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\title{Actions of $\C^*$ and $\C_+$ on affine algebraic varieties}



\author{Shulim Kaliman}
\address{Department of Mathematics,
University of Miami, Coral Gables, FL  33124, U.S.A.}
\email{kaliman@math.miami.edu}

%\usepackage{showkeys}

\begin{document}

\maketitle

\section{Introduction}

The purpose of this paper is to present recent developments in the
study of algebraic group actions on affine algebraic varieties.
This study leads inevitably to other subjects in affine algebraic
geometry and we touch them as well but in less detail (for a wider
overview of affine algebraic geometry we refer to \cite{Zai99},
\cite{Kr96}, \cite{Mi04}). Among several beautiful results of the
last decade the Koras-Russell proof \cite{KoRu99} of the
Linearization Conjecture in dimension three has a special place.
The Conjecture claims that every algebraic $\C^*$-action on $\C^n$
is conjugate to a linear one in the group of polynomial
automorphisms of $\C^n$. It was first stated by Kambayashi
\cite{Kam} but, perhaps, the starting point was the earlier paper
of Gutwirth \cite{Gu62} who proved it for $n=2$. Using his
previous result with Kraft on linearization of semi-simple group
actions Popov showed that the Koras-Russell theorem implies that
every algebraic action of a connected reductive group on $\C^3$ is
a representation in a suitable polynomial coordinate system
\cite{Po01}. In general reductive groups admit non-linearizable
actions and for every connected reductive group different from a
torus such an action exists on $\C^n$ with sufficiently large $n$,
but the question about $\C^*$-actions in higher dimensions remains
open. In the case of the similar questions for real Euclidean
spaces or holomorphic actions there are counterexamples which will
be presented below. While discussing the steps of the solution of
the Linearization Conjecture we encounter the Koras-Russell
threefolds that are smooth complex affine algebraic varieties
diffeomorphic to $\R^6$ and equipped with obviously
non-linearizable $\C^*$-actions. The trouble was that all old
invariants capable of distinguishing $\C^2$ from smooth
contractible surfaces, were the same for these threefolds and for
$\C^3$. In particular, these threefolds were viewed as potential
counterexamples to linearization until the introduction of the
Makar-Limanov invariant \cite{ML}. For an affine algebraic variety
$X$ this invariant $\AK (X)$ is the subring of the ring of regular
functions on $X$ that consists of functions invariant under any
$\C_+$-action on $X$. If $X$ is a Koras-Russell threefold then
$\AK (X)$ includes non-constant functions which shows that $X$ is
not isomorphic to $\C^3$ since $\AK (\C^n )=\C$. We present the
scheme of computation of $\AK (X)$ in the case of the most vivid
of the Koras-Russell threefolds which is the Russell cubic - the
hypersurface in $\C^4$ given by the equation $x+x^2y+z^2+t^3=0$.
In particular, we see that the study of ``good guys" (reductive
group actions) requires the study of ``bad guys" ($\C_+$-actions)
that are hidden in the group of automorphisms of $X$. This leads
to the latest developments in the fourteenth Hilbert problem, and
some other results on algebraic quotients $X//\C_+$, i.e. we
explain why for any nontrivial $\C_+$-action on a smooth affine
contractible threefold $X$ the quotient is always a smooth
contractible surface. This fact contributes to another theorem in
this survey: if in addition the $\C_+$-action on such an $X$ is
free then $X$ is isomorphic to $S \times \C$ and the action is a
translation on the second factor \cite{KaSa}. In particular, every
free
%the Daigle and Russell classification \cite{DaiRu01a}
%\cite{DaiRu01b} of homogeneous $\C_+$-actions on $\C^3$, the
%the generalization of Miyanishi's theorem on algebraic quotient
%$\C^3//\C_+$ \cite{KaSa}, and the fact that every free
$\C_+$-action on $\C^3$ is a translation in a suitable polynomial
coordinate system \cite{Ka04}. At the end we return to
$\C^*$-actions on affine algebraic varieties with nontrivial
topology and present the coming classification of $\C^*$-actions
on smooth affine algebraic surfaces.



\section{Preliminaries}
Throughout this paper $X$ will be a normal complex affine
algebraic variety, $A=\C [X ]$ will be the algebra of regular
functions on $X$, and $G$ will be an algebraic group. Recall that
a $G$-action on $X$ is a homomorphism of $G$ into the group of
bijections of $X$ which generates, therefore, a map $\Phi : G
\times X \to X$. We say that the action is algebraic (resp.
holomorphic) if $\Phi$ is a morphism (resp. a holomorphic map),
i.e. $\Phi$ is generated by a homomorphism from $G$ into the group
$ {\rm Aut} \, X$ of regular (resp. holomorphic) automorphisms of
$X$. Unless we state otherwise every action of an algebraic group
$G$ that we discuss below will be algebraic and non-degenerate
(i.e. there is a orbit of the same dimension as $G$). Consider two
examples of algebraic $\C^*$-actions and $\C_+$-actions on $\C^n$
crucial for this paper. % (where $\C^*$ is, of course, the
%multiplicative group of nonzero complex numbers while $\C_+$ is
%the additive group of complex numbers).

\bexa\label{se.2.1} (1) A linear action $\C^* \times \C^n \to
\C^n$ is given by
$$(\lambda ,  {\bar x}) \to (\lambda^{k_1}x_1, \ldots
,\lambda^{k_n}x_n)$$ where ${\bar x}=(x_1, \ldots , x_n)$ is a
coordinate system on $\C^n$, $\lambda \in \C^*$, and $k_i \in \Z$.

(2) A triangular action $\C_+ \times \C^n \to \C^n$ is given by
$$(t, {\bar x}) \to (x_1, x_2+tp_2(x_1), \ldots , x_n +tp_n(x_1,
\ldots , x_{n-1}))$$ where $ t \in \C_+$ and each $p_i$ is a
polynomial in variables $x_1, \ldots , x_{i-1}$.
%The fixed point set for this action is
%$p_2 = \ldots =p_n =0$ is a
%cylinder $Y \times \C_{x_n}$.\\[2ex]
\eexa

\brem\label{se.2.2} Elements of a triangular action are contained
in the Jonqui\`ere group of automorphisms of $\C^n$ that is
$$\cJ_n= \{ \varphi =(\varphi_1, \ldots ,\varphi_n ) \in \Aut \C^n
| \, \forall i \, \varphi_i \in \C [x_1, \ldots ,x_i] \}$$ while
elements of a linear action are contained in the intersection of
$\cJ_n$ and the subgroup of affine transformation $$\cA_n = \{
\varphi =(\varphi_1, \ldots ,\varphi_n ) \in \Aut \C^n | \,
\forall i \, \varphi_i \in \C [x_1, \ldots ,x_n], \, \, \deg
\varphi_i =1 \}.$$ \erem

\bdefi\label{se.2.3} A point $x \in X$ is a fixed point of a
$G$-action $\Phi : G \times X \to X$ on $X$ if $\Phi (g,x)=x$ for
every $g \in G$. The action is called free if it has no fixed
points (precaution: this definition of a free action is valid for
varieties  over $\C$ or other algebraically closed fields of zero
characteristics but it must be changed in the absence of algebraic
closedness). In the case of $G =\C_+$ we say that the action on
$X$ is a translation if $X$ is isomorphic to $Y \times \C$ and the
action is generated by a translation on the second factor. Of
course, each translation is free.

\edefi

\bexa\label{se.2.4} The fixed point set for the triangular action
in Example \ref{se.2.1} is given by $p_2 = \ldots =p_n =0$. That
is, it is a cylinder $Y \times \C_{x_n}$ where $\C_z$ (resp.
$\C^n_{x_1, \ldots ,x_n}$) means a line equipped with coordinate
$z$ (resp. $\C^n$ equipped with a coordinate system $(x_1, \ldots
, x_n)$). In particular, the action is free if the polynomials
$p_2, \ldots , p_n$ have no common roots. In the case of $n=2$
this means that $p_2$ is constant and, therefore, any free
triangular action on $\C^2$ is a translation. \eexa

\bdefi\label{se.2.5} Let $\Phi_i : G \times X \to X, \, i=1,2$ be
effective algebraic $G$-actions on $X$ (i.e. no proper subgroup of
$G$ acts trivially on $X$). We say that $\Phi_1$ and $\Phi_2$ are
equivalent if there exists $\alpha \in \Aut \, X$ such that
$\Phi_2 =\alpha \circ \Phi_1 \circ \alpha^{-1}$. \edefi

It is natural, for instance, to ask when a given algebraic
$\C^*$-action on $\C^n$ is equivalent to a linear one (that is, it
is linear in a suitable coordinate system). One can formulate this
question in a more general setting.

{\bf Classification Problem.} Given two effective algebraic
$G$-actions on $X$ establish whether they are equivalent. More
generally, describe
equivalence classes of effective algebraic $G$-actions on $X$.\\

The main obstacle for a solution of such a problem may lie in the
structure of $\Aut \, X$. When this structure is known the answer
may be simple. For $\C^2$, say, it works like this (e.g., see.
\cite{Kr96}).
% \bthm {\rm
%(Jung-van der Kulk)}
The group $\Aut \C^2$ is the amalgamated product $\cA_2
\ast_{\cH_2} \cJ_2$ where $\cH_n=\cA_n \cap \cJ_n$.\footnote{That
is, every $\alpha \in \Aut \, \C^2$ is a composition $\alpha =
\alpha_1 \circ \alpha_2 \circ \cdots \circ \alpha_n$ where each
$\alpha_i$ is contained either in $\cJ_2 \setminus \cA_2$ or in
$\cA_2 \setminus \cJ_2$ and $\alpha_i \in \cJ_2$ iff $\alpha_{i\pm
1} \in \cA_2$. Furthermore, this composition is unique up to
consequent changes of the following type: $\alpha_i$ is replaced
by $\alpha_i \circ \gamma$ and $\alpha_{i+1}$ is replaced by
$\gamma^{-1} \circ \alpha_{i+1}$ where $\gamma \in \cH_2$. In
particular, the number $n$ of factors is uniquely determined. }
%\ethm
Then every algebraic subgroup of $\Aut \C^2$ is of bounded length
in this amalgamated product \cite{Wr}, i.e. the number of factors
in the amalgamated decomposition of each element of this subgroup
is bounded by the same constant.  But any subgroup of bounded
length is isomorphic to a subgroup of one of the factors
\cite{Se}. Using this fact one can reprove the following results
of Gutwirth \cite{Gu62}, \cite{Gu61}, and Rentschler \cite{Re}
obtained by other methods.

\bthm\label{se.2.6} Every $\C^*$-action on $\C^2$ is equivalent to
a linear one, every $\C_+$-action on $\C^2$ is equivalent to a
triangular one. In particular, every free $\C_+$-action on $\C^2$
is a translation.\ethm

Unfortunately, the structure of the automorphism group $\Aut \,
\C^n$ is unknown for $n \geq 3$ and, furthermore, answering the
old question of Nagata, Shestakov and Umirbaev \cite{ShUm04a},
\cite{ShUm04b} proved recently that the automorphism of $\C^3$
given by
$$(x,y,z) \to (x-2y(xz+y^2)-z(xz+y^2)^2,y+z(xz+y^2), z)$$ is not a
composition of Jonqui\`ere and affine transformations. Hence one
has to use other tools when facing the Classification Problem in
higher dimensions and one of them is the notion of algebraic
quotient.

\section{Algebraic quotient and  Hilbert's Fourteenth problem}

From now on, given an algebraic $G$-action on $X$, we denote by
$A^G$ the subalgebra of $G$-invariant regular functions on $X$ and
by $X//G$ the spectrum ${\rm Spec} A^G$ which is called the
algebraic quotient.
%\bexa Let $\Phi$ be a translation on $X = Y \times \C$. Then it is
%clear that the algebraic quotient $X//\Phi \simeq Y$ and,
%therefore, coincides with the standard quotient $X/\Phi$ in which
%each orbit produces a point. In particular, in this case $X//\Phi$
%is an affine algebraic variety. \eexa
 Hilbert's fourteenth problem asks when $A^G$ is finitely
generated, or, equivalently, when $X//G$ is an affine algebraic
variety. The classical result of Nagata says that $X//G$ is affine
for a reductive $G$ but for a non-reductive $G$ the answer is
negative in general. The complete description of algebraic
quotients was obtained by Winkelmann \cite{Wi03}.

