Purely log terminal blow-ups of three-dimensional terminal singularities
S.~A.~Kudryavtsev
The purely log terminal blow-ups of three-dimensional terminal toric singularities are described.
math0404260.tex
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\title{Purely log terminal blow-ups of three-dimensional terminal singularities}
\author{S.~A.~Kudryavtsev}

\date{}
\address{Department of Algebra, Faculty of Mathematics,
Moscow State University, GSP-2 119992, Vorob'evy Gory, Moscow, Russia}

\email{kudryav@mech.math.msu.su}




\begin{document}
\begin{abstract}
The purely log terminal blow-ups of three-dimensional terminal
toric singularities are described.
\end{abstract}
\thanks{This work was done with the partial support of the
Russian Foundation for Basic Research (grant no. 02-01-00441), the President grants
(grant no. 489-2003.01 and grant MK-1285.2003.1).}
\maketitle

\section*{\bf Introduction}

The purely log terminal blow-ups of three-dimensional terminal toric singularities are
studied in this paper. The main result is the following theorem.

\begin{theorem1}
Let $f\colon (Y,E)\to (X\ni P)$ be a purely log terminal blow-up, where
$(X\ni P)$ is a three-dimensional terminal toric singularity
and $f(E)=P$. Then either $f$ is a toric morphism up to analytic isomorphism
$(X\ni P)\cong (X\ni P)$, or $f$ is described in section
$\ref{example}$.
\end{theorem1}

This theorem gives the decision of the following problem in a case of
toric singularities.
\begin{problem}
Describe the class of all log del Pezzo
surfaces and generic $\PP^1$-fibrations which can be the exceptional
divisors of some purely log terminal blow-ups of
three-dimensional terminal singularities.
\end{problem}

This problem is a very important one in studying three-dimensional
Mori contractions.
In the theory of complements there is the inductive approach to classify
the three-dimensional extremal contractions
(see, for example, \cite{PrSh}, \cite{PrEll}, \cite{Kud2}).
Nowadays the problem above-mentioned is the only one difficulty to obtain new results
(see \cite{Kud2}).
We can try to avoid this difficulty
\cite[Example 7.8]{Sh2}.
In practice it is impossible to make it because of
the great amount of calculations.

Note that the theorem above-mentioned is
the logarithmic version of main theorem of the paper
\cite{Kud4}.

I am grateful to Professors  Yu.G.~Prokhorov and I.A.~Cheltsov for valuable advices.



\section{\bf Purely log terminal blow-ups and their properties}
All varieties considered are defined over
$\CC$, the complex number field.
The main definitions, notations and notions used in the paper are
given in \cite{Koetal}. The germ of a variety
$X$ at a point $P$ is denoted by $(X\ni P)$.

\begin{definition} \label{defplt}
Let $(X \ni P)$ be a log canonical singularity and let $f\colon Y \to X$ be its blow-up.
Suppose that the exceptional locus of $f$ consists of only one irreducible divisor:
$\Exc f=E$. Then $f\colon (Y,E) \to (X \ni P)$
is called a {\it purely log terminal blow-up} if the divisor
$K_Y+E$ is purely log terminal and $-E$ is $f$-ample.
\end{definition}

\begin{remark} Definition
\ref{defplt} implicitly requires that the divisor
$E$ be $\QQ$-Cartier. Hence $Y$ is a
$\QQ$-gorenstein variety.
By inversion of adjunction
$K_E+\Diff_E(0)=(K_Y+E)|_E$ is a kawamata log terminal
divisor.
\end{remark}

\begin{proposition}\cite[Corollary 2.11]{Pr2}\label{normal}
Using the notations of Definition $\ref{defplt}$, the variety $f(E)$ is normal.
\begin{proof}
Let $D_Y\in |-n(K_Y+E)|$ be a general element for $n\gg 0$.
Then $f(E)$ is a minimal center of $\LCS(X,\frac1nD)$, where
$D=f(D_Y)$. Hence $f(E)$ is a normal variety by \cite[Theorem 1.9]{Kaw}.
\end{proof}
\end{proposition}

\begin{remark}\label{pltpropert}
Using the notations of Definition \ref{defplt}, we have the following properties
of purely log terminal blow-ups.
\begin{enumerate}

\item[1)] If $(X \ni P)$ is a log terminal singularity then
$-(K_Y+E)$ is a $f$-ample divisor. The purely log terminal blow-up of
log terminal singularity always exists under the assumption that
the log minimal model program holds \cite[Theorem 1.5]{Kud1} (see theorem \ref{inductive}).

\item[2)] If $(X \ni P)$ is a strictly log canonical singularity then
$a(E,0)=-1$. The purely log terminal blow-up of strictly
log canonical singularity exists if and only if
there is only one exceptional divisor with discrepancy
$-1$ under the assumption that
the log minimal model program holds \cite[Theorem 1.9]{Kud1}.

\item[3)] If $(X \ni P)$ is a
$\QQ$-factorial singularity then $Y$ is $\QQ$-factorial also and
$\rho(Y/X)=1$ \cite[Remark 2.2]{Pr2}.
Hence, for $\QQ$-factorial singularity we can omit the requirement that
$-E$ be $f$-ample in
Definition \ref{defplt} because it holds automatically.

\item[4)]  Let $f_i\colon (Y_i,E_i)\to (X \ni P)$ be two purely log terminal blow-ups.
If $E_1$ and $E_2$ define the same discrete valuation of the function field
$K(X)$ then
the blow-ups $f_1$ and $f_2$ are isomorphic \cite[Remark 2.2]{Pr2}.

\item[5)] See the papers \cite{Pr2} and \cite{Kud1} about the inductive
properties of purely log terminal blow-ups.

\end{enumerate}
\end{remark}


\begin{proposition} \label{lemma2}\cite[Corollary 1.13]{Kud1} $($cf.
\cite[Proposition 1.2]{Kud4}$)$
Notation as in definition
$\ref{defplt}$. Assume that we do not require $-E$ to be ample
over $X$. Then such blow-up differs from a purely log terminal blow-up  by a
small flopping contraction
$($with respect to the canonical divisor$)$.
\end{proposition}

\begin{example} \label{two-dim} 1)
Let $f\colon (Y,E) \to (X \ni P)$ be a toric blow-up of toric $\QQ$-gorenstein singularity,
where $\Exc f=E$ is an irreducible divisor. Then
$K_Y+E$ is a purely log terminal divisor. The blow-up
$f$ can differ from a purely log terminal blow-up (in the case of  non-$\QQ$-factorial
singularity only) by a small flopping contraction only by proposition \ref{lemma2}.

2) A two-dimensional toric singularity is a cyclic quotient singularity. A purely
log terminal blow-up is toric always \cite[Example 2.4]{Pr2}.

3) The two-dimensional non-toric log terminal singularities have the types
$\DDD_n$ and $\EEE_i$,
where $i=6,7,8$. The purely log terminal blow-up is unique for these singularities and
it is the extraction of a central vertex of minimal resolution graph \cite[Example 2.4]{Pr2}.
\end{example}

\begin{theorem}\cite[Theorem 1.5]{Kud1}, \cite[Proposition 2.9]{Pr2}\label{inductive}
Let $X$ be  a kawamata log terminal variety and let $D\ne 0$ be a boundary on $X$ such that
$(X,D)$ is log canonical, but not purely log terminal.
Suppose LMMP is true or $\dim X\le 3$.
Then there exists an inductive blow-up $f:Y\to X$
such that:
\begin{enumerate}
\item the exceptional locus of $f$ contains only one irreducible divisor
$E\ (\Exc(f)=E)$;
\item $K_Y+E+D_Y=f^*(K_X+D)$ is log canonical;
\item $K_Y+E+(1-\varepsilon)D_Y$ is purely log terminal and anti-ample over $X$
for any $\varepsilon>0$;
\item if $X$ is $\QQ$-factorial then $Y$ is also
$\QQ$-factorial and $\rho(Y/X)=1$.
\end{enumerate}
\end{theorem}



\section{\bf Non-toric purely log terminal blow-ups}\label{example}
In this section we construct the examples of non-toric purely log terminal blow-ups.
The case of non-singular point, the case of cyclic singularity and
the case of ordinary double point are
considered in items \ref{ex1}--\ref{calcul}, \ref{ex2} and \ref{ex3}--\ref{ex3end} respectively.



\subs\label{ex1}
At first we consider the general construction of purely log terminal blow-ups.
Then we prove its existence in special cases. After that our special cases
are studied in detail.
\par
Let $(X\ni P)=(\CC^3\ni 0)$ and $g\colon Z\to (X\ni P)$ be a weighted blow-up
with weights
$(\beta_1,\beta_2,\beta_3)$, $\Exc g=S$.
Consider some non-singular curve $\Gamma$ on
$S$, which is not a toric subvariety of $Z$.
Let $h\colon\widetilde Y\to Z$ be a blow-up of $Z$ along a curve $\Gamma$,
$\Exc h=\widetilde E$ is an irreducible divisor. We assume that
$h$ is a toric morphism near a
generic point of $\Gamma$.
In detail,
a generic point $Q$ of $\Gamma$ lies in a smooth locus of $Z$. Let $H$ be a general hyperplane
section passing through $Q$. Then $h$ induces a purely log terminal blow-up
$h|_{\widetilde H}\colon\widetilde H\to H$, where $\widetilde H$ is a proper transform of $H$.
The morphism
$h|_{\widetilde H}$ is a weighted blow-up with weights $(\alpha_1,\alpha_2)$.
To be definite, assume that $h^*S=\widetilde S+\alpha_1\widetilde E$.



Let $\widetilde Y\to Y$ be the divisorial contraction of $\widetilde S$ to a point, where
$\widetilde S$ is the proper transform of $S$.
We obtain a purely log terminal blow-up
$f\colon Y\to (X\ni P)$, where $\Exc f=E$.
The morphism $f$ is non-toric.

\subs
Now we write our cases. Equality (1) gives the precise definition of a curve $\Gamma$.
\par
{\it Type} $\AAA_n$) Put
$(\beta_1,\beta_2,\beta_3)=(\frac{a_2+a_3}k,a_2d_1,a_3d_1),$ where
$k\mid (a_2+a_3)$ and $\Gamma\sim\OO_{\PP\big(\frac{a_2+a_3}k,a_2,a_3\big)}(a_2+a_3);$
\par
{\it Type}
$\DDD_{2k+2})$ Put $(\beta_1,\beta_2,\beta_3)=(2,2k,2k+1)$, $S=\PP(1,k,2k+1)$,
$\Gamma\sim\OO_{\PP(1,k,2k+1)}(2k+1)$, where $k\ge 1$.
\par
{\it Type}
$\DDD_{2k+1})$ Put $(\beta_1,\beta_2,\beta_3)=(2,2k-1,2k)$, $S=\PP(1,2k-1,k)$,
$\Gamma\sim\OO_{\PP(1,2k-1,k)}(2k)$, where $k\ge 2$.
\par
{\it Type}
$\EEE_6)$ Put $(\beta_1,\beta_2,\beta_3)=(3,4,6)$, $S=\PP(1,2,1)$,
$\Gamma\sim\OO_{\PP(1,2,1)}(2)$.
\par
{\it Type}
$\EEE_7)$ Put $(\beta_1,\beta_2,\beta_3)=(4,6,9)$, $S=\PP(2,1,3)$,
$\Gamma\sim\OO_{\PP(2,1,3)}(3)$.
\par
{\it Type}
$\EEE_8)$ Put $(\beta_1,\beta_2,\beta_3)=(6,10,15)$, $S=\PP^2$,
$\Gamma\sim\OO_{\PP^2}(1)$.

Consider a divisor
$$
D=\sum_{i=1}^2\frac12\{\varphi(x_1,x_2,x_3)+\psi_i(x_1,x_2,x_3)=0\}
$$
on $X$, where the polynomial $\varphi$ is quasihomogeneous with respect to
weights
$(\beta_1,\beta_2,\beta_3)$ and defines a corresponding Du Val singularity.
The polynomials $\psi_i$ are general and have the large degrees.
We have
\begin{equation}
\Gamma=D_Z|_S=\widetilde{\{\varphi=0\}}|_S,
\end{equation}
where $D_Z$ and
$\widetilde{\{\varphi=0\}}$ are the proper transforms of the divisors
$D$ and $\{\varphi=0\}$.

\subs
Let us prove the existence of construction.
\begin{lemma} The linear system $|2nD_{\widetilde Y}|$
gives a required divisorial contraction
$\widetilde Y\to Y$, where $n\gg 0$.
\begin{proof}
Let
$$
T_Z\in \Big|-m\Big(K_Z+D_Z+\frac{\alpha_1-1}{\alpha_1}S\Big)\Big|
$$
be a general element of linear system for $m\gg 0$.
This system is either very ample over $X$, or $\QQ$-linearly trivial over $X$.
In second case $T_Z=0$.
By the construction of $g$ the pair $(Z,D_Z+(1-\varepsilon)S)$ is kawamata log terminal
for any $\varepsilon>0$. Therefore,
the pair $(X,D+\frac1mT)$ is kawamata log terminal, where $T=g(T_Z)$.
We have
$a(S,D+\frac1mT)=-\frac{\alpha_1-1}{\alpha_1}\le 0$ and
$a(\widetilde E,D+\frac1mT)=0$.
Therefore, by base point free theorem \cite{KMM}
the linear system $|2nD_{\widetilde Y}|$
gives a required divisorial contraction
$\widetilde Y\to Y$, where $n\gg 0$.
\end{proof}
\end{lemma}


We can prove that
a divisorial contraction
$(Y,E)\to (X\ni P)$ obtained is a purely log terminal blow-up by many ways.
For instance, we calculate the log surface $(E,\Diff_E(0))$ below.
It is kawamata log terminal.
Therefore, the pair $(Y,E)$ is purely log terminal by inversion of adjunction.

Before considering the types $\AAA_n$, $\DDD_n$, $\EEE_i$ explicitly let us prove
the following lemma.
\begin{lemma}\label{minimal}
Let $\widetilde\Gamma=\widetilde S\cap\widetilde E$. Then
$$
(\widetilde\Gamma^2)_{\widetilde E}=\alpha_2\frac{\big(K_S+\Diff_S(0)\big)\cdot\Gamma}{a(S,0)+1}
- \alpha_1(\Gamma^2)_S
$$
and
\begin{enumerate}
\item[1)]
$$
(\widetilde\Gamma^2)_{\widetilde E}= -\Big(\alpha_2\frac1{d_1}\frac{k}{a_2a_3}+
\alpha_1\frac{k(a_2+a_3)}{a_2a_3}\Big) \ \text{for type}\ \AAA_n;
$$
\item[2)]
$$
(\widetilde\Gamma^2)_{\widetilde E}= -\Big(\alpha_2\frac1{n-2}+
\alpha_1\frac{2n-2}{n-2}\Big)\ \text{for type}\
\DDD_n;
$$
\item[3)]
$$
(\widetilde\Gamma^2)_{\widetilde E}= -\Big(\alpha_2\cdot\frac16+\alpha_1\cdot 2\Big),
-\Big(\alpha_2\cdot\frac1{12}+\alpha_1\cdot \frac32\Big),
-\Big(\alpha_2\cdot\frac1{30}+\alpha_1\Big)
$$
for types $\EEE_6,\ \EEE_7,\ \EEE_8$ respectively.
\end{enumerate}
\begin{proof}
We have
\begin{gather*}
(\widetilde\Gamma^2)_{\widetilde E}=\alpha_2\widetilde S\cdot\widetilde\Gamma=
\alpha_2(S\cdot\Gamma-\alpha_1\widetilde E\cdot \widetilde \Gamma)= \alpha_2S\cdot\Gamma -
\alpha_1(\widetilde\Gamma^2)_{\widetilde S}\\ = \alpha_2S\cdot\Gamma - \alpha_1(\Gamma^2)_S.
\end{gather*}

By adjunction theorem the lemma is proved.
\end{proof}
\end{lemma}

\subs
For every type $\AAA_n$, $\DDD_n$, $\EEE_i$ the surface $\widetilde E$ is a conic bundle.
By calculations (see item \ref{calcul}) every geometrical fiber is irreducible.
The curve $\widetilde\Gamma$ is a minimal section by lemma \ref{minimal}.
The surface $E$ is obtained from $\widetilde E$ by a contraction of $\widetilde\Gamma$.

In the following diagrams we write a structure of $\widetilde E$.
All singularities of $\widetilde E$ are written.
The empty circle denotes a fiber (always with singularities of $\widetilde E$).
Every fiber $f$ and $\widetilde\Gamma$ are the toric orbits near some cyclic
singularity of type $\frac1{\gamma}(b_1,{\bf b_2})$ at a point $f\cap\widetilde\Gamma$.
The bold weight $b_2$ corresponds to $\widetilde\Gamma$.

Another section of $\widetilde E$, which does not intersect $\widetilde\Gamma$, is denoted
by $\widetilde E_0$. The variety $\widetilde Y$ has a singularity
$\frac1{\alpha_1}(-\alpha_2,1)\times \AAA^1$ at a generic point of $\widetilde E_0$.
\\
1) {\it Type} $\AAA_n$. Let $l=(a_2+a_3)/k$, $k_1=(a_2d_1,\alpha_1a_3d_1+\alpha_2)$ and
$k_2=(a_3d_1,\alpha_1a_2d_1+\alpha_2)$.
Put $b_1=(\alpha_1a_3d_1+\alpha_2)/k_1$ and $b_2=(\alpha_1a_2d_1+\alpha_2)/k_2$. Then

\begin{center}
\begin{picture}(350,70)(0,0)
\put(170,10){\fbox{$\widetilde\Gamma$}}
\put(170,15){\line(-1,0){20}}
\put(78,12){\tiny{$\frac1{a_2d_1/k_1}(l,{\bf -b_1})$}}
\put(108,15){\oval(84,20)}
\put(66,15){\line(-1,0){20}}
\put(40,15){\circle{12}}
\put(37,13){\tiny{$f_1$}}
\put(40,21){\line(0,1){20}}
\put(20,49){\tiny{$\frac1{a_2d_1/k_1}(l,b_1)$}}
\put(46,51){\oval(70,20)}
\put(184,15){\line(1,0){20}}
\put(216,12){\tiny{$\frac1{a_3d_1/k_2}(l,{\bf -b_2})$}}
\put(246,15){\oval(84,20)}
\put(288,15){\line(1,0){20}}
\put(314,15){\circle{12}}
\put(311,13){\tiny{$f_2$}}
\put(314,21){\line(0,1){20}}
\put(286,49){\tiny{$\frac1{a_3d_1/k_2}(l,b_2)$}}
\put(312,51){\oval(70,20)}
\end{picture}
\end{center}

and
$$
\Diff_E(0)=\frac{k_1-1}{k_1}f_1+\frac{k_2-1}{k_2}f_2+\frac{\alpha_1-1}{\alpha_1}\widetilde E_0.
$$

Note that the log surface $(E,\Diff_E(0))$ is toric.\\
2) {\it Type} $\DDD_n$. Let $k_1=(n-2,2\alpha_1+\alpha_2)$ and $k_2=(2,\alpha_2)$.
Put $b_1=(2\alpha_1+\alpha_2)/k_1$ and $b_2=\alpha_2/k_2$. Then
\\
\begin{center}
\begin{picture}(350,65)(0,0)
\put(160,10){\fbox{$\widetilde\Gamma$}}
\put(160,15){\line(-1,0){20}}
\put(70,12){\tiny{$\frac1{(n-2)/k_1}(1,{\bf -b_1})$}}
\put(103,15){\oval(74,20)}
\put(66,15){\line(-1,0){20}}
\put(40,15){\circle{12}}
\put(37,13){\tiny{$f_1$}}
\put(40,21){\line(0,1){20}}
\put(12,49){\tiny{$\frac1{(n-2)/k_1}(1,b_1)$}}
\put(40,51){\oval(70,20)}
\put(174,15){\line(1,0){20}}
\put(198,12){\tiny{$\frac1{2/k_2}(1,{\bf -b_2})$}}
\put(221,15){\oval(54,20)}
\put(248,15){\line(1,0){20}}
\put(274,15){\circle{12}}
\put(271,13){\tiny{$f_2$}}
\put(280,15){\line(1,0){20}}
\put(306,12){\tiny{$\frac1{2/k_2}(1,b_2)$}}
\put(325,15){\oval(50,20)}
\put(173,25){\line(1,1){22}}
\put(198,49){\tiny{$\frac1{2/k_2}(1,{\bf -b_2})$}}
\put(221,52){\oval(54,20)}
\put(248,52){\line(1,0){20}}
\put(274,52){\circle{12}}
\put(271,50){\tiny{$f_3$}}
\put(280,52){\line(1,0){20}}
\put(306,49){\tiny{$\frac1{2/k_2}(1,b_2)$}}
\put(325,52){\oval(50,20)}
\end{picture}
\end{center}

and
$$
\Diff_E(0)=\frac{k_1-1}{k_1}f_1+\frac{k_2-1}{k_2}f_2+\frac{k_2-1}{k_2}f_3+
\frac{\alpha_1-1}{\alpha_1}\widetilde E_0.
$$


3) {\it Type} $\EEE_6$. Let $k_1=(2,\alpha_2)$ and $k_2=(3,\alpha_2)$.
Put $b_i=\alpha_2/k_i$. Then
\\
\begin{center}
\begin{picture}(350,65)(0,0)
\put(160,10){\fbox{$\widetilde\Gamma$}}
\put(160,15){\line(-1,0){20}}
\put(78,12){\tiny{$\frac1{2/k_1}(1,{\bf -b_1})$}}
\put(103,15){\oval(74,20)}
\put(66,15){\line(-1,0){20}}
\put(40,15){\circle{12}}
\put(37,13){\tiny{$f_1$}}
\put(40,21){\line(0,1){20}}
\put(22,49){\tiny{$\frac1{2/k_1}(1,b_1)$}}
\put(40,51){\oval(70,20)}
\put(174,15){\line(1,0){20}}
\put(198,12){\tiny{$\frac1{3/k_2}(1,{\bf -b_2})$}}
\put(221,15){\oval(54,20)}
\put(248,15){\line(1,0){20}}
\put(274,15){\circle{12}}
\put(271,13){\tiny{$f_2$}}
\put(280,15){\line(1,0){20}}
\put(306,12){\tiny{$\frac1{3/k_2}(1,b_2)$}}
\put(325,15){\oval(50,20)}
\put(173,25){\line(1,1){22}}
\put(198,49){\tiny{$\frac1{3/k_2}(1,{\bf -b_2})$}}
\put(221,52){\oval(54,20)}
\put(248,52){\line(1,0){20}}
\put(274,52){\circle{12}}
\put(271,50){\tiny{$f_3$}}
\put(280,52){\line(1,0){20}}
\put(306,49){\tiny{$\frac1{3/k_2}(1,b_2)$}}
\put(325,52){\oval(50,20)}
\end{picture}
\end{center}

and
$$
\Diff_E(0)=\frac{k_1-1}{k_1}f_1+\frac{k_2-1}{k_2}f_2+\frac{k_2-1}{k_2}f_3+
\frac{\alpha_1-1}{\alpha_1}\widetilde E_0.
$$



4) {\it Type} $\EEE_7$. Let $k_1=(4,2\alpha_1+\alpha_2)$, $k_2=(2,\alpha_2)$ and
$k_3=(3,\alpha_2)$.
Put $b_1=(2\alpha_1+\alpha_2)/k_1$, $b_i=\alpha_2/k_i$ for $i=2,3$. Then
\\
\begin{center}
\begin{picture}(350,65)(0,0)
\put(160,10){\fbox{$\widetilde\Gamma$}}
\put(160,15){\line(-1,0){20}}
\put(78,12){\tiny{$\frac1{4/k_1}(1,{\bf -b_1})$}}
\put(103,15){\oval(74,20)}
\put(66,15){\line(-1,0){20}}
\put(40,15){\circle{12}}
\put(37,13){\tiny{$f_1$}}
\put(40,21){\line(0,1){20}}
\put(22,49){\tiny{$\frac1{4/k_1}(1,b_1)$}}
\put(40,51){\oval(70,20)}
\put(174,15){\line(1,0){20}}
\put(198,12){\tiny{$\frac1{2/k_2}(1,{\bf -b_2})$}}
\put(221,15){\oval(54,20)}
\put(248,15){\line(1,0){20}}
\put(274,15){\circle{12}}
\put(271,13){\tiny{$f_2$}}
\put(280,15){\line(1,0){20}}
\put(306,12){\tiny{$\frac1{2/k_2}(1,b_2)$}}
\put(325,15){\oval(50,20)}
\put(173,25){\line(1,1){22}}
\put(198,49){\tiny{$\frac1{3/k_3}(1,{\bf -b_3})$}}
\put(221,52){\oval(54,20)}
\put(248,52){\line(1,0){20}}
\put(274,52){\circle{12}}
\put(271,50){\tiny{$f_3$}}
\put(280,52){\line(1,0){20}}
\put(306,49){\tiny{$\frac1{3/k_3}(1,b_3)$}}
\put(325,52){\oval(50,20)}
\end{picture}
\end{center}

and
$$
\Diff_E(0)=\frac{k_1-1}{k_1}f_1+\frac{k_2-1}{k_2}f_2+\frac{k_3-1}{k_3}f_3+
\frac{\alpha_1-1}{\alpha_1}\widetilde E_0.
$$

5) {\it Type} $\EEE_8$. Let $k_1=(5,\alpha_2)$, $k_2=(2,\alpha_2)$ and $k_3=(3,\alpha_2)$.
Put $b_i=\alpha_2/k_i$. Then
\\
\begin{center}
\begin{picture}(350,65)(0,0)
\put(160,10){\fbox{$\widetilde\Gamma$}}
\put(160,15){\line(-1,0){20}}
\put(78,12){\tiny{$\frac1{5/k_1}(1,{\bf -b_1})$}}
\put(103,15){\oval(74,20)}
\put(66,15){\line(-1,0){20}}
\put(40,15){\circle{12}}
\put(37,13){\tiny{$f_1$}}
\put(40,21){\line(0,1){20}}
\put(22,49){\tiny{$\frac1{5/k_1}(1,b_1)$}}
\put(40,51){\oval(70,20)}
\put(174,15){\line(1,0){20}}
\put(198,12){\tiny{$\frac1{2/k_2}(1,{\bf -b_2})$}}
\put(221,15){\oval(54,20)}
\put(248,15){\line(1,0){20}}
\put(274,15){\circle{12}}
\put(271,13){\tiny{$f_2$}}
\put(280,15){\line(1,0){20}}
\put(306,12){\tiny{$\frac1{2/k_2}(1,b_2)$}}
\put(325,15){\oval(50,20)}
\put(173,25){\line(1,1){22}}
\put(198,49){\tiny{$\frac1{3/k_3}(1,{\bf -b_3})$}}
\put(221,52){\oval(54,20)}
\put(248,52){\line(1,0){20}}
\put(274,52){\circle{12}}
\put(271,50){\tiny{$f_3$}}
\put(280,52){\line(1,0){20}}
\put(306,49){\tiny{$\frac1{3/k_3}(1,b_3)$}}
\put(325,52){\oval(50,20)}
\end{picture}
\end{center}

and
$$
\Diff_E(0)=\frac{k_1-1}{k_1}f_1+\frac{k_2-1}{k_2}f_2+\frac{k_3-1}{k_3}f_3+
\frac{\alpha_1-1}{\alpha_1}\widetilde E_0.
$$

\subs \label{calcul}
Let us illustrate the calculations in case 1).
The other cases are analyzed similarly.

The curve $\Gamma$ passes through two different singular points $P_1$ and
$P_2$ of $Z$. Consider the first singularity
$\frac1{a_3d_1}(-1,-a_2d_1,1)$ at a point $P_1$.
Since the curve $\Gamma$ is a toric subvariety near $P_1$ we can use a toric geometry.

We refer the reader to \cite{Oda} for the basics
of a toric geometry.
Let $N\cong\ZZ^n$ be a lattice of rank $n$ and
$M=\Hom(N,\ZZ)$ its dual lattice.
For a fan $\Delta$ in $N$ the corresponding
toric variety is denoted by $T_N(\Delta)$.
For a $k$-dimensional cone $\sigma\in\Delta$ the
closure of corresponding orbit is denoted by
$V(\sigma)$. This is a closed subvariety of
codimension $k$ in  $T_N(\Delta)$.

At the point $P_1$ we have $Z=T_N(\Delta)$, where $\Delta=\{\langle e_1,e_2,e_3\rangle,\
\text{their faces}\}$ and $e_1=(1,0,0)$, $e_2=(0,1,0)$, $e_3=(\frac{a_2+a_3}k,a_2d_1,a_3d_1)$.
Note that $S=V(\langle e_3 \rangle)$, $\Gamma=V(\langle e_2, e_3 \rangle)$.
Then $\widetilde Y=T_N(\Delta')$, where
$$
\Delta'=\{\langle e_4,e_1,e_2 \rangle, \langle e_4,e_1,e_3 \rangle,\ \text{their faces}\}
$$
and
$$
e_4=\alpha_2e_2+\alpha_1e_3=\Big(\alpha_1\frac{a_2+a_3}k,\alpha_1a_2d_1+\alpha_2,
\alpha_1a_3d_1\Big).
$$
Note that $V(\langle e_4, e_1 \rangle)$ is a fiber of $\widetilde E$ over $P_1$.
By considering the cones $\langle e_4,e_1,e_2 \rangle$ and
$\langle e_4,e_1,e_3 \rangle$, it is easy to prove the requirement.
For example, it is enough to consider the first cone only. If we obtain a singularity of type
$\frac1r(\gamma_1,\gamma_2)$ of $\widetilde E$ then the another singularity of
$\widetilde E$ (over $P_1$) has a type $\frac1r(\gamma_1,-\gamma_2)$ automatically.

\subs\label{ex2} Let $(X\ni P)=(\CC^3\ni 0)/\ZZ_r(1,-1,q)$, $(r,q)=1$.
According to item \ref{part2} our problem is reduced to the case of non-singular point.
We obtain that the log surfaces $(E,\Diff_E(0))$ have the same structure as in the case of
non-singular point.
The condition on weights $(\beta_1,\beta_2,\beta_3)$ is that
$\frac1r(\beta_1,\beta_2,\beta_3)$ is a terminal singularity for some
$r$. This directly implies that we have the finite number of possibilities of $r$
for all types
$\EEE_i$.


As in item \ref{calcul} let us demonstrate the calculation of
$(E,\Diff_E(0))$.
We have $X=T_N(\Delta_X)$, where
$\Delta_X=\{\langle e_1, e_2, e_3\rangle, \ \text{their faces}\}$
and $e_1=(1,0,0)$, $e_2=(0,1,0)$, $e_3=(1,q,r)$. Also we have
$Z=T_N(\Delta_Z)$, where
$$
\Delta_Z=\{\langle e_1,e_2,b \rangle, \langle e_1,e_3,b \rangle,\langle e_2,e_3,b \rangle,
\ \text{their faces}\}
$$
and $b=(b_1,b_2,b_3)$ is a vector lying inside of a cone
$\langle e_1, e_2, e_3\rangle$.

For instance, let us see what takes place at the points
$P_1=V(\langle e_1,e_2,b \rangle)$ and
$P_2=V(\langle e_1,e_3,b \rangle)$ if we take a blow-up of
$Z$ along the curves
$C_1=V(\langle e_2,b \rangle)$ and $C_2=V(\langle e_3,b \rangle)$
respectively. For example, in case $\AAA_n$ the curve $\Gamma$ is locally isomorphic to
$C_i$ at the points $P_i$ respectively.

At a point $P_1$ we have $\widetilde Y=T_N(\Delta')$, where
$$
\Delta'=\{\langle e_4,e_1,e_2 \rangle, \langle e_4,e_1,b \rangle,\ \text{their faces}\}
$$
and $e_4=a_2e_2+a_1b=(a_1b_1,a_1b_2+a_2,a_1b_3)$.
Hence we obtain the following type of a structure of $\widetilde E$ over a point
$P_1$:
\\
\begin{center}
\begin{picture}(350,30)(0,0)
\put(270,10){\fbox{$\widetilde\Gamma$}}
\put(270,15){\line(-1,0){20}}
\put(184,12){\tiny{$\frac1{\gamma_3}(\gamma_1,{\bf -\gamma_2})$}}
\put(208,15){\oval(84,20)}
\put(166,15){\line(-1,0){20}}
\put(140,15){\circle{12}}
\put(137,13){\tiny{$f_1$}}
\put(134,15){\line(-1,0){20}}
\put(50,12){\tiny{$\frac1{\gamma_3}(\gamma_1,\gamma_2)$}}
\put(72,15){\oval(84,20)}
\end{picture}
\end{center}


At a point $P_2$ we similarly obtain that a singularity of $\widetilde Y$ outside
$\widetilde\Gamma$ is given by a cone
$\langle e_1, e_3, a_2e_3+a_1b \rangle$. Let $\theta_1q+\theta_2r=1$, where
$\theta_1$, $\theta_2\in\ZZ$. Then the following matrix
$$
M=
\begin{pmatrix}
 1  & -\theta_1 & -\theta_2\\
 0  & \theta_1 & \theta_2\\
 0  & r & -q \\
\end{pmatrix}
$$
with integer elements and determinant
$-1$ maps our cone in
$$
\langle (1,0,0), (0,1,0), (t_1,t_2,t_3)\rangle,
$$
where
\begin{gather*}
(t_1,t_2,t_3)= M(a_2e_3+a_1b)= a_2(0,1,0)+a_1Mb=\\
(a_1(b_1-b_2\theta_1-b_3\theta_2), a_1(b_2\theta_1+b_3\theta_2)+a_2,
a_1(rb_2-qb_3)).
\end{gather*}

Hence the structure of $\widetilde E$ over a point
$P_2$ is similar to one as in the case of a point $P_1$.

\subs\label{ex3} Let $(X\ni P)=(x_1x_2+x_3x_4=0\subset(\CC^4,0))$ and the toric
divisorial contraction
$g\colon Z\to (X\ni P)$ is induced by the weighted blow-up of
$\CC^4$ with weights
$(\beta_1,\beta_2,\beta_3,\beta_4)=(a_1,a_1kd_{13}d_{14},a_3d_{14},a_4d_{13})$, where
$\beta_1+\beta_2=\beta_3+\beta_4$,
$\Exc g=S$. Consider a non-singular curve
$\Gamma\sim\OO_{\PP(\beta_1,\beta_2,\beta_3,\beta_4)}(\beta_2)|_S$ on a surface
$S=(x_1x_2+x_3x_4\subset\PP(\beta_1,\beta_2,\beta_3,\beta_4))$, which is not
a toric subvariety
of $Z$.
Let $h\colon\widetilde Y\to Z$ be a blow-up of $Z$ along a curve $\Gamma$,
$\Exc h=\widetilde E$ is an irreducible divisor. We assume that
$h$ is a toric morphism near a
generic point of $\Gamma$.
In detail,
a generic point $Q$ of $\Gamma$ lies in a smooth locus of $Z$. Let $H$ be a general hyperplane
section passing through $Q$. Then $h$ induces a purely log terminal blow-up
$h|_{\widetilde H}\colon\widetilde H\to H$, where $\widetilde H$ is a proper transform of $H$.
The morphism
$h|_{\widetilde H}$ is a weighted blow-up with weights $(\alpha_1,\alpha_2)$.
To be definite, assume that $h^*S=\widetilde S+\alpha_1\widetilde E$.

Let $\widetilde Y\to Y$ be the divisorial contraction of $\widetilde S$ to a point, where
$\widetilde S$ is the proper transform of $S$.
We obtain a purely log terminal blow-up
$f\colon Y\to (X\ni P)$, where $\Exc f=E$.
The morphism $f$ is non-toric.
The existence of construction is proved similarly as in the case of non-singular point,
where

$$
D=\sum_{i=1}^2\frac12\{\varphi(x_1,x_2,x_3,x_4)+\psi_i(x_1,x_2,x_3,x_4)=0\}|_X.
$$

A quasihomogeneous polynomial
$\varphi$ is $x_1^{\beta_2/\beta_1}+x_2+\cdots$.
The curve $\Gamma$ is similarly determined from the equality
$\Gamma=D_Z|_S=\widetilde{\{\varphi=0\}}|_S$.

The following lemma is proved similarly to lemma
\ref{minimal}.
\begin{lemma}
We have
$$
(\widetilde\Gamma^2)_{\widetilde E}=-\alpha_2\frac1{\beta_1}
\Big(\frac1{\beta_3}+\frac1{\beta_4}\Big)-\alpha_1\frac{\beta_2}{\beta_1}
\Big(\frac1{\beta_3}+\frac1{\beta_4}\Big).
$$
\end{lemma}

\subs\label{ex3end}
Using the toric calculations as above-mentioned we obtain the answer.
Let $k_1=(\beta_3,\alpha_1\beta_2+\alpha_2)$ and
$k_2=(\beta_4,\alpha_1\beta_2+\alpha_2)$.
Put $b_i=(\alpha_1\beta_2+\alpha_2)/k_i$. Then
\\
\begin{center}
\begin{picture}(350,60)(0,0)
\put(170,10){\fbox{$\widetilde\Gamma$}}
\put(170,15){\line(-1,0){20}}
\put(80,12){\tiny{$\frac1{\beta_3/k_1}(\beta_1,$ {\boldmath $-b_1$}$)$}}
\put(108,15){\oval(84,20)}
\put(66,15){\line(-1,0){20}}
\put(40,15){\circle{12}}
\put(37,13){\tiny{$f_1$}}
\put(40,21){\line(0,1){20}}
\put(22,49){\tiny{$\frac1{\beta_3/k_1}(\beta_1,b_1)$}}
\put(46,51){\oval(70,20)}
\put(184,15){\line(1,0){20}}
\put(218,12){\tiny{$\frac1{\beta_4/k_2}(\beta_1,$ {\boldmath $ -b_2$}$)$}}
\put(246,15){\oval(84,20)}
\put(288,15){\line(1,0){20}}
\put(314,15){\circle{12}}
\put(311,13){\tiny{$f_2$}}
\put(314,21){\line(0,1){20}}
\put(290,49){\tiny{$\frac1{\beta_4/k_2}(\beta_1,b_2)$}}
\put(312,51){\oval(70,20)}
\end{picture}
\end{center}

and
$$
\Diff_E(0)=\frac{k_1-1}{k_1}f_1+\frac{k_2-1}{k_2}f_2+\frac{\alpha_1-1}{\alpha_1}\widetilde E_0.
$$

Note that the log surface $(E,\Diff_E(0))$ is toric.



\section{\bf Main theorem}
\begin{theorem}
Let $f\colon (Y,E)\to (X\ni P)$ be a purely log terminal blow-up, where
$(X\ni P)$ is a three-dimensional terminal toric singularity
and $f(E)=P$. Then either $f$ is a toric morphism up to analytic isomorphism
$(X\ni P)\cong (X\ni P)$, or $f$ is described in section
$\ref{example}$.
\begin{proof}
Let $f$ be not a toric morphism for any analytic isomorphism
$(X\ni P)\cong (X\ni P)$. Then there exists a toric divisorial contraction
$g\colon Z\to X$ with the following properties:

\begin{enumerate}
\item[I)] $\Exc g=S$ is an irreducible divisor;
\item[II)] $\Gamma$ is not a toric subvariety of $Z$ for any analytic isomorphism
$(X\ni P)\cong (X\ni P)$, where $\Gamma$ is the center of a divisor
$E$ on $Z$;
\item[III)] by proposition \ref{lemma2} we may assume that $-S$ is $g$-ample divisor.
\end{enumerate}

Let $D_Y\in |-nK_Y|$ be a general divisor for $n\gg 0$. Then the pair $(X,\frac1nD)$
is log canonical and
$a(E,\frac1nD)=-1$, where $D=f(D_Y)$.
Moreover, if $a(E',\frac1nD)=-1$, then $E$ and $E'$
define the same discrete valuation of the function field
$K(X)$. Hence,
$a(E,S+\frac1nD_Z)\le -1$ and $\Gamma\subset\LCS(S,\Diff_S(\frac1nD_Z))$, where
$D_Z$ is the proper transform of a divisor $D$ on $Z$.

\begin{lemma}
The set $\Gamma$ is a non-singular curve.
\begin{proof}
If $\Gamma$ is a point then the proof of \cite[Lemma 3.8]{Kud4}
implies that
$a(S,\frac1nD)\le -1$. A contradiction. Hence by proposition \ref{normal}
the set $\Gamma$ is a non-singular curve.
\end{proof}
\end{lemma}

\begin{lemma} Let $h\colon \widetilde Y\to Z$ be the inductive blow-up of a pair
$(Z,-a(S,\frac1nD)S+\frac1nD_Z)$ $($see theorem $\ref{inductive})$,
$\Exc h=\widetilde E$ and $h(\widetilde E)=\Gamma$. Then $h$ is a toric morphism near a
generic point of $\Gamma$.
\begin{proof}
A generic point $Q$ of $\Gamma$ lies in a smooth locus of $Z$. Let $H$ be a general hyperplane
section passing through $Q$. Then $h$ induces a purely log terminal blow-up
$h|_{\widetilde H}\colon\widetilde H\to H$, where $\widetilde H$ is a proper transform of $H$.
By example \ref{two-dim} the morphism $h|_{\widetilde H}$ is toric. Moreover,
$h|_{\widetilde H}$ is a weighted blow-up.
\end{proof}
\end{lemma}


\begin{lemma}\label{curve} $1)$ Let $(X\ni P)$ be a $\QQ$-factorial point.
Then the divisor
$K_S+\Diff_S(0)+\Gamma$ is purely log terminal and anti-ample.
\\
$2)$ Let $(X\ni P)$ be not a $\QQ$-factorial point.
Then there is an effective $\QQ$-divisor $\Gamma'$ such that
the divisor $K_S+\Diff_S(0)+\Gamma+\Gamma'$ is purely log terminal and anti-ample.
\begin{proof}
Let us consider the first part of lemma.
Assume that this divisor is not purely log terminal at a point
$P\in \Gamma$.
Let $h\colon \widetilde Y\to Z$ be the inductive blow-up of a pair
$(Z,-a(S,\frac1nD)S+\frac1nD_Z)$
and $\Exc h=\widetilde E$.
Then $h|_{\widetilde E}\colon\widetilde E\to h(\widetilde E)=\Gamma$ is a conic bundle.

Consider a general Weil divisor
$\Theta$ on $X$ such that
$\Theta_Z|_S=\Gamma$, where $\Theta_Z$ is the proper transform of a divisor $\Theta$.
Since a birational morphism $g$ is toric, then it is easy to prove the existence of
$\Theta$.
The divisor $K_Z+(1-\delta)S+\Theta_Z+\varepsilon H_Z$ is purely log terminal
outside the point
$P$ and strictly log canonical at the point $P$ for some $\delta>0$, $0<\varepsilon\ll 1$,
where $H_Z$ is a general hyperplane section passing through the point $P$.
Let
$$
T_Z\in |-m(K_Z+(1-\delta)S+\Theta_Z+\varepsilon H_Z)|
$$
be a general divisor of very ample linear system
$(m\gg 0)$ and
$$
K_{\widetilde Y}+\widetilde T=h^*g^*\Big(K_X+\Theta+\varepsilon H+\frac1m T\Big),
$$
where $H=g(H_Z)$ and $T=g(T_Z)$.

\begin{lemma}
We have $f\subset\LCS(\widetilde Y, \widetilde S+\widetilde\Theta_Z)$, where $f$ is a curve
lying in the fiber of a morphism $h$ over $P$ and
$\widetilde S$, $\widetilde\Theta_Z$ are the proper transforms of $S$, $\Theta$.
\begin{proof} Assume the converse.
By connectedness lemma
\cite[Theorem 17.4]{Koetal} applied to a divisor $K_{\widetilde Y}+\widetilde T$ and
a contraction $\widetilde Y\to X$, we have
$\LCS(\widetilde Y, \widetilde T)=\widetilde\Theta_Z$.
Let $Q=\LCS(\widetilde Y, \widetilde T)\cap h^{-1}(P)$.
By assumption a point $Q$ is a minimal log canonical center of $(\widetilde Y,\widetilde T)$.
The corresponding divisor with discrepancy
$-1$ and with a center at the point $Q$ is denoted by $E'$.

In the same way let us consider the (anti-ample over $X$)
divisor $K_Z+S+(1-\delta')\Theta_Z+\varepsilon' H'_Z$,
which is purely log terminal outside $P$ and $a(E',S+(1-\delta')\Theta_Z+\varepsilon' H'_Z)=-1$.
As before we write
$$
K_{\widetilde Y}+\widetilde T'=h^*g^*\Big(K_X+(1-\delta')\Theta+\varepsilon' H'+\frac1m T'\Big).
$$

By the same argument $\LCS(\widetilde Y, \widetilde T')=\widetilde S\not\ni Q$. A contradiction.
\end{proof}
\end{lemma}

Therefore the divisor $K_{\widetilde Y}+\widetilde E$ is not purely log terminal.
Hence the divisor $K_Y+E$ is not purely log terminal.
A contradiction.

Let us consider the second part of lemma.
If the divisor $\Theta$ is a $\QQ$-Cartier then $\Gamma'=0$ and the lemma
is proved similarly to the
$\QQ$-factorial case.
If the divisor $\Theta$ is not $\QQ$-Cartier then we take a general
$\QQ$-Weil divisor $\Theta'$ such that $\Theta+\Theta'$ is a $\QQ$-Cartier divisor.
By the same argument some curve $f$ in the fiber of $h$ over $P$
lies in the corresponding $\LCS$.
Since the divisor $K_{\widetilde Y}+\widetilde E+\Theta'_{\widetilde Y}$ is not purely log
terminal along $f$
for every $\Theta'$ then the divisor
$K_{\widetilde Y}+\widetilde E+\frac1nD_{\widetilde Y}$ is not purely log
terminal along $f$. Hence
the divisor $K_Y+E+\frac1nD_Y$ is not purely log
terminal.
A contradiction.
\end{proof}
\end{lemma}

According to \cite[Proposition 1.3]{Kud4}
let us consider
three cases in items
\ref{part1}, \ref{part2} and \ref{part3} respectively.

\subs \label{part1}
Let $(X\ni P)$ be a non-singular point. Then $g$ is the weighted blow-up with weights
$(\beta_1,\beta_2,\beta_3)=(a_1d_2d_3,a_2d_1d_3,a_3d_1d_2)$,
$d_i=(\beta_j,\beta_k)$, where $j,k\ne i$. We have
$$
\Big(S,\Diff_S(0)\Big)=\Big(\PP_{x_1,x_2,x_3}(a_1,a_2,a_3),
\sum_{i=1}^3\frac{d_i-1}{d_i}\{x_i=0\}\Big).
$$


\begin{lemma}\label{smooth}
Let a curve $\Gamma$ be given by a $($quasihomogeneous$)$ polynomial
$\sum b_lx_1^{l_1}x_2^{l_2}x_3^{l_3}=0$ on $S=\PP_{x_1,x_2,x_3}(a_1,a_2,a_3)$.
Then a quasihomogeneous polynomial
$\varphi=\sum b_lx_1^{d_1l_1}x_2^{d_2l_2}x_3^{d_3l_3}$ with weights $(\beta_1,\beta_2,\beta_3)$
defines a Du Val singularity
in $\CC^3_{x_1,x_2,x_3}$ and we have one of the following cases
$($up to change of variables$)$, which are
considered in section
$\ref{example}:$
\\
$\AAA_n)$ $(\beta_1,\beta_2,\beta_3)=(\frac{a_2+a_3}k,a_2d_1,a_3d_1),$ where
$k\mid (a_2+a_3)$ and $\Gamma\sim\OO_{\PP\big(\frac{a_2+a_3}k,a_2,a_3\big)}(a_2+a_3);$
\\
$\DDD_{2k+2})$ $(\beta_1,\beta_2,\beta_3)=(2,2k,2k+1),$\
$\Gamma\sim\OO_{\PP(1,k,2k+1)}(2k+1), k\ge 1;$
\\
$\DDD_{2k+1})$ $(\beta_1,\beta_2,\beta_3)=(2,2k-1,2k),$\
$\Gamma\sim\OO_{\PP(1,2k-1,k)}(2k), k\ge 2;$
\\
$\EEE_6)$ $(\beta_1,\beta_2,\beta_3)=(3,4,6),$\
$\Gamma\sim\OO_{\PP(1,2,1)}(2);$
\\
$\EEE_7)$ $(\beta_1,\beta_2,\beta_3)=(4,6,9),$\
$\Gamma\sim\OO_{\PP(2,1,3)}(3);$
\\
$\EEE_8)$ $(\beta_1,\beta_2,\beta_3)=(6,10,15),$\
$\Gamma\sim\OO_{\PP^2}(1).$

\begin{proof}
By the construction of a curve $\Gamma$ we have $\deg x^m\ge 2$ for every $x^m\in \varphi$.
It is enough to prove that $\varphi$ defines a Du Val singularity.
Assume the converse. Write $\frac1nD=\frac1t\{\varphi^t+\psi=0\}+D'$,
where $D'\ne 0$ is an effective divisor.
We can assume without loss of generality that one of the following three cases takes place.

{\it Case} 1. Let $\varphi=x_1^2+x_2^2x_3$. Then
$(\beta_1,\beta_2,\beta_3)=(2l+1,2l+1-2k,4k)$, $(2l+1,2l-2k,4k+2)$ or
$(2l,2l-2k-1,4k+2)$. The first possibility is not considered since the curve
$\Gamma$ is singular. In the remaining possibilities the curve
$\Gamma$ is tangent to
a curve $C$, where $\Diff_S(0)=\frac12C$. A contradiction with lemma
\ref{curve}.

{\it Case} 2. Let $\varphi$ does not depend on a variable $x_3$.
Then $\varphi(x_1,x_2)=\gamma_1x_1^{r\beta_2}+\gamma_2x_2^{r\beta_1}+\ldots$, where
$\gamma_i\ne 0$. Similarly to case
1 we have a  contradiction with lemma \ref{curve}.

{\it Case} 3. Let $\varphi$ depends on the variables $x_1$, $x_2$, $x_3$ and
${\bf 1}\notin \Gamma(\varphi)^0$, where $\Gamma(\varphi)^0$ denotes the interior of
Newton's polyhedron. Then we have a  contradiction
$a(S,\frac1nD) < a(S,\{\varphi=0\})\le -1$.
\end{proof}
\end{lemma}


\subs \label{part2}
Let $(X\ni P)\cong (\CC^3\ni 0)/\ZZ_r(1,r-q,q)$.
We can apply the same arguments as in item
\ref{part1}, but it is better to reduce the problem to the case of non-singular point.
Let $(X'\ni P')\to (X\ni P)$ be a natural cover
and $Y'$ be a normalization of $Y$ in the function field $K(X')$.
Then the induced divisorial contraction
$(Y',E')\to (X'\ni P')$ is purely log terminal blow-up
\cite[Proposition 20.3]{Koetal}.


Let us remark that the argument above-mentioned gives the classification of
purely log terminal blow-ups in the case, when
$(X\ni P)$ is a three-dimensional $\QQ$-factorial toric singularity
(that is, $(X\ni P)=(\CC^n\ni 0)/G$,
where $G$ is an abelian group acting freely in codimension 1).

\subs \label{part3}
Let $(X\ni P)\cong \Spec \CC[x_1,x_2,x_3,x_4]/(x_1x_2+x_3x_4)$.
It is clear that the morphism $g$ is induced by the weighted blow-up of $\CC^4$ with weights
$(\beta_1,\beta_2,\beta_3,\beta_4)$, where $\beta_1+\beta_2=\beta_3+\beta_4$ and
$(X\ni P)=(x_1x_2+x_3x_4=0\subset(\CC^4,0))$.
In particular, $(\beta_i,\beta_j,\beta_k)=1$ for all mutually distinct $i,j,k$.
Put $(\beta_1, \beta_2, \beta_3, \beta_4)=(a_1d_{23}d_{24}, a_2d_{13}d_{14},
a_3d_{14}d_{24}, a_4d_{13}d_{23})$, where $d_{ij}=(\beta_k,\beta_l)$ ($i,j,k,l$ are mutually
distinct numbers). Then
$$
\Big(S,\Diff_S(0)\Big)=\Big(\PP_{x_1,x_2,x_3,x_4}(\beta_1,\beta_2,\beta_3,\beta_4),
\sum_{i<j}\frac{d_{ij}-1}{d_{ij}}\{x_i=x_j=0\}\Big).
$$
The details of calculation can be found in the paper
\cite{Kud3}.

\begin{lemma}
Up to permutation of coordinates we have
 $d_{23}=d_{24}=1$, $a_1|a_2$ and
$\Gamma\sim\OO_{\PP(\beta_1,\beta_2,\beta_3,\beta_4)}(\beta_2)|_S$.
In particular,
$(\beta_1,\beta_2,\beta_3,\beta_4)=(a_1,a_1kd_{13}d_{14},a_3d_{14},a_4d_{13})$.
This case is
considered in section $\ref{example}$.
\begin{proof}
Note that $\rho(S)=2$ since a complement to
the open toric orbit of $S$ consists of four irreducible curves $C_i$.
Every Weil divisor of $S$ is linearly equivalent to some sum of $C_i$
with integer coefficients.
Therefore $\Gamma\sim C_1+C_2=\{x_i=0\}|_S$ for some $i$ by lemma
\ref{curve}. We can assume that $i=2$. Since the curve $\Gamma$ is irreducible then
$\Gamma=\{x_2+\gamma x_1^{\beta}+\ldots=0\}|_S$, where $\gamma\ne 0$.
Hence $\beta_1|\beta_2$.
\end{proof}
\end{lemma}


\end{proof}
\end{theorem}

\begin{thebibliography}{99}
\bibitem{KMM}
\emph{Kawamata  Y.,Matsuda  K.,Matsuki  K.} Introduction  to  the
minimal model program //  Algebraic Geometry, Sendai. Adv. Stud.
Pure Math. 1987. V. 10. P. 283--360.

\bibitem{Kaw}
\emph{Kawamata Y.} On Fujita's freeness conjecture for 3-folds and 4-folds //
Math. Ann. 1997. V. 308. P. 491-505.

\bibitem{Koetal}
\emph{Kollar J. et al} Flips and abundance for algebraic
threefolds // Ast\'erisque 1992. V. 211.

\bibitem{Kud1}
\emph{Kudryavtsev S.~A.} Purely log terminal blow-ups // Math.
Note. 2001. V. 69. No. 6. P. 814--819.

\bibitem{Kud2}
\emph{Kudryavtsev S.~A.} Three-dimensional terminal contractions // preprint of
Max-Planck-Institut. MPI 2003-114.

\bibitem{Kud3}
\emph{Kudryavtsev S.~A.}
Classification of three-dimensional exceptional log canonical hypersurface singularities.
{\small II} // Izv. Math. 2004. V. 68. No. 2.

\bibitem{Kud4}
\emph{Kudryavtsev S.~A.} Divisorial contractions of 3-folds //
e-print math.AG/0403382.

\bibitem{Oda}
\emph{Oda T.} Convex bodies and algebraic geometry. An introduction to
the theory of toric varieties // Springer-Verlag 1988.

\bibitem{Pr2}
\emph{Prokhorov Yu.~G.} Blow-ups of canonical singularities //
Algebra. Proc. Internat. Conf. on the Occasion of the 90th
birthday of A.~G.~Kurosh, Moscow, Russia, May 25-30, 1998 /
Yu.~Bahturin ed., Walter de Gruyter, Berlin, 2000. P. 301-317.

\bibitem{PrEll}
\emph{Prokhorov Yu.~G.} Classification of Mori contractions: the case of an elliptic curve //
Izvestiya Math. 2001. V. 65. No. 1. P. 75-84.

\bibitem{PrSh}
\emph{Prokhorov Yu.~G., Shokurov V.~V.} The first main theorem on complements:
from global to local //
Izvestiya Math. 2001. V. 65. No. 6. P. 1169-1196.

\bibitem{Sh2}
\emph{Shokurov V.~V.} Complements on surfaces // J. of Math. Sci.
2000. V. 102. no. 2. P. 3876--3932.

\end{thebibliography}

\end{document}

math0404260.extracted_bib
\begin{thebibliography}{99}
\bibitem{KMM}
\emph{Kawamata  Y.,Matsuda  K.,Matsuki  K.} Introduction  to  the
minimal model program //  Algebraic Geometry, Sendai. Adv. Stud.
Pure Math. 1987. V. 10. P. 283--360.

\bibitem{Kaw}
\emph{Kawamata Y.} On Fujita's freeness conjecture for 3-folds and 4-folds //
Math. Ann. 1997. V. 308. P. 491-505.

\bibitem{Koetal}
\emph{Kollar J. et al} Flips and abundance for algebraic
threefolds // Ast\'erisque 1992. V. 211.

\bibitem{Kud1}
\emph{Kudryavtsev S.~A.} Purely log terminal blow-ups // Math.
Note. 2001. V. 69. No. 6. P. 814--819.

\bibitem{Kud2}
\emph{Kudryavtsev S.~A.} Three-dimensional terminal contractions // preprint of
Max-Planck-Institut. MPI 2003-114.

\bibitem{Kud3}
\emph{Kudryavtsev S.~A.}
Classification of three-dimensional exceptional log canonical hypersurface singularities.
{\small II} // Izv. Math. 2004. V. 68. No. 2.

\bibitem{Kud4}
\emph{Kudryavtsev S.~A.} Divisorial contractions of 3-folds //
e-print math.AG/0403382.

\bibitem{Oda}
\emph{Oda T.} Convex bodies and algebraic geometry. An introduction to
the theory of toric varieties // Springer-Verlag 1988.

\bibitem{Pr2}
\emph{Prokhorov Yu.~G.} Blow-ups of canonical singularities //
Algebra. Proc. Internat. Conf. on the Occasion of the 90th
birthday of A.~G.~Kurosh, Moscow, Russia, May 25-30, 1998 /
Yu.~Bahturin ed., Walter de Gruyter, Berlin, 2000. P. 301-317.

\bibitem{PrEll}
\emph{Prokhorov Yu.~G.} Classification of Mori contractions: the case of an elliptic curve //
Izvestiya Math. 2001. V. 65. No. 1. P. 75-84.

\bibitem{PrSh}
\emph{Prokhorov Yu.~G., Shokurov V.~V.} The first main theorem on complements:
from global to local //
Izvestiya Math. 2001. V. 65. No. 6. P. 1169-1196.

\bibitem{Sh2}
\emph{Shokurov V.~V.} Complements on surfaces // J. of Math. Sci.
2000. V. 102. no. 2. P. 3876--3932.

\end{thebibliography}