NON-STABILITY OF AK-INVARIANT

TATIANA BANDMAN , LEONID MAKAR-LIMANOV

Abstract. We provide an example of a surface with non-stable AK−invariant and nontrivial fundamental group. We also compute the invariant for the cylinders over surfaces endowed with a ℂ−action which has only one singular fiber, and this fiber has only reduced components.

1. Introduction

In this note we again try to understand (see [BML1], [BML2]), what are the properties of a surface which are related to the stability of its AK-invariant. We call AK−invariant stable if

AK(S × ℂ) = AK(S).

We provide an example of a surface with non-stable AK−invariant. We also compute the invariant for the cylinders over surfaces endowed with ℂ−action which has one singular fiber, and this fiber has reduced components.

Till now the surfaces S_{n} = {x^{n}y = p(x, z)}, where p(0, v) does not have multiple roots were the only examples of smooth surfaces with unstable invariant ([D], [F], [Wi], [BML1]). In particular, the Danielewski surfaces

S_{n,m} = {x^{n}y = z^{m} − 1}

are of this kind. The following features unite all these surfaces:

1) π_{1}(S_{n}) = 1;

2) S_{n} admits a fixed-point-free ℂ^{+}−action;

3) fibering of S_{n} which corresponds to ℂ^{+}−action has only one nonconnected fiber;

4) all components of any fiber are reduced.

As far as non-stability of the invariant is concerned we show that 1) may be replaced by

The first author is partially supported by the Ministry of Absorption (Israel), the Israeli Science Foundation (Israeli Academy of Sciences, Center of Excellence Program), the Minerva Foundation (Emmy Noether Research Institute of Mathematics).

The second author is partially supported by an NSA grant and Max-Planck Institute of Mathematics.

2 TATIANA BANDMAN , LEONID MAKAR-LIMANOV

1’) π_{1}(S_{n,m}) is finite cyclic. Condition 2) is not important. However, 3) seems to be important ( see section 3) and we believe that it cannot be relaxed. 4) is not crucial: we provide an example where 4) is not satisfied.

On the other hand if 3) and 4) are satisfied then the invariant is non-stable.

Here is a more detailed description of the paper.

Let us call the fundamental group “small” if it is finite cyclic, and “large” otherwise. In section 1 we review some known facts about surfaces with large fundamental group and

remind that their invariant is stable.

In section 2 we give an example of a surface with small ( but not trivial) fundamental group and unstable invariant. To obtain this example we describe homological ℚ−plane S with a ℂ^{+}−action and with π_{1}(S) = ℤ/mℤ (where m is prime) as a quotient of a surface S_{n} by the cyclic transformation group. In case AK(S) = ℂ it was done by M. Miyanishi and K. Masuda ([MiMa1][MiMa2]).

In section 3 we prove that for a surface S with properties 3)and 4)

AK(S × ℂ^{n}) = ℂ.

2. Surfaces with large fundamental group

Definition 1. Let V be an affine variety and let G(V) be the group generated by all ℂ^{+}- actions on V. Then AK(V) ⊂ O(V) is a subring of all regular G(V)− invariant functions on V.

We want to know when

AK(S × ℂ) = AK(S).  (1)

If logarithmic Kodaira dimension \overline{k}(S) ≥ 0 it is true due to a Theorem of T. Fujita and S. Iitaka ([Ii-Fu]). So, let S be a surface with logarithmic Kodaira dimension \overline{k}(S) = −∞,

Miyanishi and Sugie ([MiSu], [Su]) proved that the equality \overline{k}(S) = −∞ is equivalent to the existence of a cylinder-like subset (c.l.s.).

Definition 2. A surface S has a cylinder-like subset U (c.l.s. U) if it has a Zariski open subset U ⊂ S isomorphic to the product C × ℂ of a smooth affine curve C and complex line.

Any c.l.s. provides ( [Ber], [Mi1], [MiSu]) a line pencil on a surface S i.e. morphism ρ : S → C into a smooth curve C, such that the fiber ρ^{−1}(z) for a general z ∈ ℂ is isomorphic to ℂ. The pencils are different if their general fibers do not coincide. Any line

NON-STABILITY OF AK-INVARIANT 3

pencil ρ over an affine curve C on a surface S corresponds to a ℂ^{+}-action φ_{ρ} on S, such that its general orbit coincides with the general fiber of the pencil (see Lemma 1), and corresponds to a locally nilpotent derivation (lnd) ∂_{ρ} of the ring O(S) of regular functions on S, such that ∂_{ρ}f = 0 if and only if f is φ_{ρ}-invariant ([ML1],[KML], [S], [Mi1]).