\bthm\label{se.3.1} For every action of an algebraic group $G$ on
$X$ its quotient is quasi-affine. Moreover, for every normal
quasi-affine algebraic variety $Y$ there is an algebraic
$\C_+$-action on some $X$ such that the algebra of
$\C_+$-invariant regular functions on $X$ is isomorphic to the
algebra of regular functions on $Y$. \ethm

The original Nagata's example of an algebraic $\C_+$-action on
$\C^n$ with a non-affine quotient was constructed for $n=32$.
Later efforts of Roberts, Daigle, and Freudenburg \cite{Ro},
\cite{Fr00}, \cite{DaiFr99} reduced this dimension to 5. For $n=3$
such a quotient is always affine by Zariski's theorem \cite{Za}
that gives a partial answer to the following version of the
Hilbert's fourteenth Problem suggested by Nagata:

{\em Is $F \cap A$ an affine domain for a subfield $F$ of ${\rm
Frac} (A)$?}

Zariski's theorem gives a positive answer when the transcendence
degree of $F$ (over $\C$) is at most $2$ and it has the following
corollary (in order to prove it one has to put $F$ equal to the
subfield of $G$-invariant rational functions on $X$).

\bthm\label{se.3.2} Suppose that the algebraic quotient $X//G$ of
an algebraic $G$-action on $X$ is of dimension 2. Then $X//G$ is
affine. In particular, for a three-dimensional $X$ its algebraic
quotients are always affine. \ethm


Recently Kuroda \cite{Ku} constructed a counterexample to the
question of Nagata in the case when the transcendence degree of $F
\cap A$ is 3 or higher. However, $F \cap A$ in his example cannot
serve as the ring of $G$-invariant functions for an algebraic
$G$-action on $X$. Thus we want to emphasize that the question
whether $\C^4//\C_+$ is affine remains open for the fans of the
fourteenth Hilbert Problem.

%\brem In any case for any nontrivial $\C_+$-action on $X$ there is
%a dense affine open subset $U$ of $X//\C_+$ such the restriction
%of the action to $\pi^{-1}(U)$ (where $\pi : X \to X//\C_+$ is the
%natural projection) is a translation.
%
%\erem

\section{Linearization Problem}

The problem we are going to discuss is when an algebraic
$G$-action on $\C^n$ is linearizable, i.e. when it is equivalent
to a representation. As we mentioned this is so for every
algebraic $\C^*$-action on $\C^2$ by the Gutwirth theorem
\cite{Gu62}.


\bthm\label{se.4.1} {\rm \cite{KoRu99}} Every $\C^*$-action on
$\C^3$ is equivalent to a linear one. \ethm

Earlier Popov and Kraft proved that an algebraic action of a
semi-simple group on $\C^3$ is equivalent to a representation
\cite{KrPo}. Combining these two results Popov got the following
\cite{Po01}.

\bthm\label{se.4.2} Every action of a connected reductive group on
$\C^3$ is linearizable, i.e. it is equivalent to a
representation.\ethm

He showed also that for a connected reductive group different from
$\C^*$ or $(\C^*)^2$ its algebraic action on $\C^4$ is always
linearizable. % (the proof is based on a theorem of Panyushev
%\cite{Pa} which in turn is based of the Kraft-Popov theorem
%\cite{KrPo}).

The fact that actions of $(\C^*)^4$ and $(\C^*)^3$ on $\C^4$ is
linearizable follows from an old result of Bialynicki-Birula
\cite{BB}. In fact, we have more \cite{De}, \cite{BeHa}.

\bthm\label{se.4.3} Let $X$ be a toric variety of dimension $n$
with a canonical action $\Phi$ of torus $T = (\C^*)^n$. Then any
other effective action of $T$ on $X$ is equivalent to $\Phi$ and,
furthermore, any effective action of $(\C^*)^{n-1}$ on $X$ is
equivalent to the action of an $(n-1)$-dimensional subtorus  of
$T$ generated by $\Phi$.

\ethm

We discuss briefly some elements of proof of the Koras-Russell
theorem (for a more detailed exposition of ideas of this proof one
can see \cite{KaKoMLRu}).

By the end of 1980's the linearization of algebraic $\C^*$-actions
on $\C^3$ was established in all cases except for the hyperbolic
one. In that case a $\C^*$-action $\Phi$ on $\C^3$ has only one
fixed point $o$ and the induced linear $\C^*$-action $\Psi$ on
$T_o \C^3 \simeq \C_{x,y,z}^3$ is hyperbolic, i.e. it is given by
$$(x,y,z) \to (\lambda^{-a} x, \lambda^by,\lambda^cz)$$ where
integers $a,b,c >0$. In particular, one can see that
$T_o\C^3//\Psi \simeq \C^2//\Z_d$ where the last quotient is the
result of a linear action on $\C^2$ of a cyclic group $\Z_d$ with
$d=a/{\rm GCD}(a,b){\rm GCD}(a,c)$. When the action $\Phi$ is
linearizable then there exists a natural isomorphism $\C^3//\Phi
\simeq \C^3//\Psi$. %Thus the action is non-linear if
%$S=\C^3//\Phi$ is not isomorphic to $\C^3//\Psi$.

Koras and Russell discovered a construction of each smooth
contractible affine algebraic threefold $X$ equipped with a
hyperbolic $\C^*$-action $\tilde \Phi$ (i.e., $\tilde \Phi$ has
only one fixed point $\tilde o$ and the induced $\C^*$-action
$\tilde \Psi$ on $T_{\tilde o}X$ is hyperbolic) such that
$X//{\tilde \Phi}$ is isomorphic to $T_{\tilde o}X//{\tilde
\Psi}$. For some of these varieties (which are called now
Koras-Russell threefolds) it was not clear whether they are
isomorphic to $\C^3$ but $\tilde \Phi$ in this case was obviously
non-linearizable. Koras and Russell proved in their previous
papers that

{\em  Every hyperbolic algebraic $\C^*$-action $\Phi$ on $\C^3$ is
linearizable provided

{\rm (1)} none of Koras-Russell threefolds are isomorphic to
$\C^3$;

{\rm (2)} $S$ is isomorphic to $\C^3//\Psi \simeq \C^2/\Z_d$ where
$\Psi$ is the induced linear action on the tangent space at the
fixed point.}

The introduction of the Makar-Limanov invariant enabled us to
remove the first obstacle \cite{ML}, \cite{KaML97b}, i.e.
Koras-Russell threefolds are, indeed, non-isomorphic to $\C^3$.
Then Koras and Russell established some properties of $S$ which
include the facts that $S$ is contractible with one singular point
$s_0$ of analytic type $\C^2//\Z_d$, and with the logarithmic
Kodaira dimension ${\bar \kappa } (S) = -\infty$. Using the
seminal paper of Fujita \cite{Fu82} and the results of Miyanishi
and Tsunoda \cite{MiTs} on the existence of affine rulings and
Platonic $\C^*$-fiber spaces for open smooth surfaces with
non-connected boundaries and a negative logarithmic Kodaira
dimension, Koras and Russell showed first that the second claim is
true under additional assumption that $S \setminus s_0$ has
logarithmic Kodaira dimension at most 1. Then using among other
tools their own deep results and the Kobayashi version of the
BMY-inequality they proved that the case when $S\setminus s_0$ is
of general type can be excluded and, therefore, removed this
additional assumption for (2)\footnote{A remarkable strengthening
of statement (2) is obtained in Gurjar's paper \cite{Gu07}. He
showed that if $X$ is a smooth contractible affine algebraic
variety that admits a dominant morphism from a Euclidean space
then for every reductive group $G$ acting on $X$ so that $X//G$ is
two-dimensional this quotient is isomorphic to the quotient of
$\C^2$ with respect to a linear action of a finite group. His
proof is based on Theorem \ref{se.4.4}.}. In fact, in their new
paper \cite{KoRu07} a stronger result is established.

\bthm\label{se.4.4} Let $S$ be a normal contractible surface of
${\bar \kappa} (S) = -\infty$ with quotient singularities only.
Then ${\bar \kappa} (S_{\rm reg}) = -\infty$. Furthermore, if $S$
has only one singular point then $S$ is isomorphic to the quotient
of $\C^2$ with respect to a linear action of a finite group. \ethm


This concludes the brief description of the proof of Linearization
Conjecture in dimension 3.

\section{The Russell cubic and the scheme of computation of the Makar-Limanov
invariant}

One of the Koras-Russell threefolds is the Russell cubic $R$ that
is the hypersurface $R$ in $\C^4_{x,y,z,t}$ given by
$x+x^2y+z^2+t^3=0$ (for construction of a general Koras-Russell
threefold we refer again to the survey \cite{KaKoMLRu}). Let us
explain first why it was difficult to distinguish $R$ from $\C^3$.

\blem\label{se.5.1} The Russell cubic is diffeomorphic to $\R^6$
as a real manifold and its logarithmic Kodaira dimension is ${\bar
\kappa} (R) = - \infty$. \elem

\bproof Note that the natural projection $\rho : R \to
\C^3_{x,z,t}$ is an affine modification such that its restriction
over $\C_{x,z,t}^3 \setminus \{ x = 0 \}\simeq \C^* \times \C^2$
is an isomorphism. In particular, ${\bar \kappa} (R) ={\bar
\kappa} (\C^* \times \C^2 )=  - \infty$. Furthermore, looking at
the exceptional divisor of $\rho$ one can see that it is an affine
modification that preserves the fundamental groups and homology
(\cite{KaZa}, Prop. 3.1 and Th. 3.1). Thus $R$ has all trivial
senior homotopy groups by the Gurewitch theorem  and $R$ is
contractible by the Whitehead theorem. Now one can apply the
Choudary-Dimca theorem that says that every smooth contractible
affine algebraic variety of dimension at least 3 is diffeomorphic
to a real Euclidean space \cite{ChDi}.
%\footnote{Choudary and Dimca reduces their
%theorem to the $h$-cobordism theorem by proving that the boundary
%of such a threefold at infinity is simply connected. It is worth
%mentioning that this property of the boundary at infinity follows
%from a stronger fact proven earlier by Andreotti, Frankel, and
%Milnor: a boundary of a simply connected Stein manifold is simply
%connected in dimensions at least 3.}.

\eproof

Ramanujam's theorem \cite{Ra} says that if a smooth contractible
surface $S$ is homeomorphic to $\R^4$ or the boundary of $S$ at
infinity is simply connected then $S$ is isomorphic to $\C^2$. The
Gurjar-Miyanishi theorem \cite{GuMi} says that a smooth
contractible surface of non-positive logarithmic Kodaira dimension
is also isomorphic to $\C^2$. Lemma \ref{se.5.1} shows that the
similar invariants cannot help us to prove that $R$ is not
isomorphic to $\C^3$. Some of contractible threefolds can be
distinguished from $\C^3$ by the absence of dominant morphisms
from Euclidean spaces \cite{KaML97a}. One of them is a
hypersurface suggested by Dimca $\{ (x,y,z,t) \in \C^4 |
x+x^4y+y^2z^3+t^5=0 \}$. But the Russell cubic admits a dominant
morphism from $\C^3$. Thus we need something new.

\bdefi\label{se.5.2} A derivation $\de$ on $A$ is locally
nilpotent (LND) if for every $a \in A$ there exists $n=n(a)$ such
that $\de^n (a)=0$. The set of locally nilpotent derivations on
$A$ is denoted by $\LND (A)$. The Makar-Limanov invariant of $A$
is $$ \AK (A)= \bigcap_{\de \in \LND (A)} \Ker \de .$$ Since $A$
is the ring of regular functions on $X$ we may write $\AK (X)$
instead of $\AK (A)$.

\edefi

\brem\label{se.5.3} There exists one-to-one correspondence between
$\C_+$-actions on $X$ and locally nilpotent derivations $\de$ on
$A$. Indeed, one can see that $\exp (\de )$ is a $\C_+$-action
while treating a $\C_+$-action as a phase flow with complex time
one can check that its generating vector field is locally
nilpotent on $A$. Hence $\AK (X)$ coincides with the ring of
regular functions on $X$ that are invariant with respect to any
$\C_+$-action. Clearly we have $\AK (\C^n )=\C$ (in order to show
that the intersection of kernels of LND consists of constants only
it suffices to consider the intersection of kernels of partial
derivatives). \erem

We give a sketch of the proof of the following \cite{ML}.

\bthm\label{se.5.4} The Makar-Limanov invariant of the Russell
cubic coincides with $\AK (R) = \C [x]|_R$, i.e. $R$ is not
isomorphic to $\C^3$. \ethm

{\em Step 1}. Introduce an associated affine variety $\hX$ and the
affine domain $\hA = \C [\hX ]$ for $X$ and $A$ with a map $A \to
\hA, \, a \to {\hat a}$ so that for every $\de \in \LND
(A)\setminus 0$ there exists a unique associated ${\hat \de} \in
\LND (\hA )\setminus 0$. Usually, in order to construct $\hA$ one
needs to consider the filtration generated by a weighted degree
function on $A$ and set $\hA$ equal to the associated graded
algebra. Another way is the geometrical construction which we
present below \cite{KaML07}.

