$A_5$  &$A_3+A_1$  &$A_2+A_1$& $2A_1$&$A_1$ & $G_0+G_5+G_4+$  \\


& & & & & & & & $P_1+G_1+G_7$ \\ \cline{5-9}

&    &  &   &$A_4$  &$A_2+A_1$& $2A_1$&$A_1$ & $G_0+G_5+G_1+$  \\


& & & & & & & & $G_4+G_1+G_7$ \\ \cline{6-9}

& &&             &      &$A_3$& $2A_1$&$A_1$ & $G_0+G_5+G_1+$  \\

 & & & & & & & & $G_1+G_4+G_7$ \\ \cline{7-9}

& &&             &&            & $A_2$&$A_1$ & $G_0+G_5+G_1+$  \\


& & & & & & & & $G_1+G_7+G_4$ \\ \cline{1-9}



$9$ & $A_7$& $A_5+A_1$ &$A_3+2A_1$&$4A_1$ & & & & $P_1+Q+P_{1,Q}$  \\ \cline{3-9}

&          & $2A_3   $ &$A_3+2A_1$&$4A_1$ & & & & $Q+P_1+P_{1,Q}$  \\
\hline



$10$ & $A_7$ & $A_5+A_1$ & $2A_2+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_2+P_1+P_3+$  \\
$11$ & & & & & & & & $G_1+G_3+G_4$ \\ \cline{4-9}


 &&   & $A_1+A_4$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_2+P_3+P_1+$  \\
& & & & & & & & $G_1+G_3+G_4$ \\ \cline{5-9}


&& &  & $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$  & $P_2+P_3+G_1+$  \\
& & & & & & & & $G_2+G_3+G_4$ \\ \cline{5-9}

&& &  & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$  & $P_2+P_3+G_0+$  \\
& & & & & & & & $P_1+G_5+G_6$ \\ \cline{6-9}

&& &  &  & $A_3$ & $A_2$ & $A_1$  & $P_2+P_3+G_0+$  \\
& & & & & & & & $G_6+G_5+P_1$ \\ \cline{7-9}

&& &  &   &   & $2A_1$ & $A_1$  & $P_2+P_3+G_0+$  \\
& & & & & & & & $G_6+P_1+G_5$ \\ \cline{3-9}


 &  & $A_2+A_4$ & $2A_2+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$  & $P_1+P_2+P_3+$  \\

& & & & & & & & $G_1+G_3+G_4$ \\ \cline{4-9}

&&             & $A_4+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_1+P_3+P_2+$  \\ 

& & & & & & & & $G_1+G_3+G_4$ \\ \cline{5-9}

&&             &   & $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_1+P_3+G_0+$  \\ 

& & & & & & & & $P_2+G_6+G_5$ \\ \cline{5-9}

&&             &           & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_1+P_3+G_1+$  \\ 

& & & & & & & & $P_2+G_3+G_4$ \\ \cline{6-9}
&&              &          &       & $A_3$ & $2A_1$ & $A_1$ & $P_1+P_3+G_1+$  \\

 & & & & & & & & $G_3+G_2+G_4$ \\ \cline{7-9}
&&            &            &       &       & $ A_2$ & $A_1$  & $P_1+P_3+G_1+$  \\

 & & & & & & & & $G_3+G_4+G_5$ \\ \cline{3-9}


& & $A_6$ & $A_4+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$  & $P_3+P_1+P_2+$  \\
& & & & & & & & $G_2+G_3+G_4$ \\ \cline{5-9}
&&  & & $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$  & $P_3+P_1+G_1+$  \\
& & & & & & & & $G_1+G_3+G_4$ \\ \cline{5-9}

&&             &           & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$  & $P_3+P_1+G_1+$  \\
& & & & & & & & $P_2+G_3+G_4$ \\ \cline{6-9}

&&              &          &       & $A_3$ & $2A_1$ & $A_1$  & $P_3+P_1+G_1+$  \\
& & & & & & & & $G_3+P_2+G_4$ \\ \cline{7-9}


&&            &            &       &       & $ A_2$ & $A_1$  & $P_3+P_1+G_1+$  \\

& & & & & & & & $G_3+G_4+G_5$ \\ \cline{1-9}

\end{tabular} 
\end{center} 

\begin{center}
\begin{tabular}{| c|c|c|c|c|c|c|c|c|}\hline


&&        & $A_5$ &   $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$  & $P_3+G_1+P_2+$  \\
& & & & & & & & $G_2+G_3+G_4$ \\ \cline{5-9}