Definition 3. Two ℂ^{+}actions are equivalent, if they induce the same fibering ρ, i.e. they have the same general orbits.

Definition 4. An affine surface is rational, if there exists a dominant regular map from ℂ^{n} into S for some n ∈ ℕ.

If in Definition 2 the surface S is rational then C ∼ ℂ or C ∼ ℙ^{1}.

For a pencil ρ over C one can find a closure \overline{S} of S and the extension \overline{ρ} : \overline{S} → \overline{C}, such, that the general fiber \overline{ρ}^{−1}(z) ≅ ℙ^{1} and any (−1) curve, contained in a fiber intersects S. For any z ∈ C the fiber ρ^{−1}(z) = ∪C′_{i} is a union of smooth disjoint curves C′_{i} ≅ ℂ^{1} ( [Mi1], Lemma 4.4.1).

Definition 5. ([Fu]) Let F_{z} = \overline{ρ^{−1}(z)} = ∑_{i=1}^{i=m} n_{i}C_{i}, where C_{i} are connected (and irreducible) components. If m = 1, n_{1} = 1, then the fiber is called non-singular. A singular fiber is either non-connected or has m = 1, n_{1} > 1. If F_{z} = ∑_{i=1}^{i=m} C_{i} (i.e. n_{i} = 1) the fiber is called reduced. The number μ(z) = gcd(n_{i}) is called the multiplicity of a fiber. If a fiber has no points in S, by definition μ = ∞.

The topology of surfaces with fibrations was studied by T. Fujita. Let k be the number of fibers with multiplicity μ > 1 (see Definition 5). We shall use the following facts ([Fu]);

I. If the fundamental group π_{1}(S) is infinite, then there is no dominant morphism ϕ : ℂ^{n} → S.

II. If the fundamental group π_{1}(S) is finite, then k ≤ 3.

III. If π_{1}(S) is finite non-abelian, then k = 3 and for all three fibers 1 < μ < ∞; IV. If π_{1}(S) is finite abelian and non-trivial, then either k = 3, π_{1}(S) ∼ ℤ/2ℤ⊕ℤ/2ℤ, μ =

2 for all three singular fibers, or k = 2, π_{1}(S) ∼ ℤ/mℤ, where m is gcd of two multiplicities. V. If π_{1}(S) is trivial, k ≤ 1. In order to use this information for computing AK(S ×ℂ), we need the following Lemmas.

4 TATIANA BANDMAN , LEONID MAKAR-LIMANOV

Lemma 1. (see also [MiMa1], Lemma 1.1) Assume that on a smooth affine surface S there is a ℂ^{+}-action ϕ. Let ρ : S → C be a regular map of S onto a smooth curve C such, that the general fiber is an orbit of ϕ. Then C is affine.

Proof of Lemma 1. Since any ℂ^{+}action on S defines an lnd in O(S) ([S]), the Lemma follows from Lemma 1 of [ML2]. According to this Lemma there should be a regular non-constant ϕ-invariant function in O(S). This function is constant along the fibers of ρ, thus it belongs to O(C). But then C cannot be compact. □

Lemma 2. Let A be an affine commutative domain. If AK(A) = A then AK(A[x]) = AK(A).

Proof of Lemma 2. Let us assume that ∂ is a non-zero locally nilpotent derivation of A[x] which is not identically zero on A. Let def(r) = deg(∂(r)) − deg(r) where deg stands for the degree relative to x. Let m = max(def(r)|r ∈ A[x]). Then m < ∞.