Let $C$ be a germ of a smooth curve at $o \in C$, $f$ be a regular
function on $C$ with a simple zero at $o$, $C^*= C \setminus o$,
$\rho : \cX \to C$ be an affine morphism such that $\cX$ is
normal, $\hX:=\rho^*(o)$ be reduced irreducible, $\cX^*: =\cX
\setminus \rho^{-1}(o) \simeq X \times C^*$ over $C^*$. Then a
nonzero LND $\de$ on $X$ defines a LND $\de^*$ on $\cX^* $. For
some $n \in \Z$ vector field $f^n\de^*$ extends to a LND $\delta$
on $\cX$ with ${\hat \de}=\delta|_{\hX} \ne 0$. We define ${\hat
a}$ via $a$ in a similar manner (that is, if $a^*=a \circ \tau$
where $\tau : \cX^* \to X$  is the natural projection then for
some $n \in \Z$ function $f^na^*$ extends to a regular function on
$\cX$ with a nonzero restriction $\hat a$ to $\hX$).

\bexa\label{se.5.5}  Consider $\rho : \cR =\{ cx+x^2y+z^2+t^3=0 \}
\to C \simeq \C$. For $c \ne 0$, $\rho^{-1}(c)\simeq R$ while
$\rho^{-1}(0) \simeq \hR =\{ x^2y+z^2+t^3 =0\}$. \eexa

Note that unlike $R$ the threefold $\hR$ has a singular line. This
line must be invariant under any $\C_+$-action. This gives hope
that it is easier to compute all associated $\C_+$-actions on
$\hR$ than $\C_+$-actions on $R$. Another encouraging fact is that
associated LND's are homogeneous (see Definition \ref{se.10.6})
which implies that with an appropriate choice of associated
objects the kernels of these LND's are generated by some variables
and functions of form $z^2 +ct^3, \, c \in \C^*$. Since it can be
shown that when $z^2 +ct^3 \in \Ker \de \setminus \{ 0 \}$ for
$\de \in \LND (A)$ then $z,t\in \Ker \de$, we disregard the last
possibility. These observations allow us to find all associated
$\C_+$-actions on $\hR$ and LND's on $\C [\hR ]$. In fact each of
these LND's is proportional to one of the vector fields (with the
coefficient of proportionality in the kernel of the LND): $2z
{\frac {\de}{\de y}} - x^2 {\frac {\de}{\de z}}$ or $3t^2 {\frac
{\de}{\de y}} - x^2 {\frac {\de}{\de t}}$.

{\em Step 2.} The rest of computation is based on the existence of
the degree function $\deg_{\de }(a) = \min \{ n|\de^{n+1}(a)=0\}$
generated by a LND $\de$ on $A\setminus 0$ (i.e., $\Ker \de =\{ a
\in A | \deg_{\de } (a)=0 \}$). The construction of the associated
objects implies the inequality $\deg_{\hat \de } ({\hat a}) \leq
\deg_{\de } (a)$. The computation of associated LND's in Step 1
shows that $\deg_{\hat \de } ({\hat y}) \geq 2$ in $\C [\hR ]$
and, therefore, $\deg_{\de } (y) \geq 2$ in $\C [R ]$.
Furthermore, using different (weighted degree functions for the
construction of) associated varieties one can extend this
inequality to a stronger fact: each $b \in \C [R ]$ with
$\deg_{\de } (b) \leq 1 $ is a restriction of $p\in \C [x,z,t]$.
Now one has to use the following simple results.

\blem\label{se.5.6} Let $\de$ be a nonzero LND on $A$. Then

{\rm (1)} for every $a \in A$, we have $a=a_2/a_1$ where $a_1 \in
\Ker \de$ and $a_2$ is from the algebra generated  over $\Ker \de$
by $b \in A$ with $\deg_{\de }(b) = 1$.

{\rm (2)} the ring $\Ker \de$ is inept, i.e. if $a_1 a_2 \in \Ker
\de \setminus 0$ then $a_1, a_2 \in \Ker \de$. \elem

%\bproof When $\de$ generates a translation choosing $b$ with $\de
%(b)=1$ one can see that each $a \in A$ coincides with an element
%of algebra generated by $b$ over $\Ker \de$. In the general case
%in order to obtain a translation one needs to consider the action
%over a dense open affine subset $U$
%of $X//\C_+$ %(Remark \ref{???})
%which may be chosen as the complement the zero locus of some
%$\alpha \in \Ker \de$. Taking localization with respect to
%$\alpha$ we obtain (1) (i.e. $a_1$ coincides with a power of
%$\alpha$). For the second statement one needs to see that the
%degree of $a_1$ and $a_2$ with respect to $\de$ must be zero since
%this is true for the degree of their product.
%
%\eproof


The description of $b$ before Lemma \ref{se.5.6} implies now that
on $R$ we have $y =p(x,z,t)/q(x,z,t)$ with $q(x,z,t) \in \Ker
\de$. On the other hand $y=-(x+z^2+t^3)/x^2$ which yields
divisibility of $q(x,z,t)$ by $x$. Hence, by Lemma \ref{se.5.6}
(2),  $x \in \Ker \de$ for every $\de$ on $\C [R ]$ and thus $x
\in \AK (R)$.

\section{Limitation of the Makar-Limanov and Derksen invariants}

\bdefi\label{se.6.1} A smooth affine algebraic variety is called
an exotic algebraic (resp. holomorphic) structure on $\C^n$ if it
is diffeomorphic to $\R^{2n}$ but not isomorphic (resp.
biholomorphic) to $\C^n$. \edefi

In particular, we proved in the previous section that the Russell
cubic is an exotic algebraic structure on $\C^3$ (as well as any
other Koras-Russell threefold \cite{KaML97b}). However, we do not
know any technique that allows to check whether $R$ is an exotic
holomorphic structure. Furthermore, the computation described
before works very poorly in dimension 4 and higher. We can check,
say, that the hypersurface $x+x^2y+z^2+t^3+u^5=0$ in $\C^5$ is an
exotic algebraic structure on $\C^4$ (see  \cite{KaML07}) but we
do not know whether $R \times \C$ is an exotic algebraic structure
on $\C^4$. In particular, $R$ is still a potential counterexample
to the following.

{\bf Cancellation Conjecture} (Zariski-Ramanujam). {\em Let $X
\times \C^k$ be isomorphic to $\C^{n+k}$. Then $ X$ is isomorphic
to $\C^n$} (for $n\leq 2$ the answer is positive \cite{Fu79}).

We still have hope that $\AK (R \times \C )$ is nontrivial (i.e.
different from $\C$) but this hope is absent in the case of other
interesting hypersurfaces \cite{KaZa}.

\blem\label{se.6.2} Let $D$ be a hypersurface in
$\C^{n+2}_{u,v,{\bar x}}$ given by $uv=p({\bar x})$ where ${\bar
x}=(x_1, \ldots ,x_n)$ and the zero fiber of the polynomial $p\in
\C [{\bar x}]$ is smooth reduced. Then $D$ has a trivial
Makar-Limanov invariant, i.e. $\AK (D)= \C$. If $p^{-1}(0)$ is
also contractible then $D$ is diffeomorphic to ${\bf R}^{2n+2}$.
\elem

\bexa\label{se.6.3} One of such contractible hypersurfaces is
given by $uv+x+x^2y+z^2+t^3=0$ in $\C^6_{x,y,z,t,u,v}$. A
topological computation shows that nonzero fibers of
$uv+x+x^2y+z^2+t^3$ are not contractible.\eexa

That is, the hypersurface from this example is a potential
counterexample to the following.

{\bf Embedding Conjecture} (Abhyankar-Sathaye). {\em Every
algebraic embedding of $\C^{n-1}$ into $\C^n$ is rectifiable. That
is, the image of $\C^{n-1}$ is a coordinate hyperplane in a
suitable polynomial coordinate system} (for $n=2$ the answer is
positive - the AMS theorem).

\brem\label{se.6.4} (1) There is another potential counterexample
to the Embedding Conjecture that would be much more spectacular;
it is the V\'en\'ereau polynomial\footnote{Freudengurg informed us
that slightly more complicated polynomials with similar properties
were discovered earlier by Bhadwadekar and Dutta \cite{BhDu} and a
general method of constructing such polynomials will appear in
\cite{DaiFr07}.} $v_1:=y+x(xz+y(yt+z^2))=y+x^2z+xy^2t+xyz^2$ on
$\C^4$. All fibers of this polynomial are isomorphic to $\C^3$ but
nobody has found a way to rectify its zero fiber. In smaller
dimensions the similar problem was solved positively by Sathaye
\cite{Sat} (in combination with \cite{BaCoWr}). Even more, every
polynomial on $\C^3$ with infinite number of fibers isomorphic to
$\C^2$ is a variable in a suitable polynomial coordinate system
(see \cite{DaiKa}, \cite{Ka02}, \cite{Mi87}). It is worth
mentioning that if one considers the V\'en\'ereau polynomial on
$\C^n$ (whose coordinate system contains $x,y,z,t$ as variables)
then for a sufficiently large $n$ the zero fiber of $v_1$ becomes
rectifiable. This is a consequence of the Asanuma \cite{As87} and
Bass-Connell-Wright theorems \cite{BaCoWr}.

(2) There is a more general form of this conjecture which says
that each algebraic embedding $\C^k \hookrightarrow \C^{n+k}$ is
rectifiable. This conjecture is true for $n \geq k+2$ \cite{Ka91}
(see also \cite{Sr} and Nori, unpublished). In particular, each
algebraic embedding $\C \hookrightarrow \C^n$ is rectifiable for
$n \ne 3$ (for $n=2$ this is the statement of the AMS theorem and
for $n \geq 4$ it was proven earlier by Craighero \cite{Cr} and
Jelonek \cite{Je}). It is also known that each algebraic embedding
$\C \hookrightarrow \C^3$ is rectifiable by means of analytic
automorphisms of $\C^3$ \cite{Ka92}. Precaution: there are proper
holomorphic embeddings $\C \hookrightarrow \C^n, \, n \geq 2$
non-rectifiable by holomorphic automorphisms. \erem


In fact, $D$ from Lemma \ref{se.6.2} is $m$-transitive for any $m
>0$ (see \cite{KaZa}) and it has the Anders\'en-Lempert property,
i.e. the Lie algebra generated by completely integrable algebraic
vector fields coincides with Lie algebra of all algebraic vector
fields on it \cite{KaKu}. This implies the validity of the
Oka-Grauert-Gromov principle for $D$ and, in particular, each
point of $D$ has a (Fatou-Bieberbach) neighborhood biholomorphic
to a Euclidean space but it is still unknown whether $D$ itself is
biholomorphic to a Euclidean space.

\bdefi\label{se.6.5} Derksen suggested a modification of the
Makar-Limanov invariant which is easier to compute in some cases
\cite{DeKr} (it almost eliminates Step 2 in the computation for
the Russell cubic). His new invariant ${\rm Dr} (A)$ is the
subring of $A$ generated by kernels of all nonzero LND's on $A$.
We say that this invariant is trivial if ${\rm Dr} (A)=A$. This is
so, of course, in the case of $A = \C^{[n]}$.

\edefi

The Derksen and Makar-Limanov invariants are not equivalent, i.e.
one may be trivial while the other is not \cite{CrMa}. For the
Russell cubic $R$ the Derksen invariant coincides with the subring
of $\C [R]$ that consists of the restrictions of all polynomials
from $\C [x,z,t]$ to $R$ but in the case of other hypersurfaces in
this section it is as ineffective as the Makar-Limanov invariant.

\section{Asanuma's construction and other counterexamples to the Linearization Problem}


The first example of a non-linearizable action of $O(2)$ on $\C^4$
was constructed by Schwarz \cite{KrSh}. As we mentioned in the
introduction for every connected reductive group different from a
torus there exists a non-linearizable action  on some $\C^n$ where
$n$ depends on the group \cite{Kn}. For finite groups
non-linearizable actions on $\C^4$ were constructed by
Jauslin-Moser, Masuda, Petrie, and Freudenburg \cite{MaPe},
\cite{MaMJPe}, \cite{FrMJ}. These examples of non-linearizable
$G$-actions were extracted from the existence of non-trivial
$G$-vector bundles on representation spaces of $G$. New ideas were
brought by Asanuma \cite{As99}.

\bthm\label{se.7.1} There exists a non-linearizable algebraic
${\bf R}^*$-action on ${\bf R}^5$. \ethm

Applying the technique of Asanuma, Derksen and Kutzschebauch
constructed a counterexample to linearization in the holomorphic
category \cite{DeKu}.