&&        &  & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$  & $P_3+G_1+G_2+$  \\

& & & & & & & & $P_2+G_3+G_4$ \\ \cline{6-9}

&&        & &             & $A_3$     & $2A_1$ & $A_1$  & $P_3+G_1+G_2+$  \\


 & & & & & & & & $G_3+G_2+G_4$ \\ \cline{7-9}

&&        & &             &           & $A_2$ & $A_1$ & $P_3+G_1+G_2+$  \\
 
& & & & & & & & $G_3+G_4+G_5$ \\ \cline{1-9}



$12$   &  $2A_3+A_1$ &  &   &  &  &   &  & \\ \hline
$13$   &  $D_6+A_1$ & $A_5+A_1$ & $A_3+2A_1$ & $4A_1$ &  &  &  & $P_1+G_3+G_0$ \\ \cline{3-9}
 $14$  &    & $D_4+2A_1$ & $A_3+2A_1$ & $4A_1$ &   &   &  & $G_0+P_1+G_3$  \\ \hline


$15$  &  $A_5+A_2$ & $3A_2$ &   &   &  &  &  & $P_1$ \\

$16$ & & & & & & & &  \\ \hline

$17$   &  $D_4+3A_1$ & $6A_1$  &   &  &  &  &   & $G_0$ \\ \hline


$18$   &  $E_6+A_1$ & $E_6$ & $D_5$ & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_3+P_1+G_1+$  \\
$19$ & & & & & & & & $G_4+P_2+G_2$ \\ \cline{6-9}
$20$&&&     &   & $A_3$ & $2A_1$  & $A_1$ & $P_3+P_1+G_1+$  \\
 & & & & & & & & $P_2+G_4+G_2$ \\ \cline{7-9}
&&&    & & & $A_2$ & $A_1$ &$P_3+P_1+G_1+$  \\
 & & & & & & & & $P_2+G_2+G_4$ \\ \cline{5-9}
&&&    & $D_4$ & $A_3$ & $2A_1$ & $A_1$& $P_3+P_1+P_2+$  \\
 & & & & & & & & $G_2+G_5+G_1$ \\ \cline{7-9}
&&&     & &  & $A_2$ & $A_1$& $P_3+P_1+P_2+$  \\
 & & & & & & & & $G_2+G_1+G_5$ \\ \cline{3-9}
&    & $D_5+A_1$ & $D_5$ & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_2+P_3+G_2+$  \\
& & & & & & & & $G_5+p_1+G_1$ \\ \cline{6-9}
&&&                      &                     & $A_3$ & $2A_1$ & $A_1$ & $P_2+P_3+G_2+$  \\
& & & & & & & & $P_1+G_5+G_1$
\\ \cline{7-9}
&&&    &  &       & $A_2$  &  $A_1$ & $P_2+P_3+G_2+$  \\
& & & & & & & & $P_1+G_1+G_5$ \\ \cline{5-9}




&&&  & $D_4$ & $A_3$ & $2A_1$ & $A_1$& $P_2+P_3+P_1+$  \\ 

& & & & & & & & $G_2+G_5+G_1$ \\ \cline{7-9}

&&&         & &           & $A_2$ & $A_1$  & $P_2+P_3+P_1+$  \\

& & & & & & & & $G_2+G_1+G_5$ \\ \cline{4-9}


&    &   & $A_4+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_2+G_2+G_5+$  \\

& & & & & & & & $P_3+P_1+G_1$ \\ \cline{5-9}

&    &    && $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_2+G_2+G_0+$  \\

& & & & & & & & $G_3+G_6+G_4$ \\ \cline{5-9}



&    &   &           & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $P_2+G_2+P_3+$  \\

& & & & & & & & $G_5+P_1+G_1$ \\ \cline{6-9}

&    &       &       &     & $A_3$ & $2A_1$ & $A_1$ & $P_2++G_2+P_3+$  \\
& & & & & & & & $P_1+G_5+G_1$ \\ \cline{7-9}
&    & &             &      &   & $A_2$ & $A_1$ & $P_2++G_2+P_3+$  \\
& & & & & & & & $P_1+G_1+G_5$ \\ \cline{1-9}