Indeed, A is finitely generated. Let y_{1}, ..., y_{k} be a generating set of A. If a ∈ A then a = p(y_{1}, ..., y_{k}) where p is a polynomial. So ∂(a) = ∑_{i} ∂(y_{i})p_{i} where p_{i} are partial derivatives of p relative to y_{i}. Since all p_{i} are elements of A, we may conclude that def(a) = deg(∂(a)) ≤ max(deg(∂(y_{i}))) is uniformly bounded. Now let r ∈ A[x]. Then r = ∑_{i=0}^{n} r_{i}x^{i} where r_{i} ∈ A. So ∂(r) = ∑_{i=0}^{n} ∂(r_{i})x^{i} + ∑_{i=0}^{n} r_{i}x^{i−1}∂(x). Therefore deg(∂(r)) ≤ max(n + max(deg(∂(r_{i})), n − 1 + deg(∂(x))) and so def(r) = deg(∂(r)) − n is uniformly bounded. By our definition of m we can write ∂(a) = ∑_{i=0}^{m} ∂_{i}(a)x^{i} for any element a ∈ A. We also can write ∂(x) = ∑_{i=0}^{m+1} e_{i}x^{i} where e_{i} ∈ A. Let us define a new derivation ϵ on A[x] by ϵ(a) = ∂_{m}(a)x^{m} and ϵ(x) = e_{m+1}x^{m+1}. It is easy to check that it is indeed a derivation and that this derivation must be locally nilpotent because otherwise ∂ cannot be locally nilpotent. Let d(r) denote degree of r ∈ A[x] relative to ϵ (see [FLN], [KML]; this degree is defined by the following formula: d(r) = max{d|ϵ^{d}(r) ≠ 0} and is non-negative for non-zero r). Then d(x)−1 = d(e_{m+1})+(m+1)d(x). Since this equality is not possible for non-negative d(x) and d(e_{m+1}) we can conclude that e_{m+1} = 0. So x is a constant for ϵ and ∂_{m} is a locally nilpotent derivation on A, which implies AK(A) ≠ A. □

All these facts give us the following

Proposition 1. Let S be an affine smooth surface with rational closure and W = S × ℂ. Then:

NON-STABILITY OF AK-INVARIANT 5

I. AK(W) = AK(S), if π_{1}(S) is infinite;

II. AK(W) = AK(S) = O(S) if π_{1}(S) is finite and non-abelian;

III. AK(W) = AK(S) = O(S) if π_{1}(S) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ;

Proof of Proposition 1. We prove Proposition case by case.

I. If π_{1}(S) is infinite, then S is not rational [Fu]. On the other hand, if AK(S) = ℂ , then the orbit G(S) contains an image of ℂ^{2} under a dominant map([BML1]). Hence, AK(S) ≠ ℂ and only two cases are possible.

a) If AK(S) = O(S), (no actions at all), by Lemma 2, AK(W) = AK(S) = O(S).

b) If AK(S) ≠ O(S), then AK(S) = O(C), where C is a curve in Definition 2. Since the general orbit O_{W} of G(W) contains an image of ℂ^{2} under a dominant map

([BML1]) and there is no dominant map from ℂ^{2} into S , the image of O_{W} under the projection of W onto S is not dense, i.e. it coincides with an orbit of a ℂ− action on S. Hence, AK(W) = O(C) = AK(S).

II. If π_{1}(S) is finite and non-abelian, then, due to [Fu], k = 3 and there is no fiber with μ = ∞ for any linear pencil. By Lemma 1, we get that there are no ℂ^{+} actions on such surfaces ^{1} and, by Lemma 2, AK(W) = AK(S) = O(S).

III. π_{1}(S) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ can happen only if k = 3 and μ = 2, i.e. there are no ℂ^{+} actions on such surfaces as well. □

3. Surfaces with small fundamental group

The surfaces with finite cyclic fundamental group are much more complicated from AK−point of view. We start with three examples of surfaces S_{i}, i = 1, 2, 3, with π_{1}(S_{i}) = ℤ/2ℤ and AK−invariant of all three possible types.

In the discussion of Examples 2 and 3 we will be using the following properties of locally nilpotent derivations. Let A be a ring with a filtration and let \overline{A} be the corresponding graded ring. It is known (see section 5 in [KML]) that:

1Another proof of this fact was explained to authors to R.V.Gurjar in private communication

6 TATIANA BANDMAN , LEONID MAKAR-LIMANOV

1) under some mild conditions (which are satisfied in our examples) a non-zero locally nilpotent derivation ∂ of A induces a non-zero locally nilpotent derivation \overline{∂} on \overline{A}, so that the kernel \overline{A}\overline{∂} of \overline{∂} contains \overline{A^{∂}};

2) a non-zero locally nilpotent derivation ∂ of A allows to define on A a function d(a) = max{d|∂^{d}(a) ≠ 0} (here a is a non-zero element of A). This function is called the degree function relative to ∂ since it possesses all usual properties of a degree function ([FLN]);

3) if a non-zero product is a ∂ constant then all the factors are also ∂ constants (see [FLN]).

Example 1. S_{1} ⊂ ℂ^{5} is defined by equations

y^{2} = x(z − 1), t^{2} = u(z − 1),
yu = tz, yz = xt, xu = z^{2}.

It has two non-equivalent actions, which correspond to the following locally nilpