math0207156.tex
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\begin{document}
\begin{center} {\bf FREE $\C_+$-ACTIONS ON $\C^3$ ARE TRANSLATIONS}
\\[3ex] {\bf SHULIM KALIMAN} \footnote{ The author was partially
supported by the NSA grant MDA904-00-1-0016.}\\[3ex] Department of
Mathematics, University of Miami \\ Coral Gables, Florida 33124,
USA \\ (e-mail: kaliman@math.miami.edu )\\[5ex]
\end{center}
ABSTRACT. {\small Let $X'$ be a smooth contractible
three-dimensional affine algebraic variety with a free algebraic
$\C_+$-action on it such that $S =X'//\C_+$ is smooth. We prove
that $X'$ is isomorphic to $S \times \C$ and the action is induced
by a translation on the second factor. As a consequence we show
that any free algebraic $\C_+$-action on $\C^3$ is a translation
in a suitable coordinate system.} \\[3ex]

{\bf 1. Introduction.} In 1968 Rentschler [Re] proved that every
algebraic action of the additive group $\C_+$ of complex numbers
on $\C^2$ is triangular in a suitable polynomial coordinate
system. This implies that a free $\C_+$-action on $\C^2$ (i.e. an
action without fixed points) can be viewed as a translation. In
1984 Bass [Ba] found a $\C_+$-action on $\C^3$ which is not
triangular in any coordinate system. But the question about free
$\C_+$-actions on $\C^3$ remained open (e.g., see [Sn] , [Kr],
[DaFr], [DeFi]). While working on this problem we consider a more
general situation when there is a nontrivial algebraic
$\C_+$-action on a complex three-dimensional affine algebraic
variety $X'$ such that its ring of regular functions is factorial
and $H_3(X')=0$. By a theorem of Zariski [Za] the algebraic
quotient $X'//\C_+$ is isomorphic to an affine surface $S$. Let
$\pi : X' \to S$ be the natural projection. Then there is a curve
$\G \subset S$ such that for $E= \pi^{-1}( \G )$ the variety $X'
\setminus E$ is isomorphic to $( S \setminus \G ) \times \C$ over
$S \setminus \G$. The study of morphism $\pi |_E : E \to \G$ is
central for this paper. As an easy consequence of the Stein
factorization one can show that $\pi |_E = \theta \circ \vartheta$
where $\vartheta : E \to Z$ is a morphism into a curve $Z$ with
general fibers isomorphic to $\C$, and $\theta : Z \to \G$ is a
quasi-finite morphism. A more delicate fact (Proposition 3.2) is
that $\theta$ is, actually, finite and, furthermore, if in
addition $X'$ is smooth and $H_2(X')=0$, then each irreducible
component $Z^1$ of $Z$, such that $\theta |_{Z^1}$ is not
injective, is rational and $\theta (Z^1)$ is a polynomial curve. A
geometrical observation (Proposition 4.2) shows that in the case
of a free $\C_+$-action and a smooth $S$ such a component $Z^1$
cannot exist. If the restriction of $\theta$ to any irreducible
component of $Z$ is injective then $\G$ can be chosen empty
(Theorem 5.4), i.e. $X'$ is isomorphic to $S \times \C$ over $S$.
In combination with  Miyanishi's theorem [Miy80] this yields the
long-expected result. \vs

\thm 1. {\em Every free algebraic $\C_+$-action on $\C^3$ is a
translation in a suitable polynomial coordinate system.} \vs
% That is, in this coordinate system there exists a vector $\bar{y} \in
%\C^3$ such that for every $\bar{x} \in \C^3$ and $t \in \C$ we
%have $\Phi_t (\bar{x}) =\bar{x} +t\bar{y}$.} \vs

{\em Acknowledgments.} It is our pleasure to thank P. Bonnet, D.
Daigle, G. Freudenburg, K.-H. Fieseler, and M. Zaidenberg for
useful consultations.\vs

\noin {\bf 2. %Affine modifications and $\C_+$-actions
Decomposition $\pi |_E = \theta \circ \vartheta$. \vs }


\lemma 2.1. {\em Let $X'$ be a factorial affine algebraic variety
of dimension 3 with a nontrivial $\C_+$-action on it and $\pi : X'
\to S$ be the quotient morphism. Then $S$ is a factorial affine
algebraic surface, $\pi^{-1}(s_0)$ is either a curve or empty for
any $s_0 \in S$, and $E=\pi^{-1}(\G )$ is a nonempty irreducible
surface for every closed irreducible curve $\G \subset S$.
Furthermore, if $\G =g^* (0)$ for a regular function $g$ on $S$
then $E =(g\circ \pi )^* (0)$.}

\proof By [Za] $S$ is an affine algebraic surface. Suppose that
$E$ is the union of a surface $E'$ and a curve $C$ where $E'$ and
$C$ may be empty. Assume $\pi (E')$ is not dense in $\G$. Consider
a rational function $h$ on $S$ with poles on $\G$ only. Let $e$ be
the product of $h$ and a regular function that does not vanish at
general points of $\G$ but vanishes on $\pi (E')$ with
sufficiently large multiplicity such that $e \circ \pi$ vanishes
on $E' \setminus C$. As $X'$ is factorial, by deleting
singularities in codimension 2 we see that $e \circ \pi$ is a
regular function on $X'$. As it is invariant under the action, $e$
must be regular. Contradiction. Thus $\pi (E')$ is dense in $\G$.
Since $X'$ is factorial $E'$ is the zero fiber of a regular
function $f$ on $X'$. As $f$ does not vanish on a general fiber
$\pi^{-1}(s) \simeq \C$ it is constant on each general fiber.
Hence $f$ is invariant under the action, i.e. $f=g \circ \pi$
where $g$ a regular function on $S$ which implies that $g$ is a
defining function for $\G$ (i.e. $S$ is factorial) and,
furthermore, $E=E' \ne \emptyset$. If for a regular function $f_1$
on $X'$ its zero fiber is an irreducible component of $E$ then
$f_1$ is again invariant under the action, i.e. $f_1=g_1 \circ
\pi$. As $g_1$ vanishes on $\G$ only it must be proportional to
$g$. Thus $E$ is irreducible and we get the last statement as
well. If we assume that $\pi^{-1}(s_0)$ is a surface for $s_0 \in
S$ the similar argument yields a regular function on $S$ that
vanishes at a finite set only which is impossible.
% As $X'$ is factorial this surface coincides with the
%zero fiber of a regular function $h$ on $X'$. As $h$ does not
%vanish on a general fiber $\pi^{-1}(s) \simeq \C$ it is constant
%on each general fiber. Hence $h$ is invariant under the action and
%may be viewed as a function on $S$. Since this function does not
%vanish anywhere but at $s_0$ we get a contradiction.
%Now (2) follows from [Miy01, Chap. 3, Th. 2.6.1].
\qed

If we have a nontrivial $\C_+$-action on a three-dimensional
affine algebraic variety $X'$ with a quotient morphism $\pi : X'
\to S=X'//\C_+$ then it is well-known that there is a closed curve
$\G \subset S$ such that for $ \breve{S}=S\setminus \G$ and $E =
\pi^{-1}(\G )$ the variety $X' \setminus E = \pi^{-1}( \breve{S})$
is naturally isomorphic to $\breve{S } \times \C$ where the
projection to the first factor corresponds to $\pi$. Let $\sigma_0
: X' \setminus E \to \breve{S} \times \C$ be an isomorphism over $
\breve{S}$, i.e. $\sigma_0 = ( \pi |_{X' \setminus E}, h_0)$ where
$h_0$ is a regular function on $X' \setminus E$. There is some
freedom in the choice of $h_0$ and, therefore, $\sigma_0$. \vs

