Resolution of singularities for a family of third-order differential systems with small meromorphic solution spaces
Yusuke Sasano
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\title{Resolution of singularities 
for a family of third-order differential systems 
with small meromorphic solution spaces }


\author{Yusuke Sasano}



\address{Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 657-8501, Japan}
\email{sasano@math.kobe-u.ac.jp}
%\keywords{Painlev\'e equations}
%\date{Febryary, 1, 2004}
\maketitle


\section{abstract}
\begin{quote}
Regarding the resolution of singularities for the differential equations of Painlev\'e type, there are important differences between the second-order Painlev\'e equations and those of higher order. Unlike the second-order case, in higher order cases there may exist some meromorphic solution spaces with codimension 2. In this paper, we will give an explicit global resolution of singularities for a 3-parameter family of third-order differential systems with meromorphic solution spaces of codimension 2.
\end{quote}



\section{Introduction} 


In 1979, K. Okamoto constructed the spaces of initial conditions of Painlev\'e equations, which can be considered as the parametrized spaces of all solutions, including the meromorphic solutions [1]. They are constructed by means of successive blowing-up procedures at singular points. For Painlev\'e equations, the dimension of the space of meromorphic solutions through any singular point is always codimension one. But, in the case of higher order Painlev\'e equations, the space of meromorphic solutions through a singular point may be of codimension greater than or equal to 2 [5]. In this paper, we will give an explicit resolution of singularities for a 3-parameter family of third-order differential systems with meromorphic solution spaces of codimension 2. For second-order Painlev\'e equations, we can obtain the entire space of initial conditions by adding subvarieties of codimension 1 (equivalently, of dimension 1) to the space of initial conditions of holomorphic solutions [3]. But in the case of third-order differential equations, we need to add codimension 2 subvarieties to the space in addition to codimension 1 subvarieties. In order to resolve singularities, we need to both blow up and blow down. Moreover, to obtain a smooth variety by blowing-down, we need to resolve for a pair of singularities.



\section{ Statement of main results} 


Let us consider a 3-parameter family of third-order differential system:



\begin{equation}
  \left\{
  \begin{aligned}
   \frac{dx}{dt} &=x(t-x-2z)+\alpha_1\\
   \frac{dy}{dt} &=y(-t+y+2z)+\alpha_2\\
   \frac{dz}{dt} &=z(t-2y-z)+\alpha_3.
   \end{aligned}
  \right. 
\end{equation}
Here, $x,y$ and $z$ denote unknown complex variables and $\alpha_1,\alpha_2$ and $\alpha_3$ are constant parameters. The last two equations of this system $(1)$ are equivalent to the fourth Painlev\'e equations with unknown variables $y$ and $z$. For each solution $z(t)$ of the Painlev\'e IV equations, the first equations of this system $(1)$ give Riccati equations with unknown variable $x$.




From the viewpoint of symmetry, it is interesting to point out that our differential system $(1)$ is invariant under the following transformations:


        $$ s_1: (x,y,z,t;\alpha_1,\alpha_2,\alpha_3) \rightarrow (x,y-\frac{\alpha_3}{z},z,t;\alpha_1,\alpha_2+\alpha_3,-\alpha_3),$$

        $$ s_2: (x,y,z,t;\alpha_1,\alpha_2,\alpha_3) \rightarrow (x-\frac{\alpha_2}{y},y,z+\frac{\alpha_2}{y},t;\alpha_1+\alpha_2,-\alpha_2,\alpha_3+\alpha_2).$$

In order to characterize this differential system $(1)$, we can prove the following two propositions by direct calculation.

{\bf Notation.}

$\bullet$ $W=<s_1,s_2>$,

$\bullet$ ${\cal V}:=\{(dx/dt=f_1(x,y,z,t;a_i), \ dy/dt=f_2(x,y,z,t;b_i), \ dz/dt=f_3(x,y,z,t,c_i))\},$
\noindent
where $f_i(X,Y,Z,t;e_i)=e_1X^2+e_2Y^2+e_3Z^2+e_4XY+e_5XZ+e_6YZ+e_7tX+e_8tY+e_9tZ+e_{10}$ with $e_i \in {\Bbb C} \ (1 \leq i \leq 10)$,

$\bullet$ ${\cal H} \cong {\Bbb P}^2( \subset {\Bbb P}^3)$: boundary divisor of ${\Bbb P}^3$.


\begin{Proposition}
The differential systems in $\cal V$ with symmetry under $W$ are written as follows$:$

\begin{equation}
  \left\{
  \begin{aligned}
   \frac{dx}{dt} &=a_1x^2+(a_1-c_3)z^2+a_5xz+a_6yz+a_7tx+(a_7+b_8)tz\\
                 & \hspace{9cm} +\alpha_2(b_2-a_6)+\alpha_3a_6+a_{10}\\
   \frac{dy}{dt} &=b_2y^2+(a_5-2a_1)xy+(2a_1-a_5-2c_3)yz+b_8ty+\alpha_2(a_5-2a_1)+\alpha_2c_3\\
   \frac{dz}{dt} &=c_3z^2+(2a_1-a_5)xz-2b_2yz-b_8tz+(\alpha_2-\alpha_3)b_2.
   \end{aligned}
  \right. 
\end{equation}
\end{Proposition}

\begin{Proposition}
The last two equations of this system $(2)$ are equivalent to the fourth Painlev\'e systems with unknown variables $y$ and $z$ if and only if

$$a_5=2a_1.$$
\end{Proposition}


\begin{Remark}
{\rm
After we recall the notion of local index in $\S$ 3, we will see that the condition $a_5=2a_1$ is equivalent to the following condition $(A)$:

$(A)$ \ The differential system $(2)$ has local index $(-a_1,-a_1,-a_1)$ at the accessible singular point $P_3=\{[z_0:z_1:z_2:z_3]=[0:1:0:0]\} \in {\cal H} \subset {\Bbb P}^3$. Here $[z_0:z_1:z_2:z_3]$ is the homogeneous coordinate of ${\Bbb P}^3$.
\rm}
\end{Remark}


\begin{Remark}
{\rm
In 1998, Noumi and Yamada [2] proposed the system of type $A_4^{(1)}$, which can be considered as a 4-parameter family of fourth-order coupled Painlev\'e IV system in dimension 4, and which is given as follows:

\begin{equation*}
  \left\{
  \begin{aligned}
   \frac{dx}{dt} &=x^2+2xy+2xw-tx+\beta_1\\
   \frac{dy}{dt} &=-y^2-2xy-2yw+ty+\beta_2\\
   \frac{dz}{dt} &=z^2+2zw+2xy-tz+\beta_1+\beta_3\\
   \frac{dw}{dt} &=-w^2-2zw+tw+\beta_4.
   \end{aligned}
  \right. 
\end{equation*}
This system reduces to the system $(1)$ by setting $x=0$ and $\beta_1=0$.
\rm}
\end{Remark}

In order to consider the phase space for the system $(1)$, let us take the compactification ${\Bbb P}^3$ of ${\Bbb C}^3$ with the natural embedding $(x,y,z)=(z_1/z_0,z_2/z_0,z_3/z_0)$. Moreover we denote the hypersuface of ${\Bbb P}^3$ by $ {\cal H}=\{z_0=0\} \simeq {\Bbb P}^2 \subset {\Bbb P}^3$. Fixing parameter $\alpha_i$ and setting $B={\rm Spec \rm} {\Bbb C} [t] \cong {\Bbb C}$, consider the product ${\Bbb P}^3 \times B \cong {\Bbb P}^3 \times \Bbb C$ and extend the regular vector field on ${\Bbb C}^3 \times B$ to a rational vector field $\tilde v$ on ${\Bbb P}^3 \times B$. By direct calculation, this rational vector field $\tilde v$ has seven accessible singular points on the boundary divisor ${\cal H} \times \{t\} \subset {\Bbb P}^3 \times \{t\}$ for each $t \in B$ (See Figure 1).



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\begin{Theorem}



After a series of explicit blowing-ups and blowing-downs of ${\Bbb P}^3 \times B$, we obtain a smooth projective family of $3$-fold $\pi : {\mathcal X} \lra B$ and a birational morphism $\varphi : {\mathcal X} {\cdots \lra} {\Bbb P}^3 \times B$ which make the following diagram commutative$:$

$$
\begin{array}{ccc}
{\mathcal X} & \stackrel {\varphi} {\cdots \ra} & {\Bbb P}^3 \times B\\ 
\pi \downarrow \quad & & \downarrow\\ B & =& B,\\
\end{array}
$$
and satisfy the following conditions$:$

\begin{enumerate}


\item The phase space $\cal X$ over $B={\Bbb C}$ for the vector field $v$ in $(1)$ is obtained by gluing eight copies of ${\Bbb C}^3 \times {\Bbb C}:$



\begin{center}
${U_0} \times {\Bbb C}={{\Bbb C}^3} \times {\Bbb C} \ni {(x,y,z,t)},$

${U_j} \times {\Bbb C}={{\Bbb C}^3} \times {\Bbb C} \ni {(x_j,y_j,z_j,t)}$ $(j=1,2,....,7),$
\end{center}


\vspace{0.2cm}


\noindent
via the following rational transformations$:$

\vspace{0.2cm}



$1)$ $x_1=\frac{1}{x+z}, \ y_1=(yz+\alpha_2)z, \ z_1=\frac{1}{z},$

$2)$ $x_2=x+z,\ y_2=(yz+\alpha_2)z,\ z_2=\frac{1}{z}$,

$3)$ $x_3=\frac{1}{x}, \ y_3=y, \ z_3=z,$

$4)$ $x_4=(xz-\alpha_1)z, \ y_4=((y+z-t)z+1-\alpha_2-\alpha_3)z, \ z_4=\frac{1}{z},$

$5)$ $x_5=\frac{1}{x}, \ y_5=\frac{1}{y}, \ z_5=(yz-\alpha_3)y,$

$6)$ $x_6=x, \ y_6=\frac{1}{y}, \ z_6=(yz-\alpha_3)y,$

$7)$ $x_7=\frac{1}{z(xz-\alpha_1)},\ y_7=z((y+z-t)z+1-\alpha_2-\alpha_3), \ z_7=\frac{1}{z}$.

\item Each fiber $({\cX} \setminus {\cD})_t$ of the morphism $\pi : {\cX} 
\setminus {\cD} \lra B$ has a stratification

$$
({\cX} \setminus {\cD})_t= {\Bbb C}^3 \sqcup {\Bbb C}^2 \sqcup 
{\Bbb C}^2 \sqcup 
{\Bbb C}^2 \sqcup {\Bbb C}^2 \sqcup 
{\Bbb C}^1 \sqcup {\Bbb C}^1 \sqcup {\Bbb C}^1,
$$
where ${\Bbb C}^3$ is the original affine open subset of ${\Bbb P}^3 \setminus {\cal H}$. The three-dimensional space is a parameter space of the holomorphic solutions, and each two-dimensional or one-dimensional space is a parameter space of a two-parameter or a one-parameter family of meromorphic solutions, respectively.


\end{enumerate}
\end{Theorem}



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\end{picture}%
\label{fig:chazy17}
\caption{}
\end{figure}




\begin{Remark}
{\rm
The surface defined by $x=0$ is the minimal compactification of the space of initial conditions of Painlev\'e IV equation (see Figure 2). Actually, the ordinary differential system $(1)$ can be reduced to Painlev\'e IV equation if one sets $x \equiv 0$ and $\alpha_1=0$. However, the total differential system (1) is not a product of the fourth Painlev\'e equation and a Riccati equation.

\rm}
\end{Remark}



\begin{Remark}
{\rm
It is still an open question whether we can resolve the accessible singularities for any differential system in $(2)$ with the condition $a_5=2a_1$.
\rm}
\end{Remark}







\section{ Accessible singularity}

\vspace{0.2cm}



Let us review the notion of accessible singularity [4]. Let $B$ be a connected open domain in $\Bbb C$ and $\pi : {\mathcal W} \lra B$ a smooth proper holomorphic map. We assume that ${\mathcal H} \subset {\mathcal W}$ is a normal crossing divisor which is flat over $B$. Let us consider a rational vector field $\tilde v$ on $\mathcal W$ satisfying the condition

$$
\tilde v \in H^0({\cal W},\Theta_{\cal W}(-\log{\cH})({\cH})).
$$

\noindent
Fixing $t_0 \in B$ and $P \in {\mathcal W}_{t_0}$, we can take a local coordinate system $(x_1,x_2,...,x_n)$ of ${\cal W}_{t_0}$ centered at $P$ such that ${\cH}_{smooth}$ can be defined by the local equation $x_1=0$.
Since $\tilde v \in H^0({\cal W},\Theta_{\cal W}(-\log{\cH})({\cH}))$, we can write down the vector field $\tilde v$ near $P=(0,0,...,0,t_0)$ as follows:


\begin{equation*}
\boxed{%
\tilde v= \frac{\partial}{\partial t}+a_1 
\frac{\partial}{\partial x_1}+\frac{a_2}{x_1} 
\frac{\partial}{\partial x_2}+.....+\frac{a_n}{x_1} 
\frac{\partial}{\partial x_n}
}%
\end{equation*}

\noindent
This vector field defines the following system of differential equations

\begin{equation}
  \left\{
  \begin{aligned}
   \frac{dx_1}{dt} &=a_1(x_1,x_2,....,x_n,t)\\
   \frac{dx_2}{dt} &=\frac{a_2(x_1,x_2,....,x_n,t)}{x_1}\\
   .\\
   .\\
   .\\
   \frac{dx_n}{dt} &=\frac{a_n(x_1,x_2,....,x_n,t)}{x_1}.
   \end{aligned}
  \right. 
\end{equation}

\noindent
Here $a_i(x_1,...,x_n,t)$ $(i=1,2,...,n)$ are holomorphic functions near $P=(0,0,..,0,t_0).$




\begin{Definition}


With the above notation, assume that the rational vector field $\tilde v$ on $\cal W$ satisfies the condition
$$
\tilde v \in H^0({\cal W},\Theta_{\cal W}(-\log{\cH})({\cH})).
$$
We say that $\tilde v$ has an accessible singular point at $P=(0,0,...,0,t_0)$ if

$$
x_1=0 \ {\rm and \rm} \ a_i(0,0,....,0,t_0)=0 \ {\rm for \rm} \ {\rm every \rm} \ i, \ 2 \leq i \leq n.
$$

\end{Definition}


If $P \in {\cH}_{smooth}$ is not an accessible singular point, all solutions of the ordinary differential equation passing through $P$ are vertical solutions, that is, the solutions are contained in the fiber ${\cal W}_{t_0}$ over $t=t_0$. If $P \in {\cH}_{smooth}$ is an accessible singular point, there may be a solution of $(3)$ which passes through $P$ and goes into the interior ${\cal W}-{\cH}$ of ${\cal W}$.


Let us recall the notion of local index [4]. When we construct the phase spaces of the higher order Painlev\'e equations, an object that we call the local index is the key for determining when we need to make a blowing-up of an accessible singularity or a blowing-down to a minimal phase space. In the case of equations of higher order with favorable properties, for example the systems of type ${A_4}^{(1)}$ [2], the local index at the accessible singular point corresponds to the set of orders that appears in the free parameters of formal solutions passing through that point [5].


\vspace{0.2cm}


\begin{Definition}
Let $v$ be an algebraic vector field which is given by $(3)$ and $(X,Y,Z)$ be a boundary coordinate system in a neighborhood of an accessible singular point $P=(0,0,0,t)$. Assume that the system is written as

\begin{equation*}
  \left\{
\begin{aligned}
   \frac {dX}{dt} &=a+f_1(X,Y,Z,t)\\
   \frac {dY}{dt} &=\frac{bY+f_2(X,Y,Z,t)}{X}\\
   \frac {dZ}{dt} &=\frac{cZ+f_3(X,Y,Z,t)}{X}
\end{aligned}
  \right. 
\end{equation*}
near the accessible singular point $P$, where $a,b$ and $c$ are nonzero constants. We say that the vector field $v$ has local index $(a,b,c)$ at $P$ if $f_1(X,Y,Z,t)$ is a polynomial which vanishes at $P=(0,0,0,t)$ and $f_i(X,Y,Z,t)$ $(i=2,3)$ are polynomials of order $2$ in $X,Y,Z$. Here $f_i \in {\Bbb C}[X,Y,Z,t]$ $(i=1,2,3).$

\end{Definition}



\vspace{0.2cm}
\begin{Remark}
{\rm
We are interested in the case with local index $(1,\frac{b}{a},\frac{c}{a}) \in {\Bbb Z}^3$. If each component of $(1,\frac{b}{a},\frac{c}{a})$ has the same sign, we may resolve the accessible singularity by blowing-up finitely many times. But when different signs appear, we may need to both blow up and blow down.
\rm}
\end{Remark}


In our case, there exist 7 accessible singular points of this system on the boundary divisor of ${\Bbb P}^3$.  They are listed as follows.


\vspace{0.5cm}

\begin{center}
\begin{tabular}{|c||c|c|c|c|} \hline 
Singular point & $P_1$ & $P_2$ & $P_3$ & $P_4$ \\ \hline 
$[z_0:z_1:z_2:z_3]$& $[0: 0: 0: 1]$& $[0: -1: 0: 1] $& $[0:1:0:0]$ & $[0:0: -1:1]$ \\ \hline 
Type (dim. of sol.)    &  $\circ$ (dim. 1) & $\bullet$ (dim. 2)  & $\bullet$ (dim. 2)  &
$\bullet$ (dim. 2)  \\ \hline
Type of local index & (-1,+3,+1) & (+1,+3,+1) & (+1,+1,+1) & (-3,-3,-1)   \\ \hline
\end{tabular}

\begin{tabular}{|c||c|c|c|} \hline 
Singular point  & $P_5$  & $P_6$ &  $P_7$ \\ \hline 
$[z_0:z_1:z_2:z_3]$   &$[0:-1: 1:0]$ & $[0:0:1:0]$ & $[0:-3:-1:1]$ \\  \hline 
Type (dim. of sol.)    & $\circ$ (dim. 1)    &  $\bullet$ (dim. 2)     & $\star$ (dim. 1)  \\ \hline
Type of local index & (+1,-1,-3) & (-1,-1,-3) & (+3,-3,-1)      \\ \hline
\end{tabular}
\end{center}

\vspace{0.5cm}

\noindent
Here, we remark that there are 3 types of accessible singularities.  
We denote the mark by $\bullet$ an accessible singular point into which two dimensional meromorphic solutions flow, and the marks by $\circ$ and $\star$ accessible singular points into which one dimensional meromorphic solutions flow.




\section{ Resolution of accessible singularities}

Let $P$ be an accessible singular point in the boundary divisor $\cal H$. Rewriting the systems in a local coordinate system, the right hand side of each differential equation has poles along $\cal H$. If we resolve the accessible singular point $P$ and the right hand side of each differential equation becomes holomorphic in new coordinate system, then we can use Cauchy's existence and uniqueness theorem of solutions.

{\bf Notation}

$(u,v,w)=(\frac{x}{z},\frac{y}{z},\frac{1}{z}), \ (p,q,r)=(\frac{1}{x},\frac{y}{x},\frac{z}{x}), \ (l,m,n)=(\frac{x}{y},\frac{1}{y},\frac{z}{y}), \ (\tilde u,v,w)=(u+1,v,w).$


{\bf 4.1. Resolution of accessible singular points $P_1,P_2$ and $P_3$ }


Let us start by summarizing the steps which are needed to resolve the accessible singular points of $\tilde v$ at $ P_1:(u,v,w)=(0,0,0),  P_2:(\tilde u,v,w)=(0,0,0) \ and \ P_3:(p,q,r)=(0,0,0)$.



\begin{enumerate}
\item{\bf Step 1}: We will blow up at two points $P_2, P_3$ (see Figure 3).
\item{\bf Step 2}: We will blow up along the curve $C_1$ (see Figure 3).
\item{\bf Step 3}: We will blow down the surface $F$ (see Figure 3).
\item{\bf Step 4}: We will blow up along the curve $C_4$ (see Figure 3).
\item{\bf Step 5}: We will blow up along the curve $C_5$ (see Figure 3).
\end{enumerate}



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\end{picture}%
\label{fig:third1}
\caption{}
\end{figure}




In a neighborhood of $P_1$, the system is written as

\begin{equation*}
  \left\{
  \begin{aligned}
   \frac {du}{dt} &=\frac{-u-u^2+2uv}{w}+\alpha_1w-\alpha_3uw\\
   \frac {dv}{dt} &=-2tv+\frac{3v+3v^2}{w}+\alpha_2w-\alpha_3vw\\
   \frac {dw}{dt} &=1+2v-tw-\alpha_3w^2.
   \end{aligned}
  \right. 
\end{equation*}



In a neighborhood of $P_2$, the system is written as

\begin{equation*}
  \left\{
  \begin{aligned}
   \frac {d{\tilde u}}{dt} &=\frac{\tilde u-{\tilde u}^2-v(2-2\tilde u)}{w}+\alpha_3w+
   \alpha_1w-\alpha_3{\tilde u}w\\
   \frac {dv}{dt} &=-2tv+\frac{3v+3{v}^2}{w}+\alpha_2w-\alpha_3vw\\
   \frac {dw}{dt} &=-\alpha_3w^2-tw+2v+1.\\ 
   \end{aligned}
  \right. 
\end{equation*}



In a neighborhood of $P_3$, the system is written as

\begin{equation*}
  \left\{
  \begin{aligned}
   \frac {dp}{dt} &=-\alpha_1p^2-tp+2r+1\\
   \frac {dq}{dt} &=\frac{q+q(-2tp+q+4r)}{p}+\alpha_2p-\alpha_1pq\\
   \frac {dr}{dt} &=\frac{r+r(r-2q)}{p}-\alpha_1pr+\alpha_3p.\\
   \end{aligned}
  \right. 
\end{equation*}













{\bf 4.1.1.} We blow up at two points $P_2, P_3$.

We blow up at the point $P_2$


$$
U_1=\frac{\tilde u}{w} \;, \;\;\; V_1=\frac{v}{w} \;, \;\;\; W_1=w.
$$


We blow up at the point $P_3$ 

$$
x_3=p \;, \;\;\; y_3=\frac{q}{p} \;, \;\;\; z_3=\frac{r}{p}.
$$
We have resolved the accessible singular point $P_3$.


\vspace{0.2cm}



{\bf 4.1.2.} We blow up along the curve $C_1=\{(u,v,w)| v=w=0\} \subset {\Bbb P}^1$




$$
u_1=u \;, \;\;\; v_1=\frac{v}{w} \;, \;\;\; w_1=w.
$$
In a neighborhood of $\{(u_1,v_1,w_1)=(0,0,0)\}$, the system is written as

\begin{equation*}
  \left\{
  \begin{aligned}
   \frac {du_1}{dt} &=2u_1v_1+\frac{-u_1-(u_1)^2}{w_1}+\alpha_1w_1-\alpha_3
   u_1w_1\\
   \frac {dv_1}{dt} &=-tv_1+(v_1)^2+\frac{2v_1}{w_1}+\alpha_2\\
   \frac {dw_1}{dt} &=1-tw_1+2v_1w_1-\alpha_3(w_1)^2.
   \end{aligned}
  \right. 
\end{equation*}




{\bf 4.1.3.} We blow down the surface $F=\{(u_1,v_1,w_1)|w_1=0\} \cong {\Bbb P}^1 \times {\Bbb P}^1$


$$
u_2=\frac{w_1}{u_1+1} \;, \;\;\; v_2=v_1 \;, \;\;\; w_2=w_1.
$$
In a neighborhood of $\{(u_2,v_2,w_2)=(0,0,0)\}$, the system is written as


\begin{equation}
  \left\{
  \begin{aligned}
   \frac {du_2}{dt} &=1-tu_2+\frac{2(u_2)^2v_2}{w_2}-\alpha_1(u_2)^2
   -\alpha_3(u_2)^2\\
   \frac {dv_2}{dt} &=-tv_2+(v_2)^2+\frac{2v_2}{w_2}+\alpha_2\\
   \frac {dw_2}{dt} &=1-tw_2+2v_2w_2-\alpha_3(w_2)^2.
   \end{aligned}
  \right. 
\end{equation}




The resolution process from Step 2 to Step 3 is well-known as flop. In order to resolve the accessible singular point $P_1$ and obtain a holomorphic coordinate system, we need to blow down the surface $F \cong {\Bbb P}^1 \times {\Bbb P}^1$ along the ${\Bbb P}^1$-fiber. After we blow down the surface $F$, the system $(4)$ has the local index $(0,+2,+1)$ at the point $\{(u_2,v_2,w_2)=(0,0,0)\}$.




{\bf 4.1.4.} We blow up along the curve $C_4=\{(u_2,v_2,w_2)| v_2=w_2=0\} \subset {\Bbb P}^1$


$$
u_3=u_2 \;, \;\;\; v_3=\frac{v_2}{w_2} \;, \;\;\; w_3=w_2.
$$

{\bf 4.1.5.} We blow up along the curve $C_5=\{(u_3,v_3,w_3)| v_3+\alpha_2=w_3=0\} \subset {\Bbb P}^1$


$$
x_1=u_3 \;, \;\;\; y_1=\frac{v_3+\alpha_2}{w_3} \;, \;\;\; z_1=w_3.
$$
We have resolved the accessible singular point $P_1$.



\vspace{0.3cm}



{\bf 4.2. Resolution of accessible singular points $ P_5:(l,m,n)=(0,-1,0)$ and $P_6:(l,m,n)=(0,0,0)$ }





We can resolve accessible singular points $P_5, P_6$ by the same way of 4.1. By the birational transformation of 4.1 and 4.2, we have resolved five accessible singular points $P_1, P_2, P_3, P_5$ and $P_6$. There are only two accessible singular points $P_4, P_7$ in ${\Bbb F}_1$-surface. Here ${\Bbb F}_1$-surface is obtained by blowing up one time at a point in ${\Bbb P}^2$. These two accessible singular points are on the same fiber in ${\Bbb F}_1$-surface (See figure 4).




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% 3 2317 1308 2317 1357 499 326
% $P_7$
\put(23.1700,-9.5700){\makebox(0,0)[lb]{$P_7$}}%
% STR 2 0 3 0
% 3 1918 1298 1918 1347 499 326
% $l_1$
\put(19.1800,-9.4700){\makebox(0,0)[lb]{$l_1$}}%
% STR 2 0 3 0
% 3 2163 1426 2163 1477 499 326
% $m_1$
\put(21.6300,-10.7700){\makebox(0,0)[lb]{$m_1$}}%
% STR 2 0 3 0
% 3 1430 1012 1430 1089 473 13
% $C_6$
\put(14.3000,-6.8900){\makebox(0,0)[lb]{$C_6$}}%
% VECTOR 0 0 3 0
% 2 1728 1356 1884 1356
% 
\special{pn 20}%
\special{pa 1728 956}%
\special{pa 1884 956}%
\special{fp}%
\special{sh 1}%
\special{pa 1884 956}%
\special{pa 1817 936}%
\special{pa 1831 956}%
\special{pa 1817 976}%
\special{pa 1884 956}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 1681 1356 1688 1520
% 
\special{pn 20}%
\special{pa 1681 956}%
\special{pa 1688 1120}%
\special{fp}%
\special{sh 1}%
\special{pa 1688 1120}%
\special{pa 1705 1053}%
\special{pa 1686 1067}%
\special{pa 1665 1054}%
\special{pa 1688 1120}%
\special{fp}%
% STR 2 0 3 0
% 3 1742 1298 1742 1347 471 13
% $L_1$
\put(17.4200,-9.4700){\makebox(0,0)[lb]{$L_1$}}%
% STR 2 0 3 0
% 3 1450 1454 1450 1505 471 13
% $M_1$
\put(14.5000,-11.0500){\makebox(0,0)[lb]{$M_1$}}%
% VECTOR 2 0 3 0
% 2 1572 1137 1681 1207
% 
\special{pn 8}%
\special{pa 1572 737}%
\special{pa 1681 807}%
\special{fp}%
\special{sh 1}%
\special{pa 1681 807}%
\special{pa 1636 754}%
\special{pa 1636 778}%
\special{pa 1614 788}%
\special{pa 1681 807}%
\special{fp}%
% LINE 0 0 3 0
% 2 4852 1233 4852 1590
% 
\special{pn 20}%
\special{pa 4852 833}%
\special{pa 4852 1190}%
\special{fp}%
% LINE 0 0 3 0
% 2 4852 1240 6076 1240
% 
\special{pn 20}%
\special{pa 4852 840}%
\special{pa 6076 840}%
\special{fp}%
% LINE 0 0 3 0
% 2 6076 1249 6076 1606
% 
\special{pn 20}%
\special{pa 6076 849}%
\special{pa 6076 1206}%
\special{fp}%
% LINE 0 0 3 0
% 2 4852 1598 6076 1598
% 
\special{pn 20}%
\special{pa 4852 1198}%
\special{pa 6076 1198}%
\special{fp}%
% ELLIPSE 0 0 0 0
% 4 4852 1439 4897 1474 4845 1509 4845 1509
% 
\special{pn 20}%
\special{sh 0.600}%
\special{ar 4852 1039 45 35  0.0000000 6.2831853}%
% LINE 0 0 3 0
% 10 5514 1455 5580 1455 5580 1455 5511 1497 5511 1497 5548 1420 5548 1420 5568 1500 5568 1500 5519 1457
% 
\special{pn 20}%
\special{pa 5514 1055}%
\special{pa 5580 1055}%
\special{fp}%
\special{pa 5580 1055}%
\special{pa 5511 1097}%
\special{fp}%
\special{pa 5511 1097}%
\special{pa 5548 1020}%
\special{fp}%
\special{pa 5548 1020}%
\special{pa 5568 1100}%
\special{fp}%
\special{pa 5568 1100}%
\special{pa 5519 1057}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 4852 1447 4858 1612
% 
\special{pn 20}%
\special{pa 4852 1047}%
\special{pa 4858 1212}%
\special{fp}%
\special{sh 1}%
\special{pa 4858 1212}%
\special{pa 4876 1145}%
\special{pa 4856 1159}%
\special{pa 4836 1146}%
\special{pa 4858 1212}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 4868 1447 5024 1447
% 
\special{pn 20}%
\special{pa 4868 1047}%
\special{pa 5024 1047}%
\special{fp}%
\special{sh 1}%
\special{pa 5024 1047}%
\special{pa 4957 1027}%
\special{pa 4971 1047}%
\special{pa 4957 1067}%
\special{pa 5024 1047}%
\special{fp}%
% VECTOR 0 0 3 0
% 4 5523 1463 5367 1457 5543 1457 5543 1577
% 
\special{pn 20}%
\special{pa 5523 1063}%
\special{pa 5367 1057}%
\special{fp}%
\special{sh 1}%
\special{pa 5367 1057}%
\special{pa 5433 1080}%
\special{pa 5420 1059}%
\special{pa 5434 1040}%
\special{pa 5367 1057}%
\special{fp}%
\special{pa 5543 1057}%
\special{pa 5543 1177}%
\special{fp}%
\special{sh 1}%
\special{pa 5543 1177}%
\special{pa 5563 1110}%
\special{pa 5543 1124}%
\special{pa 5523 1110}%
\special{pa 5543 1177}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 4716 1233 4824 1302
% 
\special{pn 8}%
\special{pa 4716 833}%
\special{pa 4824 902}%
\special{fp}%
\special{sh 1}%
\special{pa 4824 902}%
\special{pa 4779 849}%
\special{pa 4779 873}%
\special{pa 4757 883}%
\special{pa 4824 902}%
\special{fp}%
% STR 2 0 3 0
% 3 4460 1138 4460 1216 473 13
% $C_6$
\put(44.6000,-8.1600){\makebox(0,0)[lb]{$C_6$}}%
% LINE 2 2 3 0
% 2 4852 1447 6067 1447
% 
\special{pn 8}%
\special{pa 4852 1047}%
\special{pa 6067 1047}%
\special{dt 0.045}%
\special{pa 6067 1047}%
\special{pa 6066 1047}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 4868 1368 6058 1368
% 
\special{pn 8}%
\special{pa 4868 968}%
\special{pa 6058 968}%
\special{dt 0.045}%
\special{pa 6058 968}%
\special{pa 6057 968}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 4868 1312 6076 1312
% 
\special{pn 8}%
\special{pa 4868 912}%
\special{pa 6076 912}%
\special{dt 0.045}%
\special{pa 6076 912}%
\special{pa 6075 912}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 4852 1519 6067 1519
% 
\special{pn 8}%
\special{pa 4852 1119}%
\special{pa 6067 1119}%
\special{dt 0.045}%
\special{pa 6067 1119}%
\special{pa 6066 1119}%
\special{dt 0.045}%
% STR 2 0 3 0
% 3 3900 1414 3900 1494 2 0
% $\supset$
\put(39.0000,-10.9400){\makebox(0,0)[lb]{$\supset$}}%
% LINE 0 0 3 0
% 6 1759 2297 1759 2804 1759 2297 2442 2180 2442 2174 2966 2564
% 
\special{pn 20}%
\special{pa 1759 1897}%
\special{pa 1759 2404}%
\special{fp}%
\special{pa 1759 1897}%
\special{pa 2442 1780}%
\special{fp}%
\special{pa 2442 1774}%
\special{pa 2966 2164}%
\special{fp}%
% LINE 0 0 3 0
% 6 2442 2174 2442 2648 1759 2804 2442 2648 2442 2648 2966 3032
% 
\special{pn 20}%
\special{pa 2442 1774}%
\special{pa 2442 2248}%
\special{fp}%
\special{pa 1759 2404}%
\special{pa 2442 2248}%
\special{fp}%
\special{pa 2442 2248}%
\special{pa 2966 2632}%
\special{fp}%
% LINE 0 0 3 0
% 2 2966 2564 2966 3032
% 
\special{pn 20}%
\special{pa 2966 2164}%
\special{pa 2966 2632}%
\special{fp}%
% LINE 0 0 3 0
% 2 3561 2349 3561 2888
% 
\special{pn 20}%
\special{pa 3561 1949}%
\special{pa 3561 2488}%
\special{fp}%
% LINE 0 0 3 0
% 2 3561 2340 4708 2340
% 
\special{pn 20}%
\special{pa 3561 1940}%
\special{pa 4708 1940}%
\special{fp}%
% LINE 0 0 3 0
% 2 3570 2881 4717 2881
% 
\special{pn 20}%
\special{pa 3570 2481}%
\special{pa 4717 2481}%
\special{fp}%
% LINE 0 0 3 0
% 2 4717 2340 4717 2881
% 
\special{pn 20}%
\special{pa 4717 1940}%
\special{pa 4717 2481}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 3119 2634 3425 2634
% 
\special{pn 20}%
\special{pa 3119 2234}%
\special{pa 3425 2234}%
\special{fp}%
\special{sh 1}%
\special{pa 3425 2234}%
\special{pa 3358 2214}%
\special{pa 3372 2234}%
\special{pa 3358 2254}%
\special{pa 3425 2234}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 1623 2610 1130 2610
% 
\special{pn 20}%
\special{pa 1623 2210}%
\special{pa 1130 2210}%
\special{fp}%
\special{sh 1}%
\special{pa 1130 2210}%
\special{pa 1197 2230}%
\special{pa 1183 2210}%
\special{pa 1197 2190}%
\special{pa 1130 2210}%
\special{fp}%
% LINE 0 0 3 0
% 10 2660 2571 2725 2571 2725 2571 2657 2613 2657 2613 2694 2536 2694 2536 2714 2616 2714 2616 2664 2573
% 
\special{pn 20}%
\special{pa 2660 2171}%
\special{pa 2725 2171}%
\special{fp}%
\special{pa 2725 2171}%
\special{pa 2657 2213}%
\special{fp}%
\special{pa 2657 2213}%
\special{pa 2694 2136}%
\special{fp}%
\special{pa 2694 2136}%
\special{pa 2714 2216}%
\special{fp}%
\special{pa 2714 2216}%
\special{pa 2664 2173}%
\special{fp}%
% LINE 0 0 3 0
% 10 4692 2634 4757 2634 4757 2634 4688 2676 4688 2676 4726 2599 4726 2599 4745 2679 4745 2679 4696 2637
% 
\special{pn 20}%
\special{pa 4692 2234}%
\special{pa 4757 2234}%
\special{fp}%
\special{pa 4757 2234}%
\special{pa 4688 2276}%
\special{fp}%
\special{pa 4688 2276}%
\special{pa 4726 2199}%
\special{fp}%
\special{pa 4726 2199}%
\special{pa 4745 2279}%
\special{fp}%
\special{pa 4745 2279}%
\special{pa 4696 2237}%
\special{fp}%
% LINE 0 0 3 0
% 6 5465 2320 5465 2828 5465 2320 6148 2204 6148 2197 6672 2588
% 
\special{pn 20}%
\special{pa 5465 1920}%
\special{pa 5465 2428}%
\special{fp}%
\special{pa 5465 1920}%
\special{pa 6148 1804}%
\special{fp}%
\special{pa 6148 1797}%
\special{pa 6672 2188}%
\special{fp}%
% LINE 0 0 3 0
% 6 6148 2197 6148 2672 5465 2828 6148 2672 6148 2672 6672 3056
% 
\special{pn 20}%
\special{pa 6148 1797}%
\special{pa 6148 2272}%
\special{fp}%
\special{pa 5465 2428}%
\special{pa 6148 2272}%
\special{fp}%
\special{pa 6148 2272}%
\special{pa 6672 2656}%
\special{fp}%
% LINE 0 0 3 0
% 2 6672 2588 6672 3056
% 
\special{pn 20}%
\special{pa 6672 2188}%
\special{pa 6672 2656}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 5329 2634 4836 2634
% 
\special{pn 20}%
\special{pa 5329 2234}%
\special{pa 4836 2234}%
\special{fp}%
\special{sh 1}%
\special{pa 4836 2234}%
\special{pa 4903 2254}%
\special{pa 4889 2234}%
\special{pa 4903 2214}%
\special{pa 4836 2234}%
\special{fp}%
% LINE 0 0 3 0
% 10 6646 2801 6712 2801 6712 2801 6643 2843 6643 2843 6680 2766 6680 2766 6700 2846 6700 2846 6651 2803
% 
\special{pn 20}%
\special{pa 6646 2401}%
\special{pa 6712 2401}%
\special{fp}%
\special{pa 6712 2401}%