\bthm\label{se.7.2} There exist non-linearizable holomorphic
$\C^*$-actions on $\C^n$ for any $n \geq 4$. \ethm

\brem\label{se.7.3} Asanuma's construction works if there are
non-rectifiable algebraic embeddings of a line into a Euclidean
space. Say, one can find such an embedding $\R \hookrightarrow
\R^3$ since every unbounded knot in $\R^3$ is isotopic to an
algebraic embedding $\R \hookrightarrow \R^3$ \cite{Sh}. There are
also non-rectifiable proper holomorphic embeddings $\C
\hookrightarrow \C^n, \, n \geq 2$ (e.g., see \cite{FoGlRo},
\cite{Ka92}). As we mentioned before it is unknown whether there
exists an algebraic embedding $\C \hookrightarrow \C^3$
non-rectifiable by algebraic automorphisms. However we consider
this construction in the case of complex algebraic varieties since
it may have very interesting consequences (its reformulation in
the real, higher dimensional, or holomorphic cases is obvious
which yields Theorems \ref{se.7.1} and \ref{se.7.2}). \erem

\bdefi\label{se.7.4} Let $f \in A \setminus 0$, $D=f^{*}(0)
\subset X$, and $C$ be a closed subvariety of $D$ whose defining
ideal in $A$ is $I$. We suppose for simplicity that $D$ and $C$
are reduced irreducible, that $X,D$, and $C$ are smooth, and that
$C$ is at least of codimension 2 in $X$. Then one can consider the
blowing-up ${\rm Bl}_C(X)$. We call the variety $Y={\rm Bl}_C(X)
\setminus D'$ the affine modification of $X$ with center $C$ along
$D$ (where $D'$ is the strict transform of $D$ in ${\rm Bl}_C
(X)$).\edefi

The following easy facts can be found in \cite{KaZa}.

\bprop\label{se.7.5} {\rm (1)} The affine modification $Y$ is a
smooth affine variety with the algebra of regular functions $\C [Y
]=A[I/f]$ where $A[I/f]$ is the algebra generated over $A$ by
rational functions $b_1/f, \ldots , b_n/f$ such that $f,b_1,
\ldots ,b_n$ are generators of $I$.

{\rm (2)} If $X, D, C$ are contractible, so is $Y$. \eprop

\brem\label{se.7.6} If $C$ is a strict complete intersection in
$X$ given by $f=b_1= \ldots =b_k=0$ then the affine modification
$Y$ is isomorphic to the subvariety of $X \times \C^k_{v_1, \ldots
, v_k}$ given by $fv_1-b_1 = \ldots =fv_k -b_k=0$. In this case we
call $Y$ a simple modification of $X$.

\erem

\bexa\label{se.7.7} Let $X=\C^{n+1}_{x_1, \ldots ,x_n,t}$, $D$ be
the hyperplane $t=0$ in $X$, and $C$ be given by $x_1= \ldots
x_k=t=0$ where $k \leq n-1$. Then by Remark \ref{se.7.6} $Y
\subset \C^{n+k+1}_{x_1, \ldots ,x_n,t,v_1 \ldots ,v_k}$ is given
by $tv_1-x_1= \ldots =tv_k-x_k=0$. In particular, $Y \simeq
\C^{n+1}$.

\eexa

\bdefi\label{se.7.8} Let $X=\C^4_{x,y,z,t}$, $f=t$, $D= \{ t=0 \}
\simeq \C^3_{x,y,z}$, and $C$ is the image of $\C$ in $D$ under a
polynomial embedding. Then the affine modification $Y$ of $X$ with
center at $C$ along $D$ is called an Asanuma fourfold. Proposition
\ref{se.7.5} and the Choudary-Dimca theorem imply that it is
diffeomorphic to $\R^8$. \edefi



Asanuma's construction is based on the following elegant fact.

\blem\label{se.7.9} Let $X,D,C,Y$ be as in Definition
\ref{se.7.4}. Set $\tX=X\times \C, \tD =D\times \C$, and ${\tilde
C} = C \times 0 \subset \tD$. Then the affine modification $\tY$
of $\tX$ with center at $ \tilde{C}$ along $\tD$ is isomorphic to
$Y \times \C$.

\elem

\bproof Let $f$ and $I$ be as in Definition \ref{se.7.4} and $f,
b_1, \ldots , b_k$ be generators of $I$. That is, $\C [Y ]$ is
generated over $A$ by $b_1/f, \ldots , b_k/f$. Hence $\C [Y \times
\C_v]$ is generated over $A$ by $b_1/f, \ldots , b_k/f$ and an
independent variable $v$. Then $f, b_1, \ldots ,b_k ,u$ are
generators of the defining ideal of $\tilde{C}$ in $\tX = X \times
\C_u$. Hence $\C [\tY ]$ is generated over $A$ by $b_1/f , \ldots
,b_k/f$, and $u/f$. Sending $v\to u/f$ we get an isomorphism $\C
[\tY ] \simeq \C [Y \times \C]$.

\eproof

\bprop\label{se.7.10} Let $X=\C^4$, $D\simeq \C^3$ be a coordinate
hyperplane in $X$ and $C \subset D$ be isomorphic to $\C$. Suppose
that $Y$ is the affine modification of $X$ with center at $C$
along $D$. Then

{\rm (1)} $Y \simeq \C^4$ provided $C$ is rectifiable in $D$;

{\rm (2)} $Y\times \C \simeq \C^5$; % and it is isomorphic to the
%affine modification of $X \times \C_u$ along $D \times \C_u$ with
%center at the curve $C \times 0$;

{\rm (3)} $Y$ is biholomorphic to $\C^4$;

{\rm (4)} in the case of a non-rectifiable $C$ there exist
non-linearizable $\C^*$-actions on $Y$ and on $\C^5$. \eprop

\bproof Example \ref{se.7.7} implies (1). By Remark \ref{se.6.4}
every algebraic embedding of $\C$ into $\C^4$ is rectifiable.
Hence (2) follows from Lemma \ref{se.7.9} and Example
\ref{se.7.7}. Statement (3) follows from the same argument since
every algebraic embedding of $\C$ into $\C^3$ is rectifiable by a
holomorphic automorphism. In (4) consider first the $\C^*$-action
on $\C^4_{x,y,z,t}$ given by $(x,y,z,t) \to (x,y,z, \lambda t)$
and a neighborhood $U$ of any point $\zeta \in C$ which is of form
$U=U_0 \times \C_t$ where $U_0 \subset \C^3_{x,y,z}$ is open in
the standard topology. We restrict the $\C^*$-action to $U$ and
since $C$ is smooth we can suppose that it is given in $U$ by
${\tilde y}={\tilde z}=t=0$ where $({\tilde x}, {\tilde y},
{\tilde z})$ is a local holomorphic coordinate system on $U_0$ (in
particular, $\zeta =(c_0, 0, 0, 0)$). Let $\sigma : Y \to X$ be
the natural projection. Then $\sigma^{-1}(U)$ is a hypersurface in
$U \times \C_{v_1,v_2}^2$ given by $tv_1-{\tilde y}=tv_2 - {\tilde
z}=0$, i.e. the action can be lifted to a $\C^*$-action $\Phi$ on
$\sigma^{-1}(U)$ as $( {\tilde x}, {\tilde y}, {\tilde z}, t,
v_1,v_2) \to ( {\tilde x}, {\tilde y}, {\tilde z}, \lambda t,
\lambda^{-1} v_1, \lambda^{-1} v_2)$ (clearly, $\Phi$ can be
extended to $Y$). The only $\C^*$-orbit in $X$, whose proper
transforms are non-closed $\C^*$-orbits in $Y$, are of form
${\tilde x}-c_0={\tilde y}={\tilde z}=0$ and the closures of these
orbits contain fixed points of $\Phi$ (given by $t=v_1=v_2=0$).
The algebra of $\C^*$-invariant functions on $Y$ consists of
polynomials in $x, y$, and $z$, i.e. there is a natural
isomorphism between $Y//\Phi$ and $D$. Hence, if $\pi : Y \to D$
is the quotient morphism then $\pi^{-1}(\zeta )$ contains a fixed
point for $\zeta \in D$ iff $\zeta \in C$. Assume that $Y \simeq
\C^4$ and $\Phi$ is linearizable. Looking at the induced action at
fixed points one can see that $\Phi$ is equivalent to $(x,y,z,t)
\to (x,\lambda^{-1}y,\lambda^{-1}z, \lambda t)$. The quotient of
this linear action is $\C^3$ but the image of the fixed point set
in this quotient is a rectifiable line which is the desired
contradiction.


For the last statement of (4) consider the $\C^*$-action on $Y$
generated by the action $(x,y,z,t) \to (x,y,z, \lambda^2 t)$ on
$\C^4$. Extend it to the $\C^*$-action $\Psi$ on $\C^5=Y \times
\C_u$ so that $u \to \lambda^{-1}u$. Then $\C^5//\Psi \simeq
D\times \C \simeq \C^4 $ and for the quotient morphism $\rho :
\C^5 \to D \times \C$ the image of the fixed point set is $C
\times 0$ is contained in $D \times 0 =\rho (Y \times 0)$. For any
orbit $O$ of $\Psi$ the action $\Psi |_O$ is effective iff $O$ is
not in $Y \times 0$. Assume that $\Psi$ is linearizable. Looking
at the induced action at fixed points one can see that $\Psi$ is
equivalent to $(x,y,z,t,u) \to (x,\lambda^{-2}y,\lambda^{-2}z,
\lambda^2 t, \lambda^{-1} u)$. The quotient of this linear action
is $\C^4$ but the image of the orbits with non-effective action
$\Psi$ on them (resp. the fixed point set) is $\C^3$ (resp. $C'$)
and $C'$ is a rectifiable line is this $\C^3$ which is the desired
contradiction.

\eproof

\brem\label{se.7.11} If we could prove that one of Asanuma's
fourfold is an exotic algebraic structure on $\C^4$ this would
produce a counterexample to the Zariski-Ramanujam Conjecture in
dimension 4, a non-rectifiable embedding $\C \hookrightarrow
\C^3$, and an exotic algebraic structure biholomorphic to $\C^4$
(examples of exotic algebraic structures that are not exotic
analytic structures are not known). \erem

\section{Free $\C_+$-actions}

We mentioned that any $\C_+$-action on $\C^2$ is triangular and,
therefore, any free $\C_+$-action is a translation (see Theorem
\ref{se.2.6}).

There are non-triangular $\C_+$-actions on $\C^3$ \cite{Ba}, (and,
furthermore, such actions can be constructed on any $\C^n$ with $n
\geq 3$ \cite{Po87}). Indeed, consider the action $\Phi : \C_+
\times \C^3 \to \C^3$ given by
$$(t, x_1,x_2,x_3)= (x_1,x_2+tx_1u, x_3-2tx_2u-t^2x_1u^2)$$ where $u=x_1x_3+x_2^2$.
% and
%$\partial (x_1)=0, \partial (x_2)= x_1u$, and $\partial (x_3) =
%-2x_2u$ which implies that $\partial (u)=0$.
Then its fixed point set $x_1x_3+x_2^2=0$ is not a cylinder as
required for triangular actions by Remark \ref{se.2.4}.


Starting with $n=4$ not all free $\C_+$-actions on $\C^n$ are
translations since it may happen that $\C^4//\C_+$ is not
homeomorphic to the standard topological quotient $\C^4/\C_+$
(\cite{Wi90}, known also to M. Smith). In Winkelmann's example the
standard quotient $\C^4 /\C_+$ of a free action is not
Hausdorff\footnote{More precisely, the action is given by
$(t,x_1,x_2,x_3,x_4)\to (x_1,x_2+tx_1,x_3+tx_2+t^2x_1/2,
x_4+t(x_2^2-2x_1x_3-1))$. One can see that the points $(0,1,0,0)$
and $(0,-1,0,0)$ do not belong to the same orbit while for
$\varepsilon \ne 0$ the points $(\varepsilon,1,0,0)$ and
$(\varepsilon,-1,0,0)$ are in the same orbit (just take
$t=2/\varepsilon \in \C_+$).} while in the case of translations on
$X \simeq Y \times \C$ the standard topological quotient $X/\C_+$
is homeomorphic to $X//\C_+ \simeq Y$ and, therefore, is affine.

The remaining question about translations on $\C^3$ was tackled by
the following \cite{Ka04}.

\bthm\label{se.8.1} Let $\Phi$ be a $\C_+$-action on a factorial
three-dimensional $X$ with $H_2(X)=H_3(X)=0$. Suppose that the
action is free and $S=X//\Phi$ is smooth.

Then $\Phi$ is a translation, i.e. $X$ is isomorphic to $S \times
\C$ and the action is generated by a translation on the second
factor. \ethm

Indeed, since $\C^3//\C_+ \simeq \C^2$ for any nontrivial
$\C_+$-action \cite{Mi80} we have the long-expected result.

\bcor\label{se.8.2} A free $\C_+$-action on $\C^3$ is a
translation in a suitable coordinate system.

Equivalently, every nowhere vanishing (as a vector field) locally
nilpotent derivation on $\C^{[3]}$ is a partial derivative in a
suitable coordinate system.\ecor

Actually in the case of a smooth contractible $X$ the assumption
on smoothness of $S$ in Theorem \ref{se.8.1} is superfluous by
virtue of the following generalization of Miyanishi's theorem
\cite{KaSa}.