\end{tabular} 
\end{center} 

\begin{center}
\begin{tabular}{| c|c|c|c|c|c|c|c|c|}\hline



& & $A_5+A_1$ & $2A_2+A_1$ & $A_2+2A_1$  &$A_2+A_1$ & $2A_1$  &  $A_1$ & $G_0+G_3+P_2+$  \\
& & & & & & & & $G_2+G_6+G_4$ \\ \cline{4-9}

& & & $A_4+A_1$ & $A_2+2A_1$  &$A_2+A_1$ & $2A_1$  &  $A_1$ & $G_0+P_2+G_3+$  \\
& & & & & & & & $G_2+G_6+G_4$ \\ \cline{5-9}

& &  & & $A_3+A_1$  &$A_2+A_1$ & $2A_1$  &  $A_1$ & $G_0+P_2+G_2+$  \\
& & & & & & & & $G_3+G_6+G_4$ \\ \cline{5-9}


& &  & & $A_4$  &$A_2+A_1$ & $2A_1$  &  $A_1$ & $G_0+P_2+P_{-1}+$  \\
& & & & & & & & $G_3+P_1+G_1$ \\ \cline{6-9}

& &  & & &$A_3$ & $A_2$&$A_1$ & $G_0+P_2+P_{-1}+$  \\
& & & & & & & & $P_1+G_1+G_3$ \\ \cline{7-9}

& &  & &  && $2A_1$  &  $A_1$ & $G_0+P_2+P_{-1}+$  \\
& & & & & & & & $P_1+G_3+G_1$ \\ \cline{1-9}

$21$   &  $A_3+A_2+$ & $A_3+3A_1$ & $A_3+2A_1$ & $4A_1$ &  &  & & $P_4+H_1+P_2$ \\
$22$& $2A_1$ & & & & & & &  \\ \hline

$23$   &  $3A_2+A_1$ & $3A_2$ &  & &  &  &  & $P_3$ \\
$24$ & & & & & & & &  
\\ \hline

$25$&$A_6+A_1$ & $A_3+A_2+$ & $2A_2+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$  & $A_1$ & $P_2+P_4+G_1+$  \\
$26$ & &  $A_1$   & & & & & & $G_2+G_4+G_5$\\ \cline{3-9}

$27$ & & $A_5+A_1$ & $2A_2+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$  & $A_1$& $P_4+P_2+G_1+$  \\ 

& & & & & & & & $G_2+G_4+G_5$
\\ \cline{4-9}

&&   & $A_4+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$  & $A_1$& $P_4+G_2+P_2+$  \\ 

& & & & & & & & $G_2+G_4+G_5$\\ \cline{5-9}

&& &   & $A_3+A_1$ & $A_2+A_1$ & $2A_1$  & $A_1$& $P_4+G_1+G_2+$  \\ 
& & & & & & & & $G_3+G_4+G_5$\\ \cline{5-9}


&& &   & $A_4$ & $A_2+A_1$ & $2A_1$  & $A_1$& $P_4+G_1+P_1+$  \\ 
& & & & & & & & $P_2+P_3+G_6$\\ \cline{6-9}

&& &   &  & $A_3$ & $A_2$  & $A_1$& $P_4+G_1+P_1+$  \\ 
& & & & & & & & $P_2+G_6+P_2$\\ \cline{7-9}

&& && &  & $2A_1$  & $A_1$& $P_4+G_1+P_1+$  \\ 
& & & & & & & & $P_3+G_6+P_2$\\ \cline{1-9}



$28 $ &$D_5+A_2$ & $A_3+A_2+$& $2A_2+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$ & $G_0+P_3+G_3+$  \\ 

$29$ & & $A_1$ & & & & & & $G_5+P_1+H_1$ \\ \cline{3-9}

$30 $ & & $D_5+A_1$& $D_5$ & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$ &$P_4+H_1+P_3+$  \\ 

& & & & & & & & $G_3+P_2+G_1$
\\ \cline{6-9}
 & && & & $A_3$ & $2A_1$ & $A_1$ &$P_4+H_1+P_3+$  \\ 

& & & & & & & & $P_2+G_3+G_1$
\\ \cline{7-9}
&    &     &    &     &  & $A_2$ & $A_1$ &$P_4+H_1+P_3+$  \\ 

& & & & & & & & $P_2+G_1+G_3$ \\ \cline{4-9}


& & &  & $D_4$ & $A_3$ & $A_2$ & $A_1$ &$P_4+H_1+P_2+$  \\ 

& & & & & & & & $G_1+P_3+G_4$
\\ \cline{7-9}
& & & & &  & $2A_1$ & $A_1$ &$P_4+H_1+P_2+$  \\ 
& & & & & & & & $G_1+P_3+G_4$
\\ \cline{1-9}

\end{tabular} 
\end{center} 


\begin{center}
\begin{tabular}{| c|c|c|c|c|c|c|c|c|}\hline

&&         & $A_4+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_4+P_3+G_3$  \\ 
& & & & & & & & $H_1+P_2+G_1$ \\ \cline{5-9}

&&  &  & $A_3+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_4+P_3+P_1$  \\ 
& & & & & & & & $G_2+G_5+G_4$ \\ \cline{5-9}


&&  &  & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_4+P_3+H_1+$  \\ 
& & & & & & & & $G_3+P_2+G_1$

\\ \cline{6-9}

 &&  & &  & $A_3$ & $2A_1$ & $A_1$&$P_4+P_3+H_1+$  \\ 

& & & & & & & & $P_2+G_3+G_1$
\\ \cline{7-9}

 &&         &   &              &   & $ A_2$ & $A_1$ &$P_4+P_3+H_1+$  \\ 
& & & & & & & & $P_2+G_1+G_3$
\\ \cline{3-9}

 

& & $A_4+A_2$& $2A_2+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_3+G_3+P_4+$  \\ 
& & & & & & & & $H_1+P_2+G_1$
\\ \cline{4-9}

 && & $A_4+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_3+P_4+G_3+$  \\ 
& & & & & & & & $H_1+P_2+G_1$
\\ \cline{5-9}


&& &  & $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_4+P_3+P_1+$  \\ 
& & & & & & & & $G_2+G_5+G_4$
\\ \cline{5-9}

&&         &  & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_3+P_4+H_1+$  \\ 
& & & & & & & & $G_3+P_2+G_1$

\\ \cline{6-9}
 &&         &    &  & $A_3$ & $2A_1$ & $A_1$&$P_3+P_4+H_1+$  \\ 
& & & & & & & & $P_2+G_3+G_1$
\\ \cline{7-9}
 &&         &   & &   & $ A_2$ & $A_1$&$P_3+P_4+H_1+$  \\ 
& & & & & & & & $P_2+G_1+G_3$
\\ \cline{1-9}

$31$   &  $A_5+2A_1$ & $A_5+A_1$ & $A_3+2A_1$ & $4A_1$ &  &  &  & $P_3+P_2+P_1$\\ \cline{3-9}
  $32$  &    & $A_3+3A_1$ & $A_3+2A_1$ & $4A_1$ &   &  & &$P_2+P_3+P_1$ \\ \hline

$33$   &  $A_4+A_3$ & $A_3+A_2+$ & $2A_2+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_4+P_5+H_1+$  \\ 
$34$& & $A_1$ & & & & & & $H_2+P_3+G_1$ \\ \cline{3-9}

   &    & $A_4+A_2$ & $2A_2+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_5+P_1+P_4+$  \\

& & & & & & & & $G_2+P_3+H_3$ \\ \cline{4-9}
    &    &          & $A_4+A_1$ & $A_3+A_1$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_5+H_1+P_2+$  \\
& & & & & & & & $G_4+g_3+G_2$ \\ \cline{5-9}
  &    &            &      & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$& $A_1$&$P_5+H_1+P_4+$  \\
& & & & & & & & $P_2+G_3+G_2$ \\ \cline{5-9}
    &   &          &   & $A_4$ & $A_2+A_1$ & $2A_1$ & $A_1$&$P_5+H_1+H_2+$  \\
& & & & & & & & $P_4+P_3+G_1$\\ \cline{6-9}
    &    &         &   &       & $A_3$ & $2A_1$ & $A_1$&$P_5+H_1+H_2$  \\
 & & & & & & & & $P_3+P_4+G_1$ \\ \cline{7-9}
   &    &           &   &      &       & $ A_2$ & $A_1$ &$P_5+H_1+H_2$  \\

& & & & & & & & $P_3+G_1+G_2$ \\ \cline{1-9}


$35$   &  $A_4+A_2+$ & $A_3+A_2+A_1$ & $2A_2+A_1$ & $A_2+2A_1$ & $A_2+A_1$ & $2A_1$ 
 &$A_1$ & $P_6+G_1+G_2$ \\

$36$ & $A_1$& & & & & & & $G_3+P_4+H_1$  \\
$37$& & & & & & & &   \\
$38$ & & & & & & & &   \\ \hline

\end{tabular} 
\end{center} 
$$
$$
\section*{References}

\hskip 0.6cm [Artin1] \ M.Artin: {\em Some numerical criteria for contractibility of curves on algebraic surfaces}, Amer.J.Math {\bf 84} (1962), 485-496.