\lemma 2.2. {\em Let $X',E, \G ,S$, and $\sigma_0 $ be as before,
$\G =g^{-1}(0)$ for a regular function $g$ on $S$, and $X = S
\times \C$. Then $\sigma_0$ can be chosen so that there exists and
affine modification $\sigma : X' \to X$ such that $\sigma |_{X'
\setminus E} = \sigma_0$.}

\proof Note that $\sigma_0$ extends to a rational map $\sigma
=(\pi , h) : X' \to X$ where $h$ is a rational function on $X'$
whose restriction to $X'\setminus E$ coincides with $h_0$. One can
replace $h_0$ by $h_0 g^n$ (where we treat $g$ as a function on
$X'$) as $\sigma_0$ remains an isomorphism after this replacement.
Choosing $n>0$ sufficiently large we can make $h$ regular on $X'$
whence $\sigma$ becomes a regular birational morphism. \qed

\lemma 2.3. {\em Let $X'$ be a factorial affine algebraic variety
of dimension 3 with a nontrivial $\C_+$-action on it, $\pi : X'
\to S$ be the quotient morphism, $E$ and $\G$ be as before Lemma
2.2.

{\rm (1)} There exist a curve $Z$, a morphism $\vartheta : E \to
Z$ with connected general fibers, and a quasi-finite morphism
$\theta : Z \to \G$ for which $\pi |_{E} = \theta  \circ
\vartheta$. Furthermore, $\theta^{-1} (\G^1)$ is irreducible for
every irreducible component $\G^1$ of $\G$.

{\rm (2)} There exists a finite set $F \subset \G$ such that for $
\G^* = \G \setminus F$, $Z^*=\theta^{-1}(\G^* )$, and $E^*
=\pi^{-1} (\G^*)$ there is an isomorphism $E^* \simeq Z^* \times
\C$ such that under this isomorphism $\vartheta |_{E^*}$ coincides
with the projection to the first factor.}

\proof Extend $\pi |_{E} : E \to \G$ to a proper morphism $ \bar{E
} \to \bar{\G }$ of complete varieties. By the Stein factorization
theorem this extended morphism is a composition of morphisms $
\bar{E} \to \bar{Z}$ and $ \bar{Z } \to \bar{\G }$ where the first
morphism has connected fibers, $ \bar{Z}$ is a curve, and the last
morphism is finite. The restriction of these morphisms is what we
need in (1). Since $E^1 =\pi^{-1}(\G^1)$ is irreducible by Lemma
2.1, we see that $\theta^{-1}(\G^1)=\vartheta (E^1)$ is also
irreducible. As $E$ is the exceptional divisor of affine
modification $\sigma$ from Lemma 2.2, (2) follows from [Ka02, Cor.
2.2] (in the case of a free $\C_+$-action it follows also from the
fact that the general fibers of $\vartheta$ are orbits of the
action, and, therefore, isomorphic to $\C$). \qed

\rem 2.4. (1) Let the center $C$ of the affine modification
$\sigma$ from Lemma 2.2 (i.e. $C$ is the closure of $\sigma (E)$)
be a curve. Note that $\theta : Z \to \G$ factors through
quasi-finite morphisms $Z \to C$ and $C \to \G$. (2) Choosing
``bigger" $F$ and
$\G$ one can always suppose that %${\rm sing} \, S \cup {\rm sing}
%\, \G \subset F$.
$\G^* , Z^*$, and $S \setminus F$ are smooth. \vs

\noin {\bf 3. Finiteness of $\theta$. } \vs

Let $S$ and $F$ be as in Lemma 2.3. In the case of $S\simeq \C^2$
there is an isomorphism $H_0(F) \simeq H_3(S \setminus F)$: namely
we assign to each $x_0 \in F$ the 3-cycle presented by a small
sphere with center at $x_0$. %For purposes which are beyond this
%paper we want to emphasize that
The same remains true in the normal case.

\lemma 3.1. {\em Let $S$ be a normal affine surface, $\G$ be a
closed curve in $S$, $ \breve{S} =S\setminus \G$, $F$ be a finite
subset of $\G$ such that $\G^* = \G \setminus F$  and
$\tS = S \setminus F$ are smooth. %Suppose that the image of $H_2 (
%\breve{S} )$ in $H_2 (\tS )$ under the homomorphism generated by
%the natural embedding, is zero (which is always true when $S
%\simeq \C^2$ as $H_2( \tS )=0$ in this case).

{\rm (i)} There exist a natural isomorphism $\phi : H_0(F) \to H_3
(\tS )$.

{\rm (ii)} Let $\varphi : H_3( \tS ) \to H_1(\G^*)$ be the
composition of the Thom isomorphism $H_3(\tS ,  \breve{S}) \to
H_1( \G^* )$ and the natural homomorphism $H_3(\tS ) \to H_3(\tS ,
\breve{S})$, let $x_0 \in F$, $\omega = \phi (x_0)$, and $\G_j, \,
j=1, \ldots , k$ be irreducible analytic branches of $\G$ at
$x_0$. Suppose that $\gamma_j$ is a simple loop in $\G_j$ around
$x_0$ with positive orientation. Then $\varphi (\omega )=
\sum_{j=1}^k \gamma_j$ (where we treat $\gamma_j$ as an element of
$H_1(\G^* )$).}

\proof As $H_3( S)=H_4(S)=0$ [Mil, Th. 7.1] % The
%exact homology sequence of pair $(\tS , \breve{S})$ produces $0
%\to H_3(\tS ) \to H_3 (\tS , \breve{S}) \to H_2 (\breve{S}) \to
%0$, and
the exact homology sequence of pair $(S, \tS )$ shows that
$H_3(\tS ) \simeq H_4(S, \tS )$. Thus in  (i) it
suffices to prove that $H_4(S, \tS ) \simeq H_0(F)$. %and this will
%yield (i) also since $H_3 (\tS , \breve{S}) \simeq H_1 (\G^*)$ by
%Thom's isomorphism.
As $S$ may not be smooth we cannot apply Thom's isomorphism. But
by the excision theorem $H_4(S,\tS ) \simeq H_4(U, U\setminus F)$
where $U$ is a small Euclidean neighborhood of $F$ in $S$, i.e.
$U$ is a disjoint union $\bigcup_i U_i$ of neighborhoods $U_i$ of
a point $x_i$ running over $F$. Hence $H_4(S,\tS ) = \oplus_i
H_4(U_i, U_i\setminus x_i)$.  Let $B_{\epsilon}$ be the ball of
radius $\epsilon$ in $\C^n \supset S$ with center at $x_0 \in F$
and $U_0 = S \cap B_{\epsilon}$. For small $\epsilon >0$, $\omega
= S \cap \de B_{\epsilon}$ is a three-dimensional real manifold
and $U_0$ is a cone over $\omega$ [Lo]. Hence $H_4(U_0 , U_0
\setminus x_0) \simeq H_3(U_0 \setminus x_0) \simeq \Z$ and $U_0$
and $\omega$ are generators of $H_4(U_0 , U_0 \setminus x_0)$ and
$H_3(U_0 \setminus x_0)$ respectively. Thus $H_4(S,\tS ) \simeq
H_0 (F)$ which is (i). For small $\epsilon >0$ the sphere $\de
B_{\epsilon}$ and, thus, $\omega$ meet each $\G_j$ transversally
along a simple loop $\gamma_j$ around $x_0$ in $\G_j$. Hence the
description of Thom's isomorphism [Do] yields $\varphi (\omega
)=\sum_{j=1}^k \gamma_j$. \qed

\prop 3.2. {\em Let the assumption and notation of Lemma 2.3 hold,
$H_3(X')=0$, and $\tX =\pi^{-1}( \tS )$ be smooth where $\tS =
S\setminus F$.

{\rm (i)} Then $\theta$ is a finite morphism.

{\rm (ii)} If $x_0$ is a selfintersection point of $\G$ then
$\theta^{-1}(x_0)$ consists of one point $z_0$.

%{\rm (iii)} Let $x_0 \in \G$, $k$ (resp. $k'$) be the number of
%analytic branches of $\G$ at $x_0$ (resp. of $Z$ at
%$\theta^{-1}(x_0)$). Suppose that $X'$ is smooth and
%$\pi^{-1}(x_0)$ consists of $n$ irreducible components. Then
%$k'=k+n-1$.

{\rm (iii)} Let $X'$ be also smooth and $H_2(X')=0$. Let $Z^1$ be
an irreducible component of $Z$ and $\G^1 = \theta (Z^1)$. If
$\theta |_{Z^1} : Z^1 \to \G^1$ is not bijective then $\G^1$ is a
polynomial curve (i.e. its normalization is $\C$) and $Z^1$ is
rational.}

%{\rm (v)} If $X'$ and $S$ are smooth and acyclic then $\theta$
%generates an isomorphism $H_0(Z) \to H_0(\G )$.}

\proof Put $ \breve{X} = X' \setminus E$ and $
\breve{S}=S\setminus \G$, i.e. $ \breve{X } \simeq \breve{S}
\times \C$. %Note that $\pi$ generates a continuous map of pairs $(
%\tX ,\breve{X}) \to (\tS , \breve{S})$ whence
We have the following commutative diagram of the exact homology
sequences of pairs.

\begin{picture}(200,95)

\put(0,65){$\dots\longrightarrow H_{j+1}(\tX, \breve{X})
\longrightarrow H_{j}(\breve{X})\longrightarrow H_{j}(\tX)
\longrightarrow H_{j}(\tX, \breve{X}) \longrightarrow
H_{j-1}(\breve{X})\longrightarrow \dots$}

\put(63,50){$\vector(0,-1){25}$} \put(137,50){$\vector(0,-1){25}$}
\put(199,50){$\vector(0,-1){25}$} \put(122,35){$\simeq$}
\put(270,50){$\vector(0,-1){25}$}
\put(342,50){$\vector(0,-1){25}$} \put(327,35){$\simeq$}

\put(5,5){$\dots\longrightarrow H_{j+1}(\tS , \breve{S})
\longrightarrow H_{j}(\breve{S})\longrightarrow \, H_{j}(\tS ) \,
\longrightarrow \, H_{j}(\tS , \breve{S}) \, \longrightarrow \,
H_{j-1}(\breve{S}) \, \longrightarrow \dots$}

\end{picture}

\noindent which we consider for $j=3$. By Remark 2.4 we can
suppose that $\G^*$, $\tS$, and $E^*$ are smooth. As $\tX
\setminus \breve{X}= E^*$ the Thom isomorphism [Do] implies $H_i(
\tX , \breve{X}) \simeq H_{i-2}(E^*)$ and, similarly, $H_i( \tS ,
\breve{S}) \simeq H_{i-2}(\G^*)$. As $H_3( \breve{S})=0$ [Mil, Th.
7.1] and $H_i(E^*)=H_i(Z^*)$  we have

\begin{picture}(200,95)

\put(10,65){$\dots \longrightarrow \, \, \, 0 \, \, \,
\longrightarrow H_{3}(\tX ) \longrightarrow
H_{1}(Z^*)\longrightarrow H_{2}(\breve{X})\longrightarrow H_2(\tX
)\longrightarrow \dots$} \put (141,
75){$\varphi'$}\put(201,75){$\psi'$}\put(263,75){$\chi'$}
\put(60,50){$\vector(0,-1){25}$} \put(109,50){$\vector(0,-1){25}$}
\put(114,35){$\delta_3$}\put(170,50){$\vector(0,-1){25}$}
\put(175,35){$\theta_1$} \put(235,50){$\vector(0,-1){25}$}
\put(220,35){$\simeq$} \put(240,35){$\breve{\delta}_2$}
\put(295,50){$\vector(0,-1){25}$}\put(300,35){$\delta_2$}
\put(400,35){$(*)$} \put(10,5){$\dots \longrightarrow \, \, \, 0
\, \, \, \longrightarrow \, H_{3}(\tS ) \, \longrightarrow \,
H_{1}(\G^*) \longrightarrow \, \, H_{2}(\breve{S})\longrightarrow
 H_2(\tS ) \longrightarrow
\dots$}\put(142,15){$\varphi$}\put(204,15){$\psi$}
\put(266,15){$\chi$}
\end{picture}


\noindent As $H_4(X')=0$ [Mil, Th. 7.1] the exact homology
sequence of pair $(X',\tX )$ implies that $H_3(\tX )$ is
isomorphic to $H_4(X',\tX )$. By Lemma 2.1 $L= \pi^{-1}(F) = E
\setminus E^*=X' \setminus \tX$ is a curve. If $U$ is a small
Euclidean neighborhood of $L$ in $X'$ then $H_4(U,U\setminus
L)=H_4(X',\tX )$ by the excision theorem. That is, every 3-cycle
$\omega' \in H_3(\tX )$ can be chosen in $U\setminus L$. Thom's
isomorphism maps the image of $\omega'$ in $H(U, U\setminus L)$ to
a 1-cycle $\alpha \in H_1(E^* \cap U)$. As Thom's isomorphisms are
functorial under open embeddings [Do, Ch. 8, 11.5] we can treat
$\alpha$ as an element of $H_1(E^*)$. Thus the image of $\alpha$
under the homomorphism generated by the natural projection $E^*
\to Z^*$ coincides with $\beta_0 = \varphi' (\omega' ) \in
H_1(Z^*)$. As $\alpha$ is contained in a small neighborhood $U
\cap E$ of $L$ in $E$ we see that $\beta_0$ is contained in a
small neighborhood of the finite set $Z \setminus Z^*$. This
implies that ${\rm Im} \, \varphi'$ is contained in the kernel
$G'$ of the homomorphism $H_1(Z^*) \to H_1(Z)$ induced by the
natural embedding $Z^* \hookrightarrow Z$. Let $ \bar{\theta} :
\bar{Z} \to \bar{\G}$ be the extension of $\theta$ to completions
of $Z$ and $\G$. Assume that $F' = (\bar{Z}\setminus Z) \setminus
\bar{\theta}^{-1}( \bar{\G} \setminus \G )$ is not empty which is
equivalent to the fact that $\theta$ is not finite. % Hence the
%kernel $G''$ of the homomorphism $H_1(Z) \to H_1(\bar{Z} \setminus
%\bar{\theta}^{-1}( \bar{\G} \setminus \G ))$ induced by the
%natural embedding $Z \hookrightarrow \bar{Z} \setminus
%\bar{\theta}^{-1}( \bar{\G} \setminus \G )$ is not empty. Let
%$\beta_1 \in G''$ be nonzero. Then for every $\beta_0 \in G'$ we
%have $\beta=\beta_1 + \beta_0 \notin G'$ whence $\beta$ does not
%belong to ${\rm Im} \, \varphi'$.
Let $z_0 \in F'$. One can suppose that $x_0 = \theta (z_0) \in F$.
Let $\G_j , \, j=1, \ldots , k$ be irreducible analytic branches
of $\G$ at $x_0$, and let $\gamma_j$ be a simple loop in $\G_j$
around $x_0$ with positive orientation. We treat $\gamma :=
\sum_{j=1}^k \gamma_j$ as an element of $H_1(\G^* )$. %As $H_2(\tX
%)=0$ the commutativity of (*) implies that $\Im \chi =0$. Hence by
By Lemma 3.1 $H_3(\tS )$ is generated by elements $\omega$ of form
$\omega= \phi (x_0)$ and $\varphi (\omega )= \gamma$. Take a
simple loop $\beta_1$ around $z_0$ in an irreducible analytic
branch $Z_1$ of $ \bar{Z}$ at $z_0$ such that $ \bar{\theta}
(Z_1)$ is, say, $\G_1$. Take other simple loops around the points
of $ \bar{\theta}^{-1}(x_0)$ whose images under $\theta$ are
contained in $\bigcup_{j=2}^k \G_j$. Then we can construct $\beta
\notin G'$ as an integer linear combination of $\beta_1$ and these
other loops so that $\theta_1 (\beta) =m \gamma$ where $m>0$. As
$\gamma \in {\rm Im} \, \varphi$ we have $\psi (\gamma) =0$. As $
\breve{\delta}_2$ is an isomorphism we have $\psi'(\beta)=0$, i.e.
$\beta \in {\rm Im} \, \varphi'$. This contradiction implies (i).

For (ii) we need another look at ${\rm Im} \, \varphi'$. Let
$Z\setminus Z^* = \{ z_i \}$ and $G_i'$ be the kernel of the
homomorphism $H_1(Z \setminus z_i) \to H_1(Z)$ induced by the
natural embedding $Z\setminus z_i \hookrightarrow Z$. Note that
$G'$ is naturally isomorphic to $\bigoplus_i G_i'$. Moreover,
since $L$ is the disjoint union $\bigcup_i \vartheta^{-1}(z_i)$
one can choose a neighborhood $U$ of $L$ in $X'$ as a disjoint
union of neighborhoods of the curves $\vartheta^{-1}(z_i)$. Hence,
repeating the argument with the excision theorem, we get ${\rm Im}
\, \varphi' = \oplus_i (G_i' \cap {\rm Im} \, \varphi')$. Let
$x_0$ be a selfintersection point of $\G$ (i.e. the number $k$ of
irreducible analytic branches $\G_j$ of $\G$ at $x_0$ is at least
2) and assume for simplicity that $\theta^{-1} (x_0)$ consists of
two points $z_0$ and $z_1$. Note that $z_0,z_1 \in Z \setminus
Z^*$. Let $\beta' \in G_0' \cap {\rm Im} \, \varphi'$ (i.e.
$\beta' =\varphi' (\omega' )$). Then $\delta_3(\omega' )=m\omega$
where $m \in \Z$ and $\omega$ is as before. As $\varphi (m\omega)
= \theta_1 (\beta')$ we see that $\theta_1(\beta') =m \gamma$.
Similarly $\theta_1(\beta'' )$ is proportional to $\gamma$ for
every $\beta'' \in G_1' \cap {\rm Im} \, \varphi'$. On the other
hand since $x_0$ is a selfintersection point, we can find $\gamma'
\in G_0'$ and $\gamma'' \in G_1'$ such that $\theta_1 (\gamma' )$
and $\theta_1 (\gamma'' )$ are not proportional to $\gamma$ but
$\theta_1 (\gamma' +\gamma'')$ is. The commutativity of (*)
implies that $\gamma' + \gamma'' \in {\rm Ker} \, \psi' = {\rm Im}
\, \varphi'$. But the decomposition of ${\rm Im} \, \varphi'$ we
discussed before, implies that $\gamma'$ and $\gamma''$ must be in
${\rm Im} \varphi'$. This contradiction implies (ii).

%For (iii) note that the subgroup generated by $(m_j\beta^i
%-m_i\beta^j)/m_{i,j}, \, i\ne j$ is contained in ${\rm Ker} \,
%\theta_1$. As $ \check{\delta}_2$ is an isomorphism the
%commutativity of diagram (*) implies that this subgroup is
%contained in ${\rm Ker} \, \psi'$ and, therefore, in the image of
%$\varphi'$.

%As $X'$ is smooth in (iii) we can remove a finite subset from $X'$
%without spoiling diagram (*) but we can suppose then that the
%irreducible and connected components of $\pi^{-1}(x_0)$ are the
%same. For the sake of notation we consider the case when $x_0$ is
%as in (ii), i.e. $\theta^{-1}(x_0)=z_0$. Let $G_0'$ be as in the
%proof of (ii) and $G_0 \subset H_1(\G^* )$ be the kernel of the
%homomorphism $H_1 ( \G \setminus x_0) \to H_1( \G )$ generated by
%the natural embedding. It is not difficult to check that the rank
%of ${\rm Im} \, \varphi' \cap G_0'$ coincides with the number of
%connected components of $\pi^{-1}(x_0)$, which is $n$, while the
%rank of ${\rm Im} \, \varphi \cap G_0$ is 1. If the statement of
%(iii) does not hold then the ranks of $G_0'/(G_0' \cap {\rm Im} \,
%\varphi' )$ and $G_0/(G_0 \cap {\rm Im} \, \varphi )$ are
%different whence $ \breve{\delta}_2$ cannot be an isomorphism.
%This contradiction implies (iii).
%
Let $G$ be the kernel of the homomorphism $H_1( \G^* ) \to H_1 (\G
)$, induced by the natural embedding. There is a non-canonical
isomorphism $H_1(\G^* ) = G\oplus H_1(\G^{\n})$ (resp. $H_1( Z^*)=
G' \oplus H_1( Z^{\n})$) such that $\Im \varphi$ (resp. $\Im
\varphi'$) is contained in the first term and $\theta_1 (G')$ is a
subgroup of finite index in $G$. As $\breve{\delta}_2$ is an
isomorphism the ranks of $H_1( \G^{\n})$ and $H_1(Z^{\n})$ are the
same, and the ranks of $G/ \Im \varphi $ and $G' / \Im \varphi' $
coincide as well. For (iii) note that (i) and (ii) implies that
$\theta |_{Z^1} : Z^1 \to \G^1$ is bijective as soon as it is
birational. If it is not birational then the fact, that the ranks
of $H_1( \G^{\n})$ and $H_1(Z^{\n})$ are the same, implies that
the genus of $Z^1$ (and, therefore, $\G^1$) is zero. Assume that
the normalization of $\G^1$ is different from $\C$, i.e. one can
suppose that $ \bar{\G}^1 \setminus \G^1$ consists of more than
one point where $ \bar{\G}^1$ is the closure of $\G^1$ in $
\bar{\G}$. Let $ \bar{Z}^1$ be the closure of $Z^1$ in $ \bar{Z}$.
Then either $ \bar{ \theta }$ maps $\bar{Z}^1 \setminus Z^1$
bijectively onto $\bar{\G}^1 \setminus \G^1$ or it does not. In
the first case a simple loop in $\G^1$ around a point $x \in
\bar{\G}^1 \setminus \G^1$ produces $\gamma^1 \in H_1(\G^*)$ which
is not in $G + {\rm Im} \, \theta_1$ and in the second case an
integer linear combination of simple loops in $Z^1$ around points
of $ \bar{\theta}^{-1}(x)$ produces $\beta^1 \in H_1(Z^*)$ which
is in ${\rm Ker} \, \theta_1 \setminus G'$. In the first case
since $G \supset {\rm Im} \, \varphi ={\rm Ker} \, \psi$, we see
that $\varphi (\gamma^1 )$ is not in ${\rm Im }\, \breve{\delta}_2
\circ \psi' $. On the other hand, as $X'$ is smooth and
$H_2(X')=0$ we get $H_2(\tX )=0$ whence ${\rm Im} \, \psi' = H_2 (
\breve{X})$. This is a contradiction since $ \breve{\delta}_2$ is
an isomorphism. The second case is similar which concludes (iii).
\qed

%For (v) note that $H_2( \tilde{X}) =H_2 (
%\tilde{S})=H_1( \tilde{X}) =H_1 ( \tilde{S})=0$. Hence we have the
%following extension of diagram (*)
%
%\begin{picture}(200,95)
%
%\put(30,65){$\dots \longrightarrow \, \, \,  0 \, \, \,
%\longrightarrow H_{0}(Z^*)\longrightarrow
%H_{1}(\breve{X})\longrightarrow \, \, \, 0 \, \, \,
%\longrightarrow \dots$} \put(78,50){$\vector(0,-1){25}$}
%\put(130,50){$\vector(0,-1){25}$}
%\put(193,50){$\vector(0,-1){25}$}
%\put(247,50){$\vector(0,-1){25}$} \put(200,35){$\simeq$}
%\put(30,5){$\dots \longrightarrow \, \, \, 0 \, \, \,
%\longrightarrow \, H_{0}(\G^*) \longrightarrow \, \,
%H_{1}(\breve{S})\longrightarrow \, \, \, 0 \, \, \,
%\longrightarrow \dots$}
%\end{picture}
%
%\noin which yields (v).

\rem 3.3. Proposition 3.2 implies that $\pi$ is surjective which
is a generalization of the result of Bonnet [Bo] who proved
this in the case of $X' \simeq \C^3$. % Following [Da] one can see
This implies that when $X'$ is Cohen-Macaulay,  the algebra of
regular functions on $X'$ is faithfully flat over the algebra of
regular functions on $S$ (this follows from [Ma, Chap. 2, (3.J)
and Th. 3] and [Ei, Th. 18.16], see also [Da]). \vs

%(2) For purposes not related the the main result of this paper we
%want to emphasize two other facts. Let $x_0$ and $z_0$ be as in
%(ii) and let $\G_1$ be an irreducible analytic branch of $\G$ at
%$x_0$. Let $Z_1, \dots ,Z_k$ be the irreducible analytic branches
%of $\theta^{-1}(\G_1 )$ at $z_0$ and $\beta_i$ be a simple
%positively oriented loop in $Z_i$ around $z_0$ (which is viewed as
%an element of $H_1(Z^*)$). Suppose that morphism $\theta |_{Z_i} :
%Z_i \to \G_1$ is $m_i$-sheeted and $m_{i,j}$ is the greatest
%common divisor of $m_i$ and $m_j$ for $i \ne j$. Then we claim
%that the subgroup of $H_1(Z^*)$ generated by $\{ (m_j \beta_i -m_i
%\beta_j)/m_{i,j} \, | \, j \ne i \}$ is contained in ${\rm Im}
%\varphi'$. Indeed, this subgroup is contained in ${\rm Ker} \,
%\theta_1$. As $ \breve{\delta}_2$ is an isomorphism the
%commutativity of diagram (*) implies that this subgroup is
%contained in ${\rm Ker} \, \psi'$ and, therefore, in the image of
%$\varphi'$.
%
%(3) Another fact which can be proven by similar means is the
%following. Let $x_0 \in \G$, $k$ (resp. $k'$) be the number of
%analytic branches of $\G$ at $x_0$ (resp. of $Z$ at
%$\theta^{-1}(x_0)$). Suppose that $X'$ is smooth and
%$\pi^{-1}(x_0)$ consists of $n$ irreducible components. Then
%$k'=k+n-1$.\\[2ex]


\noin {\bf 4. Injectivity of $\theta$. \\} \lemma 4.1. {\em Let
the notation of Lemma 2.3 hold, $\cM \subset E$ be a germ of a
curve at a general point $x'$ of $E$ such that $\pi |_{\cM} : \cM
\to S$ is injective. Let $\cP \subset X'$ be a germ of a surface
at $x'$ that meets $E$ transversally along $\cM$. Then $\pi
|_{\cP} : \cP \to S$ is injective.}

\proof While speaking of affine modifications here and in Section
5 we use the terminology from [Ka02]. Consider the affine
modification $\sigma : X' \to X \simeq
S \times \C$ from Lemma 2.2. %Its divisor is $D =\G \times \C$. Let
%${\rm pr} : X \to S$ be the natural projection and let $\G^1,
%\ldots , \G^m$ be the irreducible components of $\G$. Then $D$
%(resp. $E$) consists of the irreducible components $D^1, \ldots ,
%D^m$ (resp. $E^1, \ldots , E^m$) such that $D^i ={\rm pr}^{-1}
%(\G^i)$ (resp. $E^i=\pi^{-1}(\G^i)$). We can suppose that
%$x_1=\sigma (x_1')$ belongs to $D^1 \setminus \bigcup_{i=2}^m
%D^i$. Deleting $\bigcup_{i=2}^m D^i$ (resp. $\bigcup_{i=2}^m E^i$)
%from $X$ (resp. $X'$) we obtains an affine modification $\sigma_1$
%of UFD's with an irreducible exceptional divisor $E^1_0= E^1
%\setminus \bigcup_{i=2}^m E^i$ and an irreducible center
%$C^1_0=\sigma (E_0^1)$.
Proposition 2.12 from [Ka02] implies that in a neighborhood of a
general point of $E$ this modification $\sigma$ is a composition
of basic modifications. % and a pseudo-basic modification. By
%definition a pseudo-basic modification is basic in a neighborhood
%of a general point of its center. Therefore, after deleting a
%hypersurface from $X\setminus \bigcup_{i=2}^m D^i$ (so that this
%hypersurface does not contain $C_0^1$) and its preimage from
%$X'\setminus \bigcup_{i=2}^m E^i$ we can suppose that $\sigma_1$
%is a composition of basic modifications.
%In a Euclidean
%neighborhood of a general point of its center
Locally, these basic modifications are nothing but usual monoidal
transformations [Ka02, Remark 2.3] (see also Remark 5.2 below). In
the case of three-dimensional manifolds and one-dimensional
centers the coordinate form of these monoidal transformations is
$(\xi_1, \xi_2 , \xi_3) \to (\xi_1, \xi_1 \xi_2 , \xi_3) : =(
\zeta_1,
\zeta_2, \zeta_3)$. %That is, the plane $\xi_1 =0$ is the
%exceptional divisor of the transformation, and the line given by
%$\zeta_1 = \zeta_2 =0$ is the center of the transformation.
If $\sigma$ is locally such a monoidal transformation, then
$\sigma$ maps $\cM$ injectively into the line $\zeta_1 = \zeta_2
=0$ since $\pi = {\rm pr} \circ \sigma$ where ${\rm pr} : X \to S$
is the natural projection. Hence $\cM$ must be given by equations
of form $\xi_1= \xi_2 -g_0 (\xi_3)=0$. As $\cP$ meets $E$
transversally along $\cM$ we see that $\cP$ is given by an
equation of form $\xi_2 -g(\xi_1 , \xi_3)$ where
$g(0,\xi_3)=g_0(\xi_3)$. Hence the image of $\cP$ is the germ of a
surface given by $\zeta_2 =\zeta_1 g(\zeta_1, \zeta_3)$. In
particular, this germ is isomorphic to $\cP$ (via the
transformation) and transversal to the plane $\zeta_1=0$.
Induction on the number of monoidal transformations in the local
decomposition of $\sigma$ implies that $\sigma |_{\cP}: \cP \to
\cQ := \sigma (\cP )$ is bijective and $\cQ$ meets $D$
transversally along a germ $\cC$ of $C=\sigma (E)$ at $x= \sigma
(x')$. As $x$ is a general point of $C$, projection ${\rm pr}$
maps $\cC$ injectively into $\G$. As $\cQ$ meets $D$ transversally
along $\cC$, morphism ${\rm pr}|_{\cQ} : \cQ \to S$ is not
ramified along $\cC$ whence this map is injective. Thus $\pi
|_{\cP} : \cP \to S$ is injective. \qed