\bthm\label{se.8.3} Let $\Phi$ be a nontrivial $\C_+$-action on a
three-dimensional smooth contractible $X$. Then the quotient
$S=X//\Phi$ is a smooth contractible surface.\ethm

This result has an application to the Van de Ven's conjecture
which says that

{\em Every smooth contractible affine algebraic manifold $X$ is
rational.}

Gurjar and Shastri established the validity of this conjecture in
the case of $\dim X =2$. Combining this result with Theorem
\ref{se.8.3} we get the following.

\bcor\label{se.8.4} If a smooth affine contractible threefold $X$
admits a nontrivial $\C_+$-action, then $X$ is rational.\ecor

\brem The rationality of a smooth affine contractible $X$ equipped
with a nontrivial $\C^*$-action was establishes by Gurjar,
Shastri, and Pradeep \cite{GuSh}, \cite{GuPr}, \cite{GuPrSh}.
\erem


\section{Elements of the proof of Theorem \ref{se.8.3}}

The crucial fact about surjectivity of the quotient morphism $\pi
: X \to S$ was established in \cite{Ka04}. Hence if $s_0$ is a
singularity of $S$ then there is a finite morphism from a germ of
a smooth surface in $X$ onto a germ of $S$ at $s_0$. This implies
that $s_0$ is at worst a quotient singularity \cite{Br}.
Furthermore, surjectivity of $\pi$ implies (via algebraic
topology) that $S$ is contractible and the link of $S$ is a
homology sphere. Thus by another Brieskorn's result $s_0$ is at
worst an $E_8$--singularity (i.e. a singularity of type
$x^2+y^3+z^5=0$) and its link is a Poincar\'e homology 3-sphere
$\cL_0$. Algebraic topology implies also that the link
$\cL_{\infty}$ of $S$ at infinity is also a homology 3-sphere.

Assume first for simplicity that $s_0$ is the only singularity of
$S$. The the part of $S$ between the links yields a cobordism
between $\cL_0$ and $\cL_{\infty}$. Moreover, surjectivity of
$\pi$ in combination with algebraic topology implies that it is a
simply connected homology cobordism. But this type of cobordisms
between $ \cL_0$ and another homology 3-sphere is forbidden by a
theorem of Taubes \cite{Ta}. Thus $S$ cannot have one singular
point.

In the case of several singularities $s_0, s_1, \ldots , s_k$ one
has to consider links $\cL_i$ around $s_i$ and drill ``holes"
joining $\cL_{\infty}$ with $\cL_1 , \ldots , \cL_k$. Then we get
again a simply connected homology cobordism between  $\cL_0$ and
another component of the boundary which is the connected sum of
$\cL_{\infty},\cL_1 , \ldots , \cL_k$. But a connected sum of
homology 3-spheres is again a homology 3-sphere. Thus we get a
contradiction with the theorem of Taubes once more.

It remains to note that being a quotient of a normal space $S$ is
normal itself, i.e. the set of its singular points is at worst
finite. Thus we get the smoothness of $S$.

\section{Partial results on classification of $\C_+$-actions on
affine contractible threefolds}

In this section $X$ is a smooth contractible affine threefold
equipped with a nontrivial $\C_+$-action and $\pi : X \to
S=X//\C_+$ is its algebraic quotient morphism. It is well-known
that there is a curve $\Gamma \subset S$ such that $\pi^{-1}(S
\setminus \Gamma )$ is isomorphic to $(S \setminus \Gamma ) \times
\C$ over $S \setminus \Gamma$. We have the following result from
\cite{Ka04}.

\bthm\label{se.10.1} Each component of the (smallest possible)
curve $\Gamma$ as before is a polynomial curve, i.e. its
normalization is $\C$. \ethm

\brem\label{se.10.2} In all known examples $\Gamma$, and
therefore, each of its components, is contractible. The author
suspects that this is always so. If this fact were correct then,
using Zaidenberg's theorem \cite{Zai88} about the absence of
curves with Euler characteristics 1 on smooth contractible
surfaces of general type, one would get the following: if
logarithmic Kodaira dimension ${\bar \kappa} (S)=2$ then $X \simeq
S \times \C$ and the action is proportional to the the natural
translation on $X$ over $S$. \erem

In the case when ${\bar \kappa} (S)=1$ one can give a complete
classification of such $\C_+$-actions based on the description of
polynomial curves in smooth contractible surfaces of logarithmic
Kodaira dimension 1. It was established by Zaidenberg, Gurjar and
Miyanishi that such a surface $S$ contains a unique curve $L$
isomorphic to a line \cite{Zai88}, \cite{GuMi}. We can claim more
\cite{KaML96}.

\bthm\label{se.10.3} Let $S$ and $L$ be as before (i.e. $S$ is
smooth contractible with ${\bar \kappa} (S)=1$ and $L$ is the line
in it). Then $L$ is the only polynomial curve in $S$. \ethm

\bproof We use the result of tom Dieck and Petrie \cite{tDPe} (see
also \cite{KaML97a}) which says that there exists an affine
modification $\rho : S \to \C^2_{x,y}$ such that (1) $\rho$ is an
isomorphism over $\C^2 \setminus P$ where $P$ is the curve given
by $x^k -y^l=0$ with $k\geq 2$ and $l\geq 2$ being relatively
prime, (2) $\rho (S)$ coincides with the union of $\C^2 \setminus
P$ and point $(1,1)$, and (3) $\rho^{-1}(1,1)=L$. Assume there is
a polynomial curve in $S$ different from $L$. Then its image $C$
in $\C^2$ is a polynomial curve that meets $P$ at most at one
point $(1,1)$. If $C$ does not meet $P$ then it is contained in a
nonzero fiber of $x^k -y^l$ which has a nonzero genus but this is
impossible for a polynomial curve. Thus $C \cap P =(1,1)$. Let $C$
be given by the zero locus of a polynomial $p(x,y)$ and $\C_z \to
\C^2, \, z\to (z^l,z^k)$ be the normalization of $P$. Then
$p(z^l,z^k)=0$ must have the only root at $z=1$ (since the image
of $z=1$ under this normalization is $(1,1)$), i.e. up to a
constant factor $p(z^l,z^k)=(z-1)^m$. Now one can check that the
right-hand side of the last equality has a zero derivative at
$z=0$ while the similar derivative of the left-hand side is
nonzero. This contradiction concludes the proof.

\eproof

\brem\label{se.10.4} Suppose that $X, \pi ,S$, and $\Gamma$ are as
in the beginning of this section. It follows from \cite{Ka04} that
the quotient morphism $\pi$ factors through a surjective affine
modification $\rho : X \to S \times \C$ which generates an
isomorphism over $(S\setminus \Gamma ) \times \C$. In the case
when $\Gamma$ is isomorphic to a line (or a union of disjoint
lines), Theorem 3.1 from \cite{Ka02} implies that $\rho$ is a
composition of a finite number of simple affine modifications (in
the sense of Remark \ref{se.7.6}) whose centers are contractible
irreducible curves (they are also smooth when $X \simeq \C^3$). In
combination with Theorem \ref{se.10.3} this gives a construction
of all smooth affine contractible threefolds equipped with a
nontrivial $\C_+$-action such that the logarithmic Kodaira
dimension of the quotient is 1. However, Freudenburg constructed a
$\C_+$-action on $\C^3$ such that $\G$ consists of two lines in $S
\simeq \C^2$ that meets transversally at one point \cite{Fr98}. It
can be shown that in this case $\rho$ cannot be presented as a
composition of simple modifications.

\erem

\bexa\label{se.10.5} (1) Consider, for instance, the $\C_+$-action
on the Russell cubic $R$ whose associated LND is $\de = 2z {\frac
{\de}{\de y}} - x^2 {\frac {\de}{\de z}}$. The kernel of $\de$ is
$\C [x,t]$. Hence the quotient space $S = \C^2_{x,t}$ and the
affine modification $\rho : R \to S \times \C$ described in Remark
\ref{se.10.4} is nothing but the natural projection $R \to
\C^3_{x,z,t}$. The role of $\Gamma \subset S$ is played by the
line $x=0$. Consider the simple affine modification $X_1 \to
\C^3_{x,z,t}$ along the $(z,t)$-coordinate plane with center at
the polynomial curve $z^2+t^3=0$. By Remark \ref{se.7.6} $X_1$ can
be viewed as the hypersurface $xv=z^2+t^3$ in $\C^4_{x,z,t,v}$.
Take the next affine modification $X_2 \to X_1$ along the divisor
$\{ x=0 \} \subset X_1$ with center at the polynomial curve
$v+1=0$. Again $X_2$ can be viewed as the hypersurface in $X_1
\times \C_y$ given by $xy=-(v+1)$. Multiplying this equality by
$x$ we get $x+x^2y +z^2 +t^3=0$, i.e. $X_2 \simeq R$ and we have
the desired decomposition of $\rho$ into simple affine
modifications.

(2) We describe Freudenburg's example as a LND on $\C [x,y,z]$.
Consider polynomials $F,G,R$ defined as follows:
$$F=xz-y^2; \, R=x^3-Fy; \, G=zF^2-2x^2yF+x^5.$$
In particular, $xG= F^3+R^2$. Consider the derivation $\partial$
such that $\partial (F) =\partial (G)=0$ (i.e. $\Ker \partial =\C
[ F,G]$) and $\partial (R)= -FG$. Since $x=(F^3+R^2)/G$,
$y=(X^3-R)/F$, and
$$z=(G-x^5+2x^2yF)/F^2= (G-2x^2R+x^5)/F^2$$ one can see that
$\partial$ is a well-defined LND on $\C^3_{x,y,z}$. The curve $\G$
is the cross $FG=0$ in the quotient $S
=\C^2_{F,G}$. % and $\theta$ is 2-sheeted over $F=0$ and  5-sheeted
%over $G=0$.
\eexa

Note that in Freudenburg's example the coordinates $F$ and $G$ in
the quotient plane $\C^2_{F,G}$ are homogeneous polynomials on
$\C^3_{x,y,z}$. This is a special case of a homogeneous LND.

\bdefi\label{se.10.6}  Consider a weighted degree function
$\omega$ on $\C [x,y,z]$ such that the weights $\omega (x), \omega
(y), \omega (z)$ are  relatively prime natural numbers, i.e.
$\omega$ generates a weighted projective space $\P^2_{\omega}$. By
Miyanishi's theorem the kernel of any nontrivial LND $\de$ on $\C
[x,y,z]$ is of form $\C [f,g]$ with $f,g \in \C [x,y,z]$. We call
a LND $\de$ homogenous if $f$ and $g$ are homogeneous with respect
to some weighted degree function $\omega$ as before.

\edefi

\bthm\label{se.10.7} {\rm (\cite{Dai} )} Let $\omega$ be as in
Definition \ref{se.10.6} and $f$ and $g \in \C [x,y,z]$ be two
$\omega$-homogeneous polynomials with relatively prime
$\omega$-degrees. Suppose that $C_f$ and $C_g$ are the curves in
$\P^2_{\omega}$ generated by zeros of $f$ and $g$ respectively.
Then the existence of a homogeneous LND on $\C [x,y,z]$ with
kernel $\C [f,g]$ is equivalent to the fact that $\P^2_{\omega}
\setminus (C_f \cup C_g) \simeq \C \times \C^*$.

\ethm

\brem\label{se.10.8} Thus the problem of classification of
homogeneous LND on $\C^{[3]}$ (and the associated homogenous
$\C_+$-actions on $\C^3$) is equivalent to the problem of finding
all curves $C_f$ and $C_g$ as in Theorem \ref{se.10.7}. Such
curves were completely classified by Daigle and Russell
\cite{DaiRu01a} and \cite{DaiRu01b} but their description is a bit
long for this survey.

\erem


\section{$\C^*$-actions on affine surfaces.}

In this section we consider the Classification Problem for
effective $\C^*$-actions on smooth affine algebraic surfaces. As
we mentioned any algebraic $\C^*$-action on $\C^2$ is equivalent
to a linear one: $(x,y) \to (\lambda^kx,\lambda^ly)$ where $(x,y)$
are coordinates on $\C^2$, $\lambda \in \C^*$, and $k,l \in \Z$.
We require that $k$ and $l$ are relatively prime since otherwise
the action is not effective. It is easy to check that any other
linear action $(x,y) \to (\lambda^nx,\lambda^my)$ is equivalent to
this one if the unordered pairs $\{ k, l\}$ and $\{n,m \}$
coincide. This yields the classification of all equivalence
classes of $\C^*$-actions on $\C^2$. Applying Theorem \ref{se.4.3}
we see that for the other smooth toric surfaces $\C \times \C^*$
and $\C^* \times \C^*$ every $\C^*$-action is also equivalent to a
linear one, i.e. we have a classification in this case as well.



In the case of a normal non-toric affine surface $S$ we need some
canonical effective $\C^*$-actions on $S$ (similar to linear
actions on $\C^2$) such that any other action is equivalent to one
of those. They are provided by the Dolgachev-Pikhman-Demazure
(DPD) presentation, introduced by Flenner and Zaidenberg
\cite{FlZa03}, \cite{FlZa05a}, which allows also to describe all
normal affine surfaces that admit nontrivial $\C^*$-actions.