[Artin2]\ M.Artin: {\em On isolated rational singularities of surfaces}, Amer.J.Math {\bf 88} (1966), 129-136.


[ATZ]\ E.Artal Bartolo, H.Tokunaga and D.-Q.Zhang: 
{\em Miranda-Persson's Problem on Extremal Elliptic $K3$ Surfaces},
Pacific J.Math., to appear.

[Brenton] \ L.Brenton: {\em On Singular complex surfaces with negative canonical bundle, with applications to singular  compactifications of  ${C}^2$ and to $3$-dimensional rational singularities}, Math.Ann. {\bf  248} (1980), 117-124.


[D] \ M.Demazure: {\em Surfaces de del Pezzo}, in: Lecture Notes in Mathematics, Vol. {\bf  777}, Springer, 1980.

[DZ]\ I.Dolgachev and D.-Q. Zhang: {\em Coble rational surfaces}, Amer.J.Math.{\bf 123}(2001), 79-114.

[Fujiki] \ A.Fujiki, R.Kobayashi, and S.Lu: {\em  On the Fundamental group of Certain Open normal Surfaces}, Saitama Math.J. {\bf  11} (1993), 15-20.


[Fu] \ M.Furushima: {\em Singular del Pezzo surfaces and analytic compactifications of $3$-dimensional complex affine space ${\bf  C}^3$}, Nagoya Math.J.{\bf  104} (1986), 1-28.

[GPZ]\ R.V.Gurjar, C.R.Pradeep and D.-Q. Zhang: {\em  On Gorenstein Surfaces 
isomorphic to ${\bf  P}^2/G$ }, preprint, 2000.


[GZ]\ R.V.Gurjar and D.-Q. Zhang:  {\em $\pi_1$ of smooth points of a log del Pezzo surface is finite: I }, J. Math. Sci. Univ. Tokyo {\bf  1} (1994), 137-180.


[GZ1]\ R.V.Gurjar and D.-Q. Zhang:  {\em $\pi_1$ of smooth points of a log del Pezzo surface is finite: II }, J. Math. Sci. Univ. Tokyo  {\bf  2} (1995), 165-196.

[Her]\ S.Herfurtner : {\em  Elliptic surfaces with four singular fibres}, Math.Ann. {\bf  291}(1991), 319-342.

[Hir]\ U.Schmickler-Hirzebruch : {\em  Elliptische Fl${\rm \ddot{a}}$chen ${\rm \ddot{u}}$ber ${\bf  P}_1{\bf  C}$ mit drei Ausnahmefasern und die hypergeometrische Differentialgleichung}. Schriftreihe des Mathematischen Instituts der Universit${\rm \ddot{a}}$t  M${\rm \ddot{u}}$nster, 2. Ser. {\bf  33}(1985).

[Kas] \ A.Kas : {\em Weierstrass normal forms and invariants of elliptic surfaces}, Trans.A.M.S. {\bf  225}(1977), 259-266.


[K1]\ K.Kodaira: {\em On compact complex 
analytic surfaces II}, Ann.Math. {\bf  77}(1963), 563-626.

[K2] \ K.Kodaira: {\em Pluricanonical systems on algebraic surfaces of general type}, J.Math.Soc.Japan Vol.{\bf  20}(1968), 170-192.

[KM] \ S.Keel and J.McKernan: {\em  Rational curves on quasi-projective surfaces}, Memoirs of the A.M.S. {\bf  669}(1999).

[KMM] \ Y.Kawamata, K.Matsuda, K.Matsuki:  {\em Introduction to the Minimal Model Problem}, Advanced Studies in Pure Mathematics {\bf  10}(1985), Algebraic Geometry, Sendai.


[M]\ R.Miranda:  {\em  Persson's list of singular fibres for a rational elliptic surface}, Math.Z.
{\bf  190}(1990), 191-211.


[MP]\ R.Miranda and U.Persson:  {\em On extremal rational elliptic surfaces}, Math.Z.
{\bf  193}(1986), 537-558.

[MP1]\ R.Miranda and U.Persson: {\em  Configurations of $I_n$ fibers
on elliptic $K3$ surfaces}, Math.Z. {\bf  201}(1989), 339-361.

[MP2] \ R.Miranda and U.Persson: {\em Mordell-Weil Groups of extremal
elliptic $K3$ surfaces}, Problems in the theory of
surfaces and their classification (Cortona, 1988), Symposia 
Mathematica, XXXII, Academic Press, London,1991, 167-192.

[M1]\ M.Miyanishi: {\em Open Algebraic Surfaces}, CRM monograph series, Amer.Math.Soc., 2000.


[MZ1]\ M.Miyanishi and D.-Q. Zhang : {\em  Gorenstein log del Pezzo surfaces of 
rank one}, J. of Algebra, {\bf  118} (1988), 63-84.

[MZ2]\ M.Miyanishi and D.-Q. Zhang : {\em Gorenstein log del Pezzo surfaces,II}, J. of Algebra, {\bf  156} (1993), 183-193.



[OS]\ K.Oguiso and T.Shioda: {\em The Mordell-Weil Lattice of a Rational Elliptic Surface}, Commentarii Mathematici, Universitatis Sancti Pauli, Vol.{\bf  40}(1991), 83-99.


[P]\ U.Persson : {\em Configurations of Kordaira fibers on rational elliptic surfaces}, Math.Z.{\bf  205} (1990), 1-47. 


[Reid]\ M.Reid : {\em Chapters on algebraic surfaces}, in "Complex Algebraic Geometry", ed. J. Koll\'{a}r, IAS/Park City Mathematics Series {\bf  3} (1997), 5-159. 

[S] \ T.Shioda: {\em Elliptic modular surfaces},J.Math.Soc.Japan
{\bf  24}(1972),20-59.

[Sh] \ T.Shioda: {\em  On the Mordell-Weil Lattices}, Commentarii
Mathematici, Universitatis Sancti Pauli, Vol.{\bf  39}(1990),
211-240.

[SZ]\ I.Shimada and D.-Q.Zhang : {\em Classification of extremal
elliptic $K3$ surfaces and fundamental groups of open $K3$ surfaces}, Nagoya Math.J.{\bf 161}(2001), 23-54.

[T1]\ S.Takayama : {\em Simple connectedness of weak Fano varities}, 
J.Alg.Geom., Vol.{\bf 9} (2000), 403-407.


[Ye]\ Q.Ye: {\em On extremal elliptic $K3$  
surfaces}, J.Korean Math.Soc. {\bf  36}(1999), 1091-1113.

[Z1]\ D.-Q. Zhang : {\em On Iitaka surfaces}, Osaka J. Math. 
{\bf  24}(1987), 417-460.

[Z2]\ D.-Q. Zhang : {\em Logarithmic del Pezzo surfaces of rank one with contractible boundaries}, Osaka J. Math.  {\bf  25}(1988), no.2, 461-497.

[Z3]\ D.-Q. Zhang : {\em  Logarithmic del Pezzo  surfaces with rational double and triple singular points}, T${\hat o}$hoku Math. J. $(2)$ {\bf  41}(1989), no.3, 399-452.

[Z4] \ D.-Q. Zhang : {\em Algebraic surfaces with nef and big anti-canonical divisor    }, Math.Proc.Cambridge Philos.Soc. 
{\bf  117}(1995),no.1,  161-163.

[Z5]\ D.-Q. Zhang : {\em The fundamental group of the smooth part of a log Fano varity}, Osaka J. Math. 
{\bf 32}(1995), no.3, 637-644.

[Z6]\ D.-Q. Zhang : {\em  Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts}, Trans.Amer.Math.Soc. 
{\bf 348}(1996), no.10, 4175-4184.
$$
$$




{\rm Departmant of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543}

{\it e-mail}: mathyeq@hotmail.com



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ex11.tex
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ex12.tex
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ex3.tex
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ex33.tex
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ex4.tex
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ex52.tex
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ex53.tex
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ex6.tex
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ex66.tex
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ex67.tex
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ex68.tex
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ex69.tex
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ex7.tex
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ex71.tex
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ex8.tex
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ex9.tex
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