\prop 4.2. {\em Let the assumption of Proposition 3.2 (iii) hold,
$S$ be smooth, and $\pi : X' \to S$ be the quotient morphism of a
free $\C_+$-action $\Phi$ on $X'$. Then $\theta$ is bijective.}

\proof Assume the contrary, i.e. there is a component $Z^1$ of the
$Z$ such that $\theta |_{Z^1}$ is not bijective. By Proposition
3.2 (i)-(ii) $\theta |_{Z^1}$ is not birational. Furthermore,
Proposition 3.2 (iii) implies that $Z$ contains a ramification
point $z_0$ of $\theta$ whose order is $m\geq 2$. Take a smooth
point $x_0'$ of $\vartheta^{-1}(z_0)$. The irreducible component
$L_0$ of $\vartheta^{-1}(z_0)$, that contains $x_0'$, is an
irreducible component of a fiber of $\pi$ and, therefore, an orbit
of $\Phi$ since $\Phi$ is free. Consider the germ $ {\cal P}$ of a
surface at $x_0'$ transversal to $L_0$. Then all orbits of $\Phi$
close to $L_0$ must be transversal to $\cP$. Let us show that this
cannot be true.

Consider $\varrho = \pi |_{{\cal P}} : {\cal P} \to \pi ({\cal P}
)$. One can see that $\varrho$ is finite (as $\cP$ does not
contain fibers of $\pi$). %Recall that $\pi ={\rm pr} \circ \sigma$
%(where $\sigma : X' \to X$ and ${\rm pr} : X \to S$ are as in
%Lemma 4.1) and
As $\pi |_E = \theta \circ \vartheta$ and $\theta$ has a
ramification point of order $m$ we see that $\varrho$ is at least
$m$-sheeted. As $S$ is smooth there must be a ramification curve
$R \subset \cP$ of morphism $\varrho$. Consider a general point
$x_1'$ of the curve $\cM=\cP \cap E$, and the germ $\cP_1$ of
$\cP$ at $x_1'$. Note that $\cP_1$ meets $E$ transversally along
the irreducible germ $\cM_1$ of $\cM$ at $x_1'$, and $\pi
|_{\cM_1}$ is injective. By Lemma 4.1, $\varrho$ is locally
injective in a neighborhood of $x_1'$, i.e. $R$ has no common
components with ${\cal M}$. Therefore, a general point $x_2' \in
R$ is contained in $X'\setminus E \simeq \breve{S} \times \C$. As
$x_2'$ is a ramification point of $\varrho$ it cannot be a point
where $\cP$ meets the fiber of the natural projection $ X'
\setminus E \to \breve{S}$ (which is a fiber of $\pi$ and,
therefore, an orbit of $\Phi$) transversally which yields the
desired contradiction. \qed
%Let $(\zeta ,\xi )$ be a coordinate system in a Euclidean
%neighborhood $U$ of $x_2=\pi (x_2') \in S$ and $t$ be a coordinate
%on $\C$ whence $(\zeta , \xi ,t)$ is coordinate system on $W:=
%\pi^{-1}(U) \simeq U \times \C_t$ with center at $x_2'$. The
%coordinate form of the vector field $\nu$ is $(0,0, h(\zeta , \xi
%) {\frac { \de }{\de t }})$ where $h(0,0) \ne 0$. By the
%Weierstrass Preparation theorem $\cP_2 := \cP \cap W$ is given by
%an equation of form $t^n +a_{n-1}(\zeta , \xi )t^{n-1} + \ldots +
%a_0(\zeta , \xi ) =0$.
%As $\varrho$ is ramified at $x_2'$ we have $n\geq 2$. The phase
%flow associated with $\nu$ moves $\cP_2$ to another surface in $W$
%given by $(t +\varepsilon h(\zeta, \xi ) +o( \varepsilon^2))^n
%+a_{n-1}(\zeta , \xi )(t +\varepsilon h(\zeta, \xi ) +o(
%\varepsilon^2))^{n-1} + \ldots  =0$. As $h(0, 0) \ne 0$ and $n
%\geq 2$ this equation has a common solution with $t^n
%+a_{n-1}(\zeta , \xi )t^{n-1} + \ldots + a_0(\zeta , \xi ) =0$
%near the origin. Hence $\Phi_{\varepsilon} (\cP )$ meets $\cP$
%which yields the desired contradiction.


%{\bf Example} 4.3. Consider $X' \simeq \C^3_{x,z,u}$ given as the
%hypersurface $xu=z^2-y$ in $\C^4_{x,y,z,u}$. Consider the locally
%nilpotent derivation on $A' = \C [X']$ given by $\de (x)=0, \de
%(z)=x, \de (u)=2z$, and the associated $\C_+$-action $\Phi$. The
%quotient space $S$ is isomorphic to $\C^2_{x,y}$, $X \simeq
%\C^3_{x,y,z}$, and $\sigma : X' \to X$, $\pi : X' \to S$ are the
%natural projections. Note that $\G$ is the $y$-axis in
%$\C^2_{x,y}$, $D$ is the plane $x=0$ in $X$, and $E$ is the
%hypersurface $x=0$ in $X'$. In particular, $E \simeq Z \times
%\C_u$ where $Z$ is the parabola given by $x=z^2-y=0$ in
%$\C^3_{x,y,z}$. Morphism $\theta : Z \to \G$ is nothing but the
%natural projection. It is 2-sheeted and the origin $o$ is the
%ramification point of $\theta$. We have the line $L_o' = \{
%x=y=z=0 \}$ in $X'$ above $o$. A surface $\cP$ transversal to
%$L_o'$ can be given by equation $u=1$ in $X'$, i.e. $z^2= x+y$ on
%$\cP$. Note that the natural projection $\varrho : \cP \to S$ is
%2-sheeted, the ramification curve $R \subset \cP$ is the line
%given by $x+y=0$. As we showed in Lemma 4.1 $R$ has no common
%component with the curve $\cM =\cP \cap E$ given by $z^2-y=x=0$ in
%$\cP$. The action $\Phi_{\varepsilon}$ is given by $(x,y,z,u) \to
%(x,y,z+ \varepsilon x, u+2\varepsilon z +\varepsilon^2 x)$ whence
%$\Phi_{\varepsilon} (\cP )$ satisfies the equation
%$(1+2\varepsilon z +\varepsilon^2 x)(z+\varepsilon x)^2=x+y$. Thus
%$\Phi_{\varepsilon} (\cP )$ meets $\cP$ along the curve given in
%$X'$ by $u=1$, $2z+\varepsilon x=0$, and $y=\varepsilon^2
%x^2/4-x$. The same remains true if we replace $\de$ by another
%locally nilpotent derivation $\de^1$ such that $\de^1 (x)=\de^1
%(y)=0$. Indeed, consider the line ${\cal N}$ in $S$ given by $x=1$
%and the surface in $X'$ above ${\cal N}$. This surface is
%isomorphic to $\C^2_{y,z}$ and its intersection with $\cP$ is the
%parabola ${\cal K}$ given by $1=z^2-y$. The restriction of $\de^1$
%to this surface is a vector field parallel to the $z$-axis.
%Clearly any perturbation of ${\cal K}$ along this vector field
%meets ${\cal K}$. \\[2ex]

\noin {\bf 5. Main theorem. \\} We want to decompose $\sigma : X'
\to X$ from Lemma 2.2 into a composition of simpler modifications
$X' \to X_1$ and $X_1 \to X$. Let $A$, $A_1$, and $A'$ be the
algebras of regular functions on $X$, $X_1$, and $X'$
respectively. In order to find $X_1$ it suffices to find an affine
domain $A_1$ which is a proper subring of $A'$ and which contains
but not equal to $A$.

\lemma 5.1. {\em Let $\sigma : X' \to X$ be an affine
modification, $C$ be its center, $D^1$ be an irreducible component
of its divisor $D$, $C^1 =C \cap D^1$. Suppose that $A' =A[I/f]$
where $I$ is an ideal in $A$ that vanish on $C$, $f=g^{n} h$, $D
=f^{-1}(0)$, and $D^1=g^{*}(0)$ while $h$ is not identically zero
on $D^1$. Let $E^1 = \sigma^{-1} (C^1)$, and the defining ideal
$\cI_{A'}(E^1)$ of $E^1$ be the principal ideal in $A'$ generated
by $g$.

{\rm (1)} Then $A[(\cI_A(C^1))^{n}/g^{n}] \subset A'$ for the
defining ideal $\cI_A(C^1)$ of $C^1$ in $A$. In particular, for
every ideal $J \subset \cI_A (C^1)$ the algebra $A_1 =A[J/g]$ is
contained in $A'$.

{\rm (2)} If $J$ in (1) coincides with $\cI_A (C^1)$ then $A'
=A_1[I_1/f_1]$ where $I_1$ is an ideal in $A_1$ and $f_1 =
g^{n-1}h$.}

\proof Note that for every $a \in (\cI_A(C^1)) ^{n}$ we have $a
\in (\cI_{A'}(E^1)) ^{n}$ (as an element of $A'$) whence $a/g^{n}
\in A'$. For the second statement of (1) note that $A_1
=A[(g^{n-1}J)/g^{n}] \subset A[(\cI_A(C^1))^{n}/g^{n}]$.

Let $g, b_1, \ldots , b_k$ be generators of $I \subset
\cI_A(C^1)$. Then in (2) $b_1/g, \ldots , b_k/g \in A_1$. The
ideal $I_1$ generated by these elements is what we need in (2).
\qed

\rem 5.2. Let $x \in C^1$ be a smooth point of both $C^1$ and $X$,
and the zero multiplicity of $g$ at $x$ be 1. In a small Euclidean
neighborhood $U$ of $X$ the defining ideal of $C^1 \cap U$ is a
complete intersection, i.e. it is generated by regular functions
$g, b_1, \ldots , b_k$ which are elements of a local coordinate
system on $U$. Hence in $U$ the modification $A \hookrightarrow
A_1=A[\cI_A (C^1)/g]$ can be viewed as a monoidal transformation.
This fact in combination with Lemma 5.1 (2) and induction on $n$
yields another way to show that modification $\sigma$ in the proof
of Lemma 4.1 is a composition of monoidal transformations in a
neighborhood of a general point of its center. \vs

\prop 5.3. {\em Let the assumption of Proposition 3.2 (iii) hold,
and $\G^1$ (resp. $Z^1$) be an irreducible components of $\G$
(resp. $Z$) such that $\theta ( Z^1) \subset \G^1$. Suppose that
$\theta |_{Z^1} : Z^1 \to \G^1$ is bijective. %, and the ring of
%germs of analytic functions at any point of $S$ is factorial.
Let $\G^{\bullet}$ be the closure of $\G \setminus \G^1$ in $S$.
Then $\pi^{-1}(S \setminus \G^{\bullet})$ is isomorphic to $(S
\setminus \G^{\bullet}) \times \C$ over $S \setminus
\G^{\bullet}$.}

\proof Let $\sigma : X' \to X \simeq_S S \times \C$ be the affine
modification from Lemma 2.2. In particular, its divisor $D=\G
\times \C$ contains an irreducible component $D^1 = \G^1 \times
\C$. Our aim is to construct another affine modification
$\sigma^{\bullet} : X' \to X$ such that its divisor is
$D^{\bullet} = \G^{\bullet} \times \C$ which would yield the
statement of this Proposition. As $S$ is factorial by Lemma 2.1,
$\G^1$ is the zero locus of a regular function $g$ on $S$. If we
treat $g$ as a regular function on $X$ we get $A' = A[I/f]$ where
$I$ and $f=g^{n} h$ are as in Lemma 5.1. In virtue of [Ka02, Lemma
2.2] one can suppose that the center $C$ of $\sigma$ is a curve.
By Remark 2.4 $\theta : Z \to \G$ factors through ${\rm pr}|_C : C
\to \G$ and $Z \to C$. Let $C^1$ be the image of $Z^1$ under the
last morphism. Then $\pr |_{C^1}: C^1 \to \G^1$ is bijective.
Hence $C^1$ is given in $D^1= \G^1 \times \C$ by $t-{\bar{b}}_1=0$
where $t$ is a coordinate on the second factor of $X\simeq S\times
\C_t$ and $ \bar{b}_1$ is a continuous rational function on
$\G^1$. Let
us show that $t-\bar{b}_1$ is regular on $D^1$. %Let $o$ be
%a singular point of $\G^1$.
Let $x_0$ be the preimage of $o \in \G^1$ in $C^1$ and $x_0' \in
\pi^{-1}(o)$ be a smooth point of $\pi^{-1}(o)$. Take the germ
$\cP$ of an analytic surface at $x_0'$ such that $\cP \cap L
=x_0'$. One can check that $\cQ = \sigma (\cP )$ is closed in a
neighborhood of $x_0$ and meets $D_1$ along the germ $\cC^1$ of
$C^1$ at $x_0$. Hence $\cQ$ is given in $X\simeq S \times \C_t$ by
zeros of a function $t^n+a_{n-1}t^{n-1} + \ldots + a_0 $ where
each $a_i$ is a germ of a continuous function at $o \in S$ which
is analytic outside $\G^1$. As $S$ is normal $a_i$ is analytic at
$o$ as well. Hence by [Ka02, Lemma 3.3] $t- \bar{b}_1$ is regular.
Let $b_1$ be an extension of $ \bar{b}_1$ to $S$. Then
$A_1=A[(t-b_1)/g]$ is the algebra of regular functions on a
hypersurface $X_1$ in $X \times \C_u$ given by equation $ug
=t-b_1$, and %. As $g$ and $b_1$ are functions on $S$ (i.e. they
%are independent of $t$) we see that
$X_1 \simeq S \times \C_u \simeq X$. By Lemma 2.1 the defining
ideal of $E^1=\sigma^{-1}(C^1)$ in $A'$ is the principal ideal
generated by $g$. As Lemma 2.3 implies that $Z^1=\theta^{-1}(\G^1
)$, we have $C^1=D^1\cap C$. Thus by Lemma 5.1 $A_1 \subset A'$.
Hence we can replace modification $\sigma : X' \to X$ by
modification $X' \to X_1\simeq_S S\times \C$ (induced by the
embedding $A_1 \hookrightarrow A'$) and all assumption of
Proposition 3.2 hold but the advantage is that now $f=g^{n-1} h$
by Lemma 5.1 (since the defining ideal of $C^1$ in $A$ is
generated by $g$ and $t - b_1$). Repeating this procedure we can
make $n=0$, i.e. we get the modification $\sigma^{\bullet} : X'
\to X$ which concludes the proof. \qed

\thm 5.4. {\em Let the assumption of Lemma 2.3 hold, $X'$ be
smooth, and $H_2(X')=H_3(X')=0$.