\brem\label{se.11.1} Their approach was generalized in \cite{AlHa}
with $\C^*$-actions replaced by actions of higher dimensional
tori. Another beautiful extension of this DPD-presentation can be
found in \cite{Ko}.

\erem

Recall that the existence of a non-trivial algebraic $\C^*$-action
$\Phi$ on $S$ is equivalent to the existence of a nontrivial
$\Z$-grading $B =\oplus_{i\in \Z} B_i= B_{\geq 0} \oplus_{B_0}
B_{\leq 0}$ of the algebra $B= \C [S ]$ of regular functions on
$S$ where each $b \in B_i$ is an $\lambda^i$-eigenvector of the
isomorphism of $B$ generated by the action of $\lambda \in \C^*$.

If $S//\Phi$ is a curve, we set $C=S//\Phi$. Otherwise there
exists only one fixed (attractive) point $o$ of $\Phi$ and we set
$C=(S\setminus o)/\Phi$. The DPD-presentation describes the
$\Z$-grading in terms of $\Q$-divisors on $C$ and it distinguishes
three types of action.

{\em Elliptic type}: $\Phi$ has an attractive fixed point $o$. %(recall
%that the point is attractive if it is contained in the closures of
%more than two orbits).
Then $C$ is a smooth projective curve and there exists an ample
${\Q}$-divisor $D$ on $C$ so that $B=\oplus_{i\geq 0} H^0(C, \cO
(\lfloor iD \rfloor )u^i$ where $u$ is an unknown and $\lfloor E
\rfloor$ is the integral part of a ${\Q}$-divisor $E$. (If
$B=\oplus_{i\leq 0} H^0(C, \cO (\lfloor iD \rfloor )u^i$ then
switching from $\Phi$ to $\Phi^{-1}$ one can make indices
nonnegative.)

{\em Parabolic type}: $\Phi$ contains a curve of fixed points.
Then this curve is smooth affine and isomorphic to $C$, and there
exists a ${\Q}$-divisor $D$ on $C$ so that $B=\oplus_{i\geq 0}
H^0(C, \cO (\lfloor iD \rfloor )u^i$. (One may need to replace
$\Phi$ by $\Phi^{-1}$ as before to get such a presentation for
$B$.)

{\em Hyperbolic type}: a finite number of fixed points none of
which are attractive. Then $C$ is affine smooth and there exist
${\Q}$-divisors $D_+$ and $D_-$ on $C$ so that $D_++D_-\leq 0$;
$B_{\geq 0} =\oplus_{i\geq 0} H^0(C, \cO (\lfloor iD_+ \rfloor
)u^i$, $B_{\leq 0} =\oplus_{i\leq 0} H^0(C, \cO (\lfloor -iD_-
\rfloor )u^i$, and $B=B_{\geq 0} \oplus_{B_0} B_{\leq 0}$. To
replace of $\Phi$ by $\Phi^{-1}$ in this case is the same as to
interchange $D_+$ and $D_-$.

Vice versa: taking the spectrum of the algebra $B$, that appears
in the description of these three types of actions, one can see
that any smooth affine curve $C$ with $\Q$-divisors $D_+$ and
$D_-$ on it such that $D_++D_-\leq 0$ (resp. a smooth affine or
projective curve $C$ with an appropriate $\Q$-divisor $D$ on it)
corresponds to a normal affine algebraic surface $S$ equipped with
an effective hyperbolic (resp. parabolic or elliptic)
$\C^*$-action.

\brem It is worth mentioning some geometrical features of $S$ in
the hyperbolic case which is the most interesting for us. Suppose
that $\{ p_i \}$ (resp. $\{ q_j \}$) is the set of points of $C$
for which $D_+ (p_i) + D_-(p_i) <0$ (resp. $D_+ (q_j) =- D_-(q_j)
\notin \Z$). By technical reasons we put $D_{\pm} (p_i)={\frac
{\mp e^{\pm}_i}{m^{\pm}_i}}$ where $( e^{\pm}_i, m^{\pm}_i)$ is a
pair of relatively prime integers and $\pm {m^{\pm}_i}>0$.  We put
also $D_{+} (q_j)={\frac {- e_j}{m_j}}$ with $m_j >0$. The
quotient morphism $\pi : S \to C= S//\C^*$ admits a proper
extension $\tS \to C$ such that $\tS \setminus S$ consists of two
sections $C_+$ and $C_-$ of this morphism.  Denote by $p^{\pm}_i$
(resp. $ q_j^{\pm}$) the image of $p_i$ (resp. $ q_j$) under the
natural isomorphism $C\simeq C_{\pm}$. Every singularity of $\tS$
is automatically a quotient singularity and the only singularities
of $\tS$ on $C_{\pm}$ are points from $\{ p^{\pm}_i, q^{\pm}_j \}$
with each $p^{\pm}_i $ being of type $(\pm m^{\pm}_i, -e^{\pm}_i)$
\footnote{Recall that a cyclic quotient singularity of type
$(m,e)$ is biholomorphic to the singularity of $\C^2/\Z_m$ where
$\Z_m$ is the cyclic group generated by a primitive $m$-root
$\zeta$ of unity that acts on $\C^2_{x,y}$ by formula $(x,y) \to
(\zeta x,\zeta^e y)$ with $e$ and $m$ being relatively prime.} and
each $q^{\pm}_j $ being of type $({m}_j, {\mp}e_j)$. A complete
description of the quotient morphism fibers and the other
singularities of $\tS$ can be also obtained from this data. Say,
every fiber $\pi^*(p_i)$ is not irreducible and every fiber $\pi^*
(q_j)$ is not reduced. \erem
% neighborhood of a point $c \in C \setminus ({\rm
%supp} \, D_+ \cup {\rm supp} \, D_-) $ the quotient morphism
%yields a locally trivial $\C^*$-fibration.

Consider an effective $\C^*$-action $\Phi'$ on our non-toric
surface $S$ and suppose that a curve $C'$ plays the same role for
$\Phi'$ as $C$ for $\Phi$. It can be shown that $\Phi$ and $\Phi'$
are of
the same type and $C \simeq C'$. %Therefore, we identify these
%curves.

\bthm\label{se.11.2} {\rm (\cite{FlZa03})} %Let $\Phi$ and $\Phi'$
%be effective $\C^*$-actions on a non-toric surface $S$, and let
%$C$ and $C'\simeq C$ be as before.
Let $D$ (resp. $D'$) be the $\Q$-divisor on $C\simeq C'$ that
appears in the DPD-presentation of $\Phi$ (resp. $\Phi'$) in the
elliptic or parabolic cases. In the hyperbolic case we denote the
similar divisors on $C$ by $D_+$ and $D_-$ (resp. $D_+'$ and
$D_-'$). Then $\Phi$ is equivalent to $\Phi'$ if and only if an
isomorphism $C \simeq C'$ can be chosen so that

{\rm (1)} $D$ is linearly equivalent to $D'$ in the elliptic and
parabolic cases;

{\rm (2)} $D_+'=D_++P$ and $D_-'=D_--P$ for a principal divisor
$P$ in the hyperbolic case.

\ethm

Thus in order to find all equivalence classes of effective
$\C^*$-actions one needs to classify all possible
DPD-presentations for a fixed surface $S$ up to a linear
equivalence (in the parabolic and elliptic case) or up to a
principal divisor (in the hyperbolic case). Flenner and Zaidenberg
established a uniqueness of DPD-presentations for non-toric
surfaces with a non-trivial Makar-Limanov invariant
\cite{FlZa05a}, \cite{FlZa05b}.

\bthm\label{se.11.2a} Let $\Phi$ and $\Phi'$ be effective
$\C^*$-actions on a non-toric affine surface $S$ such that $\AK
(S)\ne \C$. Then, replacing $\Phi$ by $\Phi^{-1}$ if necessary,
one can choose an isomorphism between the curves $C$ and $C'$ so
that $D$ and $D'$ (resp. $D_{\pm}$ and $D_{\pm}'$) in the
DPD-presentations satisfy condition (1) (resp. (2)) of Theorem
\ref{se.11.2} (in particular, there are at most two equivalence
classes of effective $\C^*$-actions on $S$). Moreover, $\Phi$ (or
$\Phi^{-1}$) is conjugated to $\Phi'$ by an element of a
$\C_+$-action\footnote{Thus, if $\AK (S)=\C [S]$ then either
$\Phi$ or $\Phi^{-1}$ is equal to $\Phi'$.}. \ethm

For a non-toric surface with a trivial Makar-Limanov invariant
effective $\C^*$-actions are automatically hyperbolic
\cite{FKZ07b} and the curve $C$ is always isomorphic to $\C$. In
this case the situation is much more complicated and Russell found
examples of such smooth surfaces with more than two equivalence
classes of effective $\C^*$-actions.

\bdefi\label{se.11.3} Let ${\F}_n\to {\PP}^1$ be a Hirzebruch
surface over ${\PP}^1$ and $L$ be its section with $L^2=k+1$. If
$L$ is ample (i.e. $k \geq n$) then ${\F}_n\setminus L$ is a
Danilov-Gizatullin surface\footnote{We use this name because of
the theorem of Danilov and Gizatullin which states that two of
such surfaces are isomorphic if and only if they have the same $k$
in their construction \cite{DaGi}. This fact was crucial for
Russell.}. \edefi

\bthm\label{se.11.4} {\rm (\cite{FKZ07b})} There are exactly $k$
equivalence classes of effective $\C^*$-actions on a surface from
Definition \ref{se.11.3}. More precisely, every action is
hyperbolic and its DPD-presentation is of form $C \simeq \C$,
$D_+=-{\frac{1}{r}} [p_0]$, $D_-=-{\frac{1}{k+1-r}} [p_1]$ for
distinct $p_0,p_1 \in C$ and $r=1, \ldots , k$. \ethm

Danilov-Gizatullin surfaces are contained in the wider class of
Gizatullin surfaces.

\bdefi A normal affine algebraic surface $S$ is a Gizatullin one
if it has a simple normal crossing completion $\bS$ such its
boundary divisor ${\bS} \setminus S$ consists of rational curves
and the dual weighted graph $\G$ of this divisor is
linear\footnote{Recall that each irreducible component of ${\bS}
\setminus S$ is viewed as a vertex of $\G$ with its weight being
the selfintersection number of the component, and two vertices are
connected by an edge if the corresponding components meet.} (this
graph is called a zigzag). If $S$ admits also a nontrivial
$\C^*$-action we say that it is a Gizatullin $\C^*$-surface.
\edefi

\brem It is worth mentioning that with the exception of $\C \times
\C^*$ the set of Gizatullin surfaces coincides with the set of
surfaces with a trivial Makar-Limanov invariant. Each of these
surfaces is also quasi-homogeneous, i.e. it contains an orbit of
the algebraic automorphism group whose complement is at most
finite (the only quasi-homogeneous surface that is not a
Gizatullin one is $\C^*\times \C^*$).  In particular, such
surfaces possess huge automorphism groups. That is why the
Classification Problem was the most difficult in this case.\erem

A simple normal crossing completion $\bS$ of a non-toric $S$ can
be always chosen so that its zigzag is standard, i.e. $\Gamma$
consists of vertices $C_i, \, i=0,1, \ldots ,n$ (where $n\geq 1$)
of weights $w_i$ such that $w_0=w_1=0$ and $w_i \leq -2$ for the
rest of indices. The standard zigzag is unique up to the reversion
of its nonzero part and smooth Gizatullin $\C^*$-surfaces
can be divided into the following three collections (\cite{FKZ07b}):\\
(1) Danilov-Gizatullin surfaces (for these surfaces $w_i
=-2$ for all $i \geq 2$ in their standard zigzags);\\
(2) special $\C^*$-Gizatullin surfaces whose standard zigzags have
$w_i
=-2$ for all $i \geq 2$ except for one index $2 \leq k \leq n$;\\
(3) the rest of smooth Gizatullin $\C^*$-surfaces.

Collection (3) was classified in \cite{FKZa}, and we have the
following uniqueness result.

\bthm\label{unique}  For every surface $S$ from the third
collection up to interchange of $D_+$ and $D_-$ any two
DPD-presentations are equivalent, i.e. their divisors satisfy
condition (2) of Theorem \ref{se.11.2}. In particular, there are
at most two equivalence classes of $\C^*$-actions. \ethm

The most difficult remaining case of the second collection will be
settled in the coming paper \cite{FKZb} which concludes the
classification of effective $\C^*$-actions on smooth affine
surfaces. Compared with the other cases the answer is really
amazing but first we need a few remarks.