\special{pa 6643 2443}%
\special{fp}%
\special{pa 6643 2443}%
\special{pa 6680 2366}%
\special{fp}%
\special{pa 6680 2366}%
\special{pa 6700 2446}%
\special{fp}%
\special{pa 6700 2446}%
\special{pa 6651 2403}%
\special{fp}%
% LINE 0 0 3 0
% 2 1589 3262 1589 3803
% 
\special{pn 20}%
\special{pa 1589 2862}%
\special{pa 1589 3403}%
\special{fp}%
% LINE 0 0 3 0
% 2 1589 3254 2736 3254
% 
\special{pn 20}%
\special{pa 1589 2854}%
\special{pa 2736 2854}%
\special{fp}%
% LINE 0 0 3 0
% 2 1598 3795 2745 3795
% 
\special{pn 20}%
\special{pa 1598 3395}%
\special{pa 2745 3395}%
\special{fp}%
% LINE 0 0 3 0
% 2 2745 3254 2745 3795
% 
\special{pn 20}%
\special{pa 2745 2854}%
\special{pa 2745 3395}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 1147 3548 1453 3548
% 
\special{pn 20}%
\special{pa 1147 3148}%
\special{pa 1453 3148}%
\special{fp}%
\special{sh 1}%
\special{pa 1453 3148}%
\special{pa 1386 3128}%
\special{pa 1400 3148}%
\special{pa 1386 3168}%
\special{pa 1453 3148}%
\special{fp}%
% LINE 0 0 3 0
% 10 2720 3548 2785 3548 2785 3548 2716 3591 2716 3591 2754 3513 2754 3513 2773 3593 2773 3593 2724 3551
% 
\special{pn 20}%
\special{pa 2720 3148}%
\special{pa 2785 3148}%
\special{fp}%
\special{pa 2785 3148}%
\special{pa 2716 3191}%
\special{fp}%
\special{pa 2716 3191}%
\special{pa 2754 3113}%
\special{fp}%
\special{pa 2754 3113}%
\special{pa 2773 3193}%
\special{fp}%
\special{pa 2773 3193}%
\special{pa 2724 3151}%
\special{fp}%
% ELLIPSE 0 0 0 0
% 4 1750 2571 1796 2605 1744 2641 1744 2641
% 
\special{pn 20}%
\special{sh 0.600}%
\special{ar 1750 2171 46 34  0.0000000 6.2831853}%
% ELLIPSE 0 0 0 0
% 4 3552 2650 3598 2685 3546 2720 3546 2720
% 
\special{pn 20}%
\special{sh 0.600}%
\special{ar 3552 2250 46 35  0.0000000 6.2831853}%
% ELLIPSE 0 0 0 0
% 4 5465 2643 5510 2677 5458 2712 5458 2712
% 
\special{pn 20}%
\special{sh 0.600}%
\special{ar 5465 2243 45 34  0.0000000 6.2831853}%
% ELLIPSE 0 0 0 0
% 4 1598 3557 1643 3591 1591 3626 1591 3626
% 
\special{pn 20}%
\special{sh 0.600}%
\special{ar 1598 3157 45 34  0.0000000 6.2831853}%
% LINE 0 0 3 0
% 6 3510 3251 3510 3758 3510 3251 4193 3134 4193 3127 4717 3517
% 
\special{pn 20}%
\special{pa 3510 2851}%
\special{pa 3510 3358}%
\special{fp}%
\special{pa 3510 2851}%
\special{pa 4193 2734}%
\special{fp}%
\special{pa 4193 2727}%
\special{pa 4717 3117}%
\special{fp}%
% LINE 0 0 3 0
% 6 4193 3127 4193 3602 3510 3758 4193 3602 4193 3602 4717 3986
% 
\special{pn 20}%
\special{pa 4193 2727}%
\special{pa 4193 3202}%
\special{fp}%
\special{pa 3510 3358}%
\special{pa 4193 3202}%
\special{fp}%
\special{pa 4193 3202}%
\special{pa 4717 3586}%
\special{fp}%
% LINE 0 0 3 0
% 2 4717 3517 4717 3986
% 
\special{pn 20}%
\special{pa 4717 3117}%
\special{pa 4717 3586}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 3374 3564 2881 3564
% 
\special{pn 20}%
\special{pa 3374 3164}%
\special{pa 2881 3164}%
\special{fp}%
\special{sh 1}%
\special{pa 2881 3164}%
\special{pa 2948 3184}%
\special{pa 2934 3164}%
\special{pa 2948 3144}%
\special{pa 2881 3164}%
\special{fp}%
% LINE 0 0 3 0
% 10 4692 3731 4757 3731 4757 3731 4688 3773 4688 3773 4726 3696 4726 3696 4745 3776 4745 3776 4696 3733
% 
\special{pn 20}%
\special{pa 4692 3331}%
\special{pa 4757 3331}%
\special{fp}%
\special{pa 4757 3331}%
\special{pa 4688 3373}%
\special{fp}%
\special{pa 4688 3373}%
\special{pa 4726 3296}%
\special{fp}%
\special{pa 4726 3296}%
\special{pa 4745 3376}%
\special{fp}%
\special{pa 4745 3376}%
\special{pa 4696 3333}%
\special{fp}%
% ELLIPSE 0 0 0 0
% 4 3510 3572 3555 3607 3503 3642 3503 3642
% 
\special{pn 20}%
\special{sh 0.600}%
\special{ar 3510 3172 45 35  0.0000000 6.2831853}%
% LINE 0 0 3 0
% 2 5286 3270 5286 3811
% 
\special{pn 20}%
\special{pa 5286 2870}%
\special{pa 5286 3411}%
\special{fp}%
% LINE 0 0 3 0
% 2 5286 3262 6434 3262
% 
\special{pn 20}%
\special{pa 5286 2862}%
\special{pa 6434 2862}%
\special{fp}%
% LINE 0 0 3 0
% 2 5295 3803 6442 3803
% 
\special{pn 20}%
\special{pa 5295 3403}%
\special{pa 6442 3403}%
\special{fp}%
% LINE 0 0 3 0
% 2 6442 3262 6442 3803
% 
\special{pn 20}%
\special{pa 6442 2862}%
\special{pa 6442 3403}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 4844 3557 5150 3557
% 
\special{pn 20}%
\special{pa 4844 3157}%
\special{pa 5150 3157}%
\special{fp}%
\special{sh 1}%
\special{pa 5150 3157}%
\special{pa 5083 3137}%
\special{pa 5097 3157}%
\special{pa 5083 3177}%
\special{pa 5150 3157}%
\special{fp}%
% LINE 0 0 3 0
% 10 6417 3557 6482 3557 6482 3557 6414 3598 6414 3598 6451 3521 6451 3521 6471 3601 6471 3601 6421 3558
% 
\special{pn 20}%
\special{pa 6417 3157}%
\special{pa 6482 3157}%
\special{fp}%
\special{pa 6482 3157}%
\special{pa 6414 3198}%
\special{fp}%
\special{pa 6414 3198}%
\special{pa 6451 3121}%
\special{fp}%
\special{pa 6451 3121}%
\special{pa 6471 3201}%
\special{fp}%
\special{pa 6471 3201}%
\special{pa 6421 3158}%
\special{fp}%
% ELLIPSE 0 0 0 0
% 4 5295 3564 5340 3599 5288 3635 5288 3635
% 
\special{pn 20}%
\special{sh 0.600}%
\special{ar 5295 3164 45 35  0.0000000 6.2831853}%
% LINE 2 2 3 0
% 2 5304 3564 6434 3564
% 
\special{pn 8}%
\special{pa 5304 3164}%
\special{pa 6434 3164}%
\special{dt 0.045}%
\special{pa 6434 3164}%
\special{pa 6433 3164}%
\special{dt 0.045}%
% VECTOR 0 0 3 0
% 2 1415 4362 959 4362
% 
\special{pn 20}%
\special{pa 1415 3962}%
\special{pa 959 3962}%
\special{fp}%
\special{sh 1}%
\special{pa 959 3962}%
\special{pa 1026 3982}%
\special{pa 1012 3962}%
\special{pa 1026 3942}%
\special{pa 959 3962}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 3207 4362 2751 4362
% 
\special{pn 20}%
\special{pa 3207 3962}%
\special{pa 2751 3962}%
\special{fp}%
\special{sh 1}%
\special{pa 2751 3962}%
\special{pa 2818 3982}%
\special{pa 2804 3962}%
\special{pa 2818 3942}%
\special{pa 2751 3962}%
\special{fp}%
% VECTOR 0 0 3 0
% 2 5040 4377 4583 4377
% 
\special{pn 20}%
\special{pa 5040 3977}%
\special{pa 4583 3977}%
\special{fp}%
\special{sh 1}%
\special{pa 4583 3977}%
\special{pa 4650 3997}%
\special{pa 4636 3977}%
\special{pa 4650 3957}%
\special{pa 4583 3977}%
\special{fp}%
% LINE 0 0 3 0
% 2 1488 4081 1488 4599
% 
\special{pn 20}%
\special{pa 1488 3681}%
\special{pa 1488 4199}%
\special{fp}%
% LINE 0 0 3 0
% 2 1488 4072 2550 4072
% 
\special{pn 20}%
\special{pa 1488 3672}%
\special{pa 2550 3672}%
\special{fp}%
% LINE 0 0 3 0
% 2 1496 4590 2557 4590
% 
\special{pn 20}%
\special{pa 1496 4190}%
\special{pa 2557 4190}%
\special{fp}%
% LINE 0 0 3 0
% 2 2557 4072 2557 4590
% 
\special{pn 20}%
\special{pa 2557 3672}%
\special{pa 2557 4190}%
\special{fp}%
% LINE 0 0 3 0
% 10 2657 4522 2717 4522 2717 4522 2654 4561 2654 4561 2688 4488 2688 4488 2707 4565 2707 4565 2661 4524
% 
\special{pn 20}%
\special{pa 2657 4122}%
\special{pa 2717 4122}%
\special{fp}%
\special{pa 2717 4122}%
\special{pa 2654 4161}%
\special{fp}%
\special{pa 2654 4161}%
\special{pa 2688 4088}%
\special{fp}%
\special{pa 2688 4088}%
\special{pa 2707 4165}%
\special{fp}%
\special{pa 2707 4165}%
\special{pa 2661 4124}%
\special{fp}%
% ELLIPSE 0 0 0 0
% 4 1737 4514 1779 4548 1731 4582 1731 4582
% 
\special{pn 20}%
\special{sh 0.600}%
\special{ar 1737 4114 42 34  0.0000000 6.2831853}%
% LINE 2 2 3 0
% 2 1503 4362 2550 4362
% 
\special{pn 8}%
\special{pa 1503 3962}%
\special{pa 2550 3962}%
\special{dt 0.045}%
\special{pa 2550 3962}%
\special{pa 2549 3962}%
\special{dt 0.045}%
% SPLINE 2 2 3 0
% 4 1486 4377 1635 4393 1729 4537 1729 4537
% 
\special{pn 8}%
\special{pa 1486 3977}%
\special{pa 1521 3975}%
\special{pa 1555 3975}%
\special{pa 1587 3978}%
\special{pa 1617 3985}%
\special{pa 1643 3998}%
\special{pa 1665 4017}%
\special{pa 1683 4042}%
\special{pa 1700 4071}%
\special{pa 1714 4102}%
\special{pa 1728 4135}%
\special{pa 1729 4137}%
\special{sp -0.045}%
% LINE 2 2 3 0
% 2 1729 4522 2538 4522
% 
\special{pn 8}%
\special{pa 1729 4122}%
\special{pa 2538 4122}%
\special{dt 0.045}%
\special{pa 2538 4122}%
\special{pa 2537 4122}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 2555 4522 2681 4522
% 
\special{pn 8}%
\special{pa 2555 4122}%
\special{pa 2681 4122}%
\special{dt 0.045}%
\special{pa 2681 4122}%
\special{pa 2680 4122}%
\special{dt 0.045}%
% SPLINE 0 0 3 0
% 4 2547 4370 2664 4415 2688 4522 2688 4522
% 
\special{pn 20}%
\special{pa 2547 3970}%
\special{pa 2583 3977}%
\special{pa 2616 3985}%
\special{pa 2644 3998}%
\special{pa 2665 4017}%
\special{pa 2678 4043}%
\special{pa 2685 4075}%
\special{pa 2687 4110}%
\special{pa 2688 4122}%
\special{sp}%
% LINE 0 0 3 0
% 2 3359 4096 3359 4613
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% SPLINE 2 2 3 0
% 4 3513 4522 3608 4582 3656 4696 3656 4696
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% LINE 2 2 3 0
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% 
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\end{picture}%
\label{fig:third2}
\caption{}
\end{figure}


\vspace{0.3cm}


{\bf 4.3. Resolution of accessible singular points $P_4$ and $P_7$ }



Let us start by summarizing the steps which are needed to resolve the accessible singular points of $\tilde v$ at $P_4$ and $P_7$:

$$ P_4=\{(u,v,w)=(0,-1,0)\}, \  P_7=\{(u,v,w)=(-3,-1,0)\}.$$




\begin{enumerate}
\item{\bf Step 6}: We will blow up along the curve $C_6$ and blow down the surface $S_1$ (see Figure 4).
\item{\bf Step 7}: We will blow up along the curve $C_7$ and blow down the surface $S_2$ (see Figure 4).
\item{\bf Step 8}: We will blow up along the curve $C_{8}$ and blow down the surface $S_3$  (see Figure 4).
\item{\bf Step 9}: We will blow up along the curve $C_{9}$ (see Figure 4).
\item{\bf Step 10}: We will blow up along the curve $C_{10}$ (see Figure 4).
\item{\bf Step 11}: We will blow up along the curve $C_{11}$ (see Figure 4).
\end{enumerate}










In a neighborhood of $P_7$, the system is written as



\begin{equation*}
  \left\{
  \begin{aligned}
   \frac {dl_1}{dt} &=\frac{3l_1-(l_1)^2-6m_1+2l_1m_1}{n_1}+n_1\alpha_1
   +3n_1\alpha_3
   -l_1n_1\alpha_3\\
   \frac {dm_1}{dt} &=\frac{-3m_1+3(m_1)^2}{n_1}+2t-2tm_1+n_1\alpha_2
   +n_1\alpha_3-m_1n_1\alpha_3\\
   \frac {dn_1}{dt} &=-1+2m_1-tn_1-\alpha_3(n_1)^2.
   \end{aligned}
  \right. 
\end{equation*}


In a neighborhood of $P_4$, the system is written as
\begin{equation*}
  \left\{
  \begin{aligned}
   \frac {dL_1}{dt} &=\frac{-3L_1-(L_1)^2+2L_1M_1}{N_1}+\alpha_1N_1
   -\alpha_3L_1N_1\\
   \frac {dM_1}{dt} &=2t-2tM_1+\frac{-3M_1+3(M_1)^2}{N_1}+\alpha_2N_1
   +\alpha_3N_1-\alpha_3M_1N_1\\
   \frac {dN_1}{dt} &=-1+2M_1-tN_1-\alpha_3(N_1)^2.
   \end{aligned}
  \right. 
\end{equation*}





{\bf 4.3.1.} We blow up along the curve $C_6=\{(L_1, M_1, N_1)| L_1=N_1=0\} \subset {\Bbb P}^1$ and blow down the surface $S_1=\{(l_1,m_1,n_1)|n_1=0\}$



$$
L_2=\frac{L_1}{N_1} \;, \;\;\; M_2=M_1 \;, \;\;\; N_2=N_1,
$$
$$
l_2=\frac{n_1}{l_1-3} \;, \;\;\;  m_2=m_1 \;, \;\;\; n_2=n_1.
$$

In a neighborhood of  $\{(l_2,m_2.n_2)=(0,0,0)\}$, the system is written as


\begin{equation*}
  \left\{
  \begin{aligned}
   \frac {dl_2}{dt} &=1-tl_2+\frac{2l_2}{n_2}-\alpha_1(l_2)^2\\
   \frac {dm_2}{dt} &=2t-2tm_2-\frac{3m_2-3(m_2)^2}{n_2}+\alpha_2n_2
   +\alpha_3n_2-\alpha_3m_2n_2\\
   \frac {dn_2}{dt} &=-1+2m_2-tn_2-\alpha_3(n_2)^2.
   \end{aligned}
  \right. 
\end{equation*}






{\bf 4.3.2.} We blow up along the curve $C_7=\{(L_2, M_2, N_2)| L_2=N_2=0\} \subset {\Bbb P}^1$ and blow down the surface $S_2=\{(l_2,m_2,n_2)|n_2=0\}$



$$
L_3=\frac{L_2}{N_2} \;, \;\;\; M_3=M_2 \;, \;\;\; N_3=N_2,
$$
$$
l_3=l_2n_2 \;, \;\;\; m_3=m_2 \;, \;\;\; n_3=n_2.
$$


In a neighborhood of  $\{(l_3,m_3,n_3)=(0,0,0)\}$, the system is written as



\begin{equation*}
  \left\{
  \begin{aligned}
   \frac {dl_3}{dt} &=-2tl_3+\frac{l_3+2l_3m_3-\alpha_2(l_3)^2}{n_3}
   +n_3-\alpha_3l_3n_3\\
   \frac {dm_3}{dt} &=2t-2tm_3-\frac{3m_3-3(m_3)^2}{n_3}+\alpha_2n_3
   +\alpha_3n_3-\alpha_3m_3n_3\\
   \frac {dn_3}{dt} &=-1+2m_3-tn_3-\alpha_3(n_3)^2.
   \end{aligned}
  \right. 
\end{equation*}


{\bf 4.3.3.} We blow up along the curve $C_{8}=\{(L_3, M_3, N_3)| L_3=N_3=0\} \subset {\Bbb P}^1$ and blow down the surface $S_3=\{(l_3,m_3,n_3)|n_3=0\}$


$$
L_4=\frac{L_3-\alpha_1}{N_3} \;, \;\;\; M_4=M_3 \;, \;\;\; N_4=N_3,
$$
$$
l_4=\frac{l_3n_3}{1-\alpha_1l_3} \;, \;\;\; m_4=m_3 \;, \;\;\; n_4=n_3.
$$








{\bf 4.3.4.} We blow up along the curve $C_{9}=\{(L_4, M_4, N_4)| M_4=N_4=0\} \cup \{(l_4, m_4, n_4)| m_4=n_4=0\} \cong {\Bbb P}^1$


$$
L_5=L_4 \;, \;\;\; M_5=\frac{M_4}{N_4} \;, \;\;\; N_5=N_4,
$$
$$
l_5=l_4 \;, \;\;\; m_5=\frac{m_4}{n_4} \;, \;\;\; n_5=n_4.
$$





{\bf 4.3.5.} We blow up along the curve $C_{10}=\{(L_5, M_5, N_5)| M_5-t=N_5=0\} \cup \{(l_5, m_5, n_5)| m_5-t=n_5=0\} \cong {\Bbb P}^1$


$$
L_6=L_5 \;, \;\;\; M_6=\frac{M_5-t}{N_5} \;, \;\;\; N_6=N_5,
$$
$$
l_6=l_5 \;, \;\;\; m_6=\frac{m_5-t}{n_5} \;, \;\;\; n_6=n_5.
$$








{\bf 4.3.6.} We blow up along the curve $C_{11}=\{(L_6, M_6, N_6)| M_6+1-\alpha_2-\alpha_3=N_6=0\} \cup \{(l_6, m_6, n_6)| m_6+1-\alpha_2-\alpha_3=n_6=0\} \cong {\Bbb P}^1$




$$
x_4=L_6 \;, \;\;\; y_4=\frac{M_6+1-\alpha_2-\alpha_3}{N_6} \;, \;\;\; z_4=N_6,
$$
$$
x_7=l_6 \;, \;\;\; y_7=\frac{m_6+1-\alpha_2-\alpha_3}{n_6} \;, \;\;\; z_7=n_6.
$$
We have resolved the accessible singular points $P_4$ and $P_7$.


\vspace{0.3cm}

{\it Acknowledgement.} The author thanks Professors M.-H. Saito, K. Takano and Y. Yamada for giving helpful suggestions and encouragement, and Professor W. Rossman for checking English.





\begin{thebibliography}{99}




\bibitem[1]{1} K. Okamoto, {\em Sur les 
feuilletages associ\'es aux \'equations du second ordre \`a points critiques fixes de P. Painlev\'e, Espaces des conditions initiales}, Japan. 
J. Math., {\bf 5}, 1979, 1--79.  


\bibitem[2]{2} M. Noumi and Y. Yamada, 
        {\em Higher order Painlev\`e equations of type ${A_l}^{(1)}$}, Funkcial. Ekvac., {\bf 41}, 1998, 483--503.


\bibitem[3]{3} T. Shioda and K. Takano, On some Hamiltonian structures of Painlev\'e systems I, Funkcial. Ekvac., {\bf 40}, 1997, 271--291. 



\bibitem[4]{4} Y. Sasano, Coupled Painlev\'e II systems in dimension 4 and the systems of type ${A_4}^{(1)}$, preprint.



\bibitem[5]{5} N. Tahara, An augmentation of the phase space of the system of type ${A_4}^{(1)}$, Kyushu J. Math. {\bf 58}, 2004, 393--425. 


\end{thebibliography}








\end{document}


math0505201.extracted_bib
\begin{thebibliography}{99}




\bibitem[1]{1} K. Okamoto, {\em Sur les 
feuilletages associ\'es aux \'equations du second ordre \`a points critiques fixes de P. Painlev\'e, Espaces des conditions initiales}, Japan. 
J. Math., {\bf 5}, 1979, 1--79.  


\bibitem[2]{2} M. Noumi and Y. Yamada, 
        {\em Higher order Painlev\`e equations of type ${A_l}^{(1)}$}, Funkcial. Ekvac., {\bf 41}, 1998, 483--503.


\bibitem[3]{3} T. Shioda and K. Takano, On some Hamiltonian structures of Painlev\'e systems I, Funkcial. Ekvac., {\bf 40}, 1997, 271--291. 



\bibitem[4]{4} Y. Sasano, Coupled Painlev\'e II systems in dimension 4 and the systems of type ${A_4}^{(1)}$, preprint.



\bibitem[5]{5} N. Tahara, An augmentation of the phase space of the system of type ${A_4}^{(1)}$, Kyushu J. Math. {\bf 58}, 2004, 393--425. 


\end{thebibliography}