{\rm (i)} Then $\G$ can be chosen so that each irreducible
component $Z^1$ of $Z$ (resp. $\G^1 =\theta (Z^1)$ of $\G$) is
rational (resp. a polynomial curve), and $\theta |_{Z^1} :Z^1 \to
\G^1$ is not bijective.

{\rm (ii)} If $S$ is smooth and the $\C_+$-action $\Phi$ on $X'$
has no fixed points then $X'$ is isomorphic to $S\times \C$ over
$S$ and the action is generated by a translation on the second
factor.}

%{\rm (iii)} If in (ii) there is also a dominant morphism $\C^3 \to
%X'$ then $X'$ is isomorphic to $\C^3$ and $\Phi$ is a translation
%in a suitable polynomial coordinate system on $X'$.}

\proof Applying Proposition 5.2 consequently to each component
$Z^1$ of $Z$, such that $\theta |_{Z^1} : Z^1 \to \G^1 =\theta
(Z^1)$ is bijective, we obtain (i) in virtue of Proposition 3.2
(iii). Proposition 4.2 and (i) imply that the quotient morphism
$\pi : X' \to S$ makes $X'$ isomorphic
to $S \times \C$ over $S$. \qed %In (iii) by Lemma 2.1 and [Miy01,
%Chap. 3, Th. 2.6.1] $S$ is either isomorphic to $\C^2_{\zeta
%,\xi}$ or to a singular surface. As $X'$ is smooth $S$ must be
%smooth, i.e. $S\simeq \C^2_{\zeta ,\xi}$ and $X'\simeq \C^3_{\zeta
%, \xi ,t}$. Hence $\Phi_{\tau} (\zeta , \xi ,t) = (\zeta , \xi ,
%t+a (\zeta , \xi ) \tau )$ where $a( \zeta , \xi )$ is a
%polynomial. The absence of fixed points implies that $a$ is
%nowhere zero, i.e. it is constant.

{\bf Proof of Theorem 1.} By [Miy80] $\C^3//\C_+ \simeq \C^2$ for
every nontrivial $\C_+$-action on $\C^3$ whence Theorem 5.4 (ii)
implies the desired conclusion. \qed

%By strengthening [Miy80] we can get a better result. \vs

%\prop 5.5. {\em Let $X'$ be a factorial affine algebraic variety
%of dimension 3 with a trivial fundamental group such that there is
%a dominant morphism $\C^3 \to X'$. Then for any nontrivial
%$\C_+$-action the quotient $S = X'//\C_+$ is isomorphic to
%$\C^2$.}
%
%\proof  Assume that $S$ is not isomorphic to $\C^2$. Then by Lemma
%2.1 and [Miy01, Chap. 3, Th. 2.6.1] $S$ is isomorphic to the
%hypersurface $x^2+y^3+z^5=0$ in $\C^3_{x,y,z}$. In particular, if
%$o$ is the origin in $\C^3$ then $S_0= S\setminus o$ has a
%nontrivial fundamental group. Consider $\sigma : X' \to X =S\times
%\C$ from Lemma 2.2. The divisor of this affine modification is of
%form $D =\G \times \C$. Let $h_i=0$ be the defining equation of an
%irreducible component $D^i=\G^i \times \C$ of $D$ (where $\G^i$ is
%an irreducible component of $\G$). That is, $h_i=g_i \circ {\rm
%pr}$ where $\G^i =g_i^*(0)$ and ${\rm pr} : X \to S$ is the
%natural projection. Lemma 2.1 implies that there is a bijection
%between the irreducible components $E^i$ of the exceptional
%divisor $E\subset X'$ of $\sigma$ and the irreducible components
%of $D$ under which the defining equation of $E^i$ is $g_i \circ
%\pi =h_i \circ \sigma =0$. Consider $X_0 =S_0 \times \C$ and
%$X_0'= \pi^{-1}(S_0)$. Then the restriction of $\sigma$ yields a
%modification $\sigma_0 : X_0' \to X_0$ of complex manifolds. The
%relation between irreducible components of $D$ and $E$, which we
%just described, implies that $\sigma_0$ generates an isomorphism
%of the fundamental groups [KaZa, Prop. 3.1]. On the other hand,
%the fundamental group of $X'$ is trivial, $X'$ is smooth
%three-dimensional, and $X_0'=X' \setminus \pi^{-1}(o)$ where
%$\pi^{-1}(o)$ is a curve, whence the fundamental group of $X_0'$
%is trivial while the fundamental group of $X_0 =S_0 \times \C$ is
%not. Contradiction. \qed
%
% By Fujita's result (e.g., see [Ka94, Prop. 3.2]) a smooth
%contractible affine algebraic variety is a factorial. Hence
%Propositions 5.4 and 5.5 imply. \vs
%
%\thm 5.6. {\em Let $X'$ be a smooth contractible affine algebraic
%variety of dimension 3 with a free $\C_+$ action on it. Then $X'$
%is isomorphic to $\C^3$ and the action is a translation in a
%suitable polynomial coordinate system.} \vs


\rem 5.5. (1) There is another way to extract Theorem 1 from
Proposition 4.2. For a bijective $\theta$ and $X'\simeq \C^3$ an
easy Euler characteristics argument shows that all fibers of
$\vartheta$ (not only the general ones) are irreducible. This, in
turn, implies that the $\C_+$-action is separated and, therefore,
it is a translation [DeFi, Kr].

(2) Consider an $n$-dimensional smooth contractible affine
algebraic variety $X'$ and a free algebraic action of a unipotent
group $U$ on it (i.e. each orbit of the action is isomorphic to
$U$). Suppose that $U$ is of dimension $n-2$ (i.e. $U$ is
isomorphic to $\C^{n-2}$ as an affine algebraic variety). By
Fujita's result (e.g., see [Ka94, Prop. 3.2]) $X'$ is factorial
and by Zariski's theorem [Za] $X'//U$ is still an affine algebraic
surface whence one can repeat all Lemmas and Propositions before
for this situation with obvious adjustments (say, in Lemma 2.3
$E^*$ will be isomorphic to $Z^* \times \C^{n-2}$). As Miyanishi's
theorem [Miy80] says that in the case of a free action $\C^n//U
\simeq \C^2$, we have the following analogue of Theorem 1: for
every free algebraic action of $U$ on $\C^n$ there is an
isomorphism $\C^n \simeq \C^2 \times U$ such that the action is
induced by the natural action on the second factor.

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