%Since the divisors $D_+$ and $D_-$ in the DPD-presentation can be
%changed by a principal divisor we suppose below that the integral
%part $\lfloor D_+ \rfloor$ of $D_+$ is zero.
It can be shown \cite{FKZ07b} that for any Gizatullin
$\C^*$-surface the support of the fractional part $\{ D_+ \}= D_+-
\lfloor D_+ \rfloor $ of $D_+$ (resp. $\{ D_- \}$ of $D_-$) is
concentrated at most at one point $p_-$ (resp. $p_+$). For special
Gizatullin surfaces these fractional parts are both nonzero and
$p_+ \ne p_-$ unless $k=2$ or $n$ in the description of collection
(2). In the case of $k=2$ or $n$ either $\{ D_- \}$ or $\{ D_+ \}$
vanishes and the support of the fractional part of $D_+ + D_-$ is
concentrated at most at one point $p$ (when $n=2$ both fractional
parts vanish).

\bthm\label{non-unique} Let $S$ and $S'$ be special Gizatullin
surfaces with the same standard zigzag and with DPD-presentations
$D_+, D_-$ and $D_+', D_-'$ on curves $C \simeq \C$ and $C' \simeq
\C$ respectively.
%Suppose also that $\lfloor D_+ \rfloor = \lfloor D_+' \rfloor =0$.
Then $S$ and $S'$ are isomorphic if and only if an isomorphism $C
\simeq C'$ can be chosen so that $\lfloor D_+ + D_-
\rfloor=\lfloor D_+'+D_-' \rfloor =: E$. In particular, (by
Theorem \ref{se.11.2}) up to automorphisms of $\C$ preserving $E$
each equivalence class of $\C^*$-actions on $S$ is determined
uniquely by the continuous parameters $p_+$ and $p_-$ in the case
of $k \ne 2$ or $n$,  and by the continuous parameter $p$ in the
case of $k=2$ or $k=n>2$. \ethm

\begin{thebibliography}{KMMH}

\bibitem[AlHa]{AlHa} K.\ Altmann, J.\ Hausen, {\em Polyhedral Divisors and Algebraic Torus
Actions}, Math. Ann. {\bf 334} (2006), no. 3, 557--607.


\bibitem[As87]{As87} T.\ Asanuma, {\em Polynomial fibre rings of algebras over Noetherian
rings}, Invent. Math. {\bf 87} (1987), no. 1, 101--127.

\bibitem[As99]{As99} T.\ Asanuma, {\em Non-linearizable algebraic $k^*$-actions on
affine spaces}, Invent. Math. {\bf 138} (1999), no. 2, 281--306.

\bibitem[Ba]{Ba} H.\ Bass, {\em Algebraic group actions on affine spaces}, Group
actions on rings (Brunswick, Maine, 1984), 1--23, Contemp. Math.,
 {\bf 43}, Amer. Math. Soc., Providence, RI, 1985.

\bibitem[BaCoWr]{BaCoWr} H.\ Bass, E.\ H.\ Connell, D.\ L.\ Wright, {\em Locally polynomial
algebras are symmetric}, Bull. Amer. Math. Soc. {\bf 82} (1976),
no. 5, 719--720.

\bibitem[BeHa]{BeHa} F.\ Berchtold, J.\ Hausen, {\em Demushkin's theorem in
codimension one}, Math. Z. {\bf 244} (2003), no. 4, 697--703.

\bibitem[BhDu]{BhDu} S. M. Bhatwadekar, A. K. Dutta, {\em On affine
fibrations}, Commutative Algebra (Trieste, 1992), 1--17, World
Sci. Publ., River Edge, NJ, 1994.

\bibitem[BB]{BB} A.\ Bia\l ynicki-Birula, {\em Remarks on the action of an algebraic torus on
$k^{n}$}, II. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom.
Phys. {\bf 15} (1967), 123--125.


\bibitem[Br]{Br}  E.~Brieskorn, {\em Rationale singularit\"aten
                       komplexer Fl\"achen}, Invent. Math. {\bf 4}
                       (1967/1968), 336--358.

\bibitem[ChDi]{ChDi} A.\ D.\ R.\ Choudary, A.\ Dimca, {\em Complex hypersurfaces diffeomorphic
to affine spaces}, Kodai Math. J. {\bf 17} (1994), no. 2,
171--178.

\bibitem[CrMa]{CrMa} A.\ Crachiola, S.\ Maubach, {\em The Derksen invariant vs. the
Makar-Limanov invariant}, Proc. Amer. Math. Soc. {\bf 131} (2003),
no. 11, 3365--3369 (electronic).

\bibitem[Cr]{Cr} P. C. Craighero, {\em A result on $m$-flats in $A\sp n\sb k$},
Rend. Sem. Mat. Univ. Padova {\bf 75} (1986), 39--46.

\bibitem[Dai]{Dai} D.\ Daigle, {\em Homogeneous locally nilpotent derivations of
$k[X,Y,Z]$},  J. Pure Appl. Algebra {\bf 128} (1998), no. 2,
109--132.

\bibitem[DaiFr99]{DaiFr99} D.\ Daigle, G.\ Freudenburg, {\em A counterexample to Hilbert's
fourteenth problem in dimension $5$}, J. Algebra {\bf 221} (1999),
no. 2, 528--535.

\bibitem[DaiFr07]{DaiFr07} D.\ Daigle, G.\ Freudenburg, {\em
Families of affine lines}, preprint, 7 p..

\bibitem[DaiRu01a]{DaiRu01a} D.\ Daigle, P.\ Russell, {\em On weighted projective planes and
their affine rulings}, Osaka J. Math. {\bf 38} (2001), no. 1,
101--150.

\bibitem[DaiRu01b]{DaiRu01b}
D.\ Daigle, P.\ Russell, I. {\em Affine rulings of normal rational
surfaces}, Osaka J.\ Math.\ {\bf 38} (2001), 37--100; II. {\em On
weighted projective planes and their affine rulings}, {\it ibid.},
101--150.

\bibitem[DaiKa]{DaiKa}
D.\ Daigle, S.\ Kaliman {\em A note on locally nilpotent
derivations and variables of $k[X,Y,Z]$}, Canadian. Math. J., (to
appear), 9 p..


\bibitem[DaGi]{DaGi}
V.\ I.\ Danilov, M.\ H.\ Gizatullin, {\em Automorphisms of affine
surfaces}, I. Math. USSR Izv. {\bf 9} (1975), 493--534; II. {\it
ibid.} {\bf 11} (1977), 51--98.

\bibitem[De]{De} A.\ S.\ Demushkin, {\em Combinatorial invariance of toric
singularities} (Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh.
{\bf 2} (1982), 80--87, 117.

\bibitem[DeKr]{DeKr} H.\ Derksen, H.\ Kraft, {\em Constructive invariant
theory}, Alg\`ebre non commutative, groupes quantiques et
invariants (Reims, 1995), 221--244, S\'emin. Congr.,  {\bf 2},
Soc. Math. France, Paris, 1997.

\bibitem[DeKu]{DeKu} H. Derksen, F. Kutzschebauch, {\em Nonlinearizable holomorphic
group actions}, Math. Ann. {\bf 311} (1998), no. 1, 41--53.

\bibitem[tDPe]{tDPe} T.\ tom Dieck, T.\ Petrie, {\em Homology planes and algebraic
curves}, Osaka J. Math. {\bf 30} (1993), no. 4, 855--886.

\bibitem[FlZa03]{FlZa03}
H.\ Flenner, M.\ Zaidenberg, {\em Normal affine surfaces with
$\C^*$-actions}, Osaka J.\ Math.\  {\bf 40} (2003),  981--1009.

\bibitem[FlZa05a]{FlZa05a} H.\ Flenner, M.\ Zaidenberg, {\em Locally nilpotent derivations
on affine surfaces with a $\C^*$-action}, Osaka J. Math. {\bf 42}
(2005), no. 4, 931--974.

\bibitem[FlZa05b]{FlZa05b} H.\ Flenner, M.\ Zaidenberg,
{\em On the uniqueness of $\C^*$-actions on affine surfaces},
Affine algebraic geometry, 97--111, Contemp. Math.,  {\bf 369},
Amer. Math. Soc., Providence, RI, 2005.


\bibitem[FKZ07a]{FKZ07a} H.\ Flenner, S.\ Kaliman, M.\ Zaidenberg,  {\em Birational transformations
of weighted graphs}, Affine algebraic geometry, 107--147, Osaka
Univ. Press, Osaka, 2007.


\bibitem[FKZ07b]{FKZ07b} H.\ Flenner, S.\ Kaliman, M.\ Zaidenberg, {\em Completions of
$\C^*$-surfaces}, Affine algebraic geometry, 149--201, Osaka Univ.
Press, Osaka, 2007.

\bibitem[FKZa]{FKZa} H.\ Flenner, Sh.\ Kaliman, M.\ Zaidenberg,
{\em Uniqueness of $\C^*$- and $\C_+$-actions on Gizatullin
surfaces}, Transfor. Groups, (to appear), 43 p..

\bibitem[FKZb]{FKZb} H.\ Flenner, Sh.\ Kaliman, M.\ Zaidenberg,
{\em Smooth Gizatullin surfaces with non-unique $\C^*$-actions},
(to appear).

\bibitem[Fr98]{Fr98} G.\ Freudenburg, {\em Actions of $G_a$ on $\A^3$ defined by
homogeneous derivations}, J. Pure Appl. Algebra  {\bf 126} (1998),
no. 1-3, 169--181.

\bibitem[Fr00]{Fr00} G.\ Freudenburg, {\em A counterexample to Hilbert's fourteenth problem
in dimension six}, Transform. Groups  {\bf 5} (2000), no. 1,
61--71.

\bibitem[FrMJ]{FrMJ} G.\ Freudenburg, L.\ Moser-Jauslin, {\em A nonlinearizable action of
$S_3$ on $\C^4$}, Ann. Inst. Fourier (Grenoble)  {\bf 52} (2002),
no. 1, 133--143.

\bibitem[FoGlRo]{FoGlRo} F.\ Forstneric, J.\ Globevnik, J.-P.\ Rosay, {\em
Nonstraightenable complex lines in $\C^2$}, Ark. Mat.  {\bf 34}
(1996), no. 1, 97--101.

\bibitem[Fu79]{Fu79} T.\ Fujita, {\em Cancellation problem of complete varieties},
Invent. Math.  {\bf 64} (1981), no. 1, 119--121.

\bibitem[Fu82]{Fu82} T. Fujita, {\em On the topology of non-complete
algebraic surfaces}, J. Fac. Sci. Univ. Tokyo, Sect.IA, {\bf 29}
(1982), 503--566.

\bibitem[Gi]{Gi} M.\ H.\ Gizatullin, I.
{\em Affine surfaces that are quasihomogeneous with respect to an
algebraic group}, Math. USSR Izv.  {\bf 5} (1971), 754--769; II.
{\em Quasihomogeneous affine surfaces}, {\it ibid.} 1057--1081.

\bibitem[Gu07]{Gu07} R. V. Gurjar {\em Two-dimensional quotients of $\C^n$ are
isomorphic to $\C^2/\Gamma$}, Transform. Groups  {\bf 12} (2007),
no. 1, 117--125.

\bibitem[GuMi]{GuMi} R.~V. Gurjar, M.\ Miyanishi, {Affine surfaces with
${\overline\kappa} \leq 1$}, Algebraic geometry and commutative
algebra, Vol. I, 99--124, Kinokuniya, Tokyo, 1988.

\bibitem[GuSh]{GuSh}   R.~V.~Gurjar, A.~R.~Shastri, {\em On the
                       rationality of complex homology $2$-cells. II},
                       J. Math. Soc. Japan {\bf 41} (1989), no. 2,
                       175--212.

\bibitem[GuPrSh]{GuPrSh} R.~V.~Gurjar, C.~R.~Pradeep, A.~R.~Shastri,
                       {\em On rationality of logarithmic $\Q$-homology
                       planes. II}, Osaka J. Math. {\bf 34} (1997), no. 3,
                      725--743.


\bibitem[GuPr]{GuPr}   R.~V.~Gurjar, C.~R.~Pradeep, {\em $\Q$-homology
                       planes are rational. III}, Osaka J. Math. {\bf 36}
                       (1999), no. 2, 259--335.

\bibitem[Gu61]{Gu61} A.\ Gutwirth, {\em An inequality for certain pencils of plane
curves}, Proc. Amer. Math. Soc.  {\bf 12} (1961), 631--638.

\bibitem[Gu62]{Gu62} A.\ Gutwirth, {\em The action of an algebraic torus on the affine
plane}, Trans. Amer. Math. Soc.  {\bf 105} (1962), 407--414.

\bibitem[Je]{Je} Z.\ Jelonek, {\em The extension of regular and rational
embeddings}, Math. Ann.  {\bf 277} (1987), no. 1, 113--120.

\bibitem[Ka91]{Ka91} S.\ Kaliman, {\em Extensions of isomorphisms between affine
algebraic subvarieties of $k^n$ to automorphisms of $k^n$}, Proc.
Amer. Math. Soc.  {\bf 113} (1991), no. 2, 325--334.

\bibitem[Ka92]{Ka92} S.\ Kaliman, {\em Isotopic embeddings of affine algebraic varieties
into $\C^n$}, The Madison Symposium on Complex Analysis (Madison,
WI, 1991), 291--295, Contemp. Math.,  {\bf 137}, Amer. Math. Soc.,
Providence, RI, 1992.

\bibitem[Ka02]{Ka02} S.\ Kaliman, {\em Polynomials with general $\C^2$-fibers are
variables}, Pacific J. Math.  {\bf 203} (2002), no. 1, 161--190.

\bibitem[Ka04]{Ka04} S.\ Kaliman, {\em Free $\C_+$-actions on $\C^3$ are
translations}, Invent. Math.  {\bf 156} (2004), no. 1, 163--173.

\bibitem[KaKoMLRu]{KaKoMLRu} S.\ Kaliman, M.\ Koras, L.\ Makar-Limanov, P.\ Russell, {\em
$\C^*$-actions on $\C^3$ are linearizable}, Electron. Res.
Announc. Amer. Math. Soc.  {\bf 3} (1997), 63--71.

\bibitem[KaKu]{KaKu} S. Kaliman, F. Kutzschebauch, {\em Density property for hypersurfaces
$UV=(\bar X)$}, Math. Zeit. , (to appear).

\bibitem[KaML96]{KaML96} S.\ Kaliman, L.\ Makar-Limanov, {\em On morphisms into
contractible surfaces of Kodaira logarithmic dimension $1$}, J.
Math. Soc. Japan  {\bf 48} (1996), no. 4, 797--810.

\bibitem[KaML97a]{KaML97a} S.\ Kaliman, L.\ Makar-Limanov, {\em Affine algebraic manifolds
without dominant morphisms from Euclidean spaces}, Rocky Mountain
J. Math.  {\bf 27} (1997), no. 2, 601--610.

\bibitem[KaML97b]{KaML97b} S.\ Kaliman, L.\ Makar-Limanov, {\em On the Russell-Koras
contractible threefolds}, J. Algebraic Geom.  {\bf 6} (1997), no.
2, 247--268.

\bibitem[KaML07]{KaML07} S. Kaliman, L.\ Makar-Limanov, {\em
AK-invariant of affine domains},  Affine algebraic geometry,
231--255, Osaka Univ. Press, Osaka, 2007.

\bibitem[KaSa]{KaSa} S.\ Kaliman, N.\ Saveliev, {\em $\C_+$-actions on
contractible threefolds}, Michigan Math. J.  {\bf 52} (2004), no.
3, 619--625.

\bibitem[KaZa]{KaZa} S.\ Kaliman, M.\ Zaidenberg, {\em Affine modifications and affine
hypersurfaces with a very transitive automorphism group},
Transform. Groups  {\bf 4} (1999), no. 1, 53--95.

\bibitem[KaZaVe]{KaZaVe} S.\ Kaliman, S.\ V\'en\'ereau, M.\ Zaidenberg, {\em Simple
birational extensions of the polynomial algebra $\C^{[3]}$},
Trans. Amer. Math. Soc.  {\bf 356} (2004), no. 2, 509--555.


\bibitem[Kam]{Kam} T.\ Kambayashi, {\em Automorphism group of a polynomial ring and
algebraic group action on an affine space}, J. Algebra  {\bf 60}
(1979), no. 2, 439--451.

\bibitem[Kn]{Kn} F.\ Knop, {\em Nichtlinearisierbare Operationen halbeinfacher Gruppen auf affinen R\"aumen} (German),
Invent. Math.  {\bf 105} (1991), no. 1, 217--220.

\bibitem[Ko]{Ko} J.\ Koll\'ar {\em Seifert $G_m$-bundles}, preprint (2004), 21 p.,
math.AG/0404386.

\bibitem[KoRu99]{KoRu99} M.\ Koras, P.\ Russell, {\em $\C^*$-actions on $C^3$: the
smooth locus of the quotient is not of hyperbolic type}, J.
Algebraic Geom.  {\bf 8} (1999), no. 4, 603--694.

\bibitem[KoRu07]{KoRu07} M.\ Koras, P.\ Russell, {\em Contractible affine surfaces with a quotient
singularity}, Transform. Groups  {\bf 12} (2007), no. 2, 293-340.

\bibitem[Kr96]{Kr96} H.\ Kraft, {\em Challenging problems on affine $n$-space},
S\'eminaire Bourbaki, Vol. 1994/95. Ast\'erisque No.  {\bf 237}
(1996), Exp. No. 802, 5, 295--317.

\bibitem[KrPo]{KrPo} H.\ Kraft, V.\ L.\ Popov {\em Semisimple group actions on
the three-dimensional affine space are linear},  Comment. Math.
Helv.  {\bf 60} (1985), no. 3, 466--479.

%\bibitem[Kr90]{Kr90} H.\ Kraft, {\em $C^*$-actions on affine space}, Operator
%algebras, unitary representations, enveloping algebras, and
%invariant theory (Paris, 1989), 561--579, Progr. Math., 92,
%Birkhäuser Boston, Boston, MA, 1990.

\bibitem[KrSh]{KrSh} H.\ Kraft, G.\ W. Schwarz, {\em Reductive group actions with
one-dimensional quotient}, Inst. Hautes \'Etudes Sci. Publ. Math.
{\bf 76} (1992), 1--97.

%\bibitem[Kr91]{Kr91} H.\ Kraft, {\em Algebraic automorphisms of affine
%space}, Proceedings of the Hyderabad Conference on Algebraic
%Groups (Hyderabad, 1989), 251--274, Manoj Prakashan, Madras, 1991.

\bibitem[Ku]{Ku} S.\ Kuroda, {\em A counterexample to the fourteenth problem of
Hilbert in dimension three}, Michigan Math. J.  {\bf 53} (2005),
no. 1, 123--132.

\bibitem[ML]{ML} L.\ Makar-Limanov, {\em On the hypersurface $x+x^2y+z^2+t^3=0$
in $\C^4$ or a $\C^3$-like threefold which is not $\C^3$}, Israel
J. Math.  {\bf 96} (1996), part B, 419--429.

\bibitem[MaPe]{MaPe} M.\ Masuda, T.\ Petrie, {\em Equivariant algebraic vector bundles
over representations of reductive groups: theory}, Proc. Nat.
Acad. Sci. U.S.A.  {\bf 88} (1991), no. 20, 9061--9064.


\bibitem[MaMJPe]{MaMJPe} M.\ Masuda, L.\ Moser-Jauslin, T.\ Petrie, {\em Equivariant
algebraic vector bundles over representations of reductive groups:
applications}, Proc. Nat. Acad. Sci. U.S.A.  {\bf 88} (1991), no.
20, 9065--9066.


\bibitem[Mi80]{Mi80} M.\ Miyanishi, {\em Regular subrings of a polynomial ring}, Osaka
J. Math.  {\bf 17} (1980), no. 2, 329--338.

\bibitem[Mi87]{Mi87} M.\ Miyanishi, {\em Algebraic characterizations of the affine
$3$-space}, Algebraic Geometry Seminar (Singapore, 1987), 53--67,
World Sci. Publishing, Singapore, 1988.

\bibitem[Mi04]{Mi04} M.\ Miyanishi, {\em Recent developments in
affine algebraic geometry: from the personal viewpoints of the
author}, Affine algebraic geometry, 307--378, Osaka Univ. Press,
Osaka, 2007.

\bibitem[MiTs]{MiTs} M.\ Miyanishi, S.\ Tsunoda, {\em Noncomplete algebraic
surfaces with logarithmic Kodaira dimension $-\infty$ and with
nonconnected boundaries at infinity}, Japan. J. Math. (N.S.) {\bf
10} (1984), no. 2, 195--242.

%\bibitem[Pa]{Pa} D.\ I.\ Panyushev, {\em Semisimple groups of automorphisms of a
%four-dimensional affine space} (Russian) Izv. Akad. Nauk SSSR Ser.
%Mat. 47 (1983), no. 4, 881--894. See also {\em A correction to my
%paper: "Semisimple groups of automorphisms of a four-dimensional
%affine space"} (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985),
%no. 6, 1336--1337.

\bibitem[Po87]{Po87} V.\ L.\ Popov, {\em On actions of $G_a$ on
$A^n$}, Algebraic groups, Utrecht 1986, 237--242, Lecture Notes in
Math.,  {\bf 1271}, Springer, Berlin, 1987.

\bibitem[Po01]{Po01} V.\ L.\ Popov, {\em On polynomial automorphisms of affine
spaces} (Russian) Izv. Ross. Akad. Nauk Ser. Mat.  {\bf 65}
(2001), no. 3, 153--174; translation in Izv. Math.  {\bf 65}
(2001), no. 3, 569--587.

\bibitem[Ra]{Ra} C.\ P.\ Ramanujam, {\em A topological characterisation of the affine
plane as an algebraic variety}, Ann. of Math. (2)  {\bf 94}
(1971), 69--88.

\bibitem[Re]{Re} R.\ Rentschler, {\em Op\'erations du groupe additif sur le plan
affine}, (French) C. R. Acad. Sci. Paris S\'er. A-B  {\bf 267}
(1968), 384--387.

\bibitem[Ro]{Ro} P.\ C.\ Roberts, {\em A prime ideal in a polynomial ring whose symbolic
blow-up is not Noetherian}, Proc. Amer. Math. Soc.  {\bf 94}
(1985), no. 4, 589--592.

\bibitem[Sat]{Sat} A. Sathaye, {\em Polynomial ring in two variables
over a D. V. R.: A criterion}, Invent. Math. {\bf 74} (1983),
159--168.

\bibitem[Se]{Se} J.-P.\ Serre, {\em Trees} (Translated from the French by John
Stillwell), Springer-Verlag, Berlin-New York, 1980.

\bibitem[Sh]{Sh} A.\ R.\ Shastri, {\em Polynomial representations of knots}, Tohoku
Math. J. (2) {\bf 44} (1992), no. 1, 11--17.

\bibitem[ShUm04a]{ShUm04a} I.\ Shestakov, U.\ Umirbaev, {\em Poisson brackets and
two-generated subalgebras of rings of polynomials}, J. Amer. Math.
Soc.  {\bf 17} (2004), no. 1, 181--196.

\bibitem[ShUm04b]{ShUm04b} I.\ Shestakov, U.\ Umirbaev, {\em The tame and the wild
automorphisms of polynomial rings in three variables}, J. Amer.
Math. Soc.  {\bf 17} (2004), no. 1, 197--227.

\bibitem[Sr]{Sr} V. Srinivas, {\em On the embedding dimension of an affine
variety}, Math. Ann.  {\bf 289} (1991), no. 1, 125--132.

\bibitem[Ta]{Ta}      C.~H.~Taubes, {\em Gauge theory on asymptotically
                        periodic $4$-manifolds}, J. Differential Geom.
                        {\bf 25} (1987), no. 3, 363--430.

\bibitem[Wi90]{Wi90} J.\ Winkelmann, {\em On free holomorphic $C$-actions on $C\sp n$ and
homogeneous Stein manifolds}, Math. Ann.  {\bf 286} (1990), no.
1-3, 593--612.

\bibitem[Wi03]{Wi03} J.\ Winkelmann, {\em Invariant rings and quasiaffine quotients}, Math.
Z.  {\bf 244} (2003), no. 1, 163--174.



\bibitem[Wr]{Wr} D.\ Wright, {\em Abelian subgroups of ${\rm Aut}\sb{k}(k[X,\,Y])$ and
applications to actions on the affine plane}, Illinois J. Math.
{\bf 23} (1979), no. 4, 579--634.

\bibitem[Za]{Za} O.\ Zariski, {\em Interpr\'etations alg\'ebrico-g\'eom\'etriques du quatorzi\`eme
probl\`eme de Hilbert} (French), Bull. Sci. Math. (2)  {\bf 78}
(1954), 155--168.

\bibitem[Zai88]{Zai88} M.\ Zaidenberg, {\em Isotrivial families of curves on affine
surfaces, and the characterization of the affine plane} (Russian)
Izv. Akad. Nauk SSSR Ser. Mat.  {\bf 51} (1987), no. 3, 534--567,
688; translation in Math. USSR-Izv.  {\bf 30} (1988), no. 3,
503--532. See also: {\em Additions and corrections to the paper:
"Isotrivial families of curves on affine surfaces, and the
characterization of the affine plane"}, (Russian) Izv. Akad. Nauk
SSSR Ser. Mat.  {\bf 55} (1991), no. 2, 444--446; translation in
Math. USSR-Izv.  {\bf 38} (1992), no. 2, 435--437.

\bibitem[Zai99]{Zai99} M.\ Zaidenberg, {\em Exotic algebraic structures on affine
spaces}, (Russian) Algebra i Analiz  {\bf 11} (1999), no. 5,
3--73; translation in St. Petersburg Math. J.  {\bf 11} (2000),
no. 5, 703--760

\end{thebibliography}




\end{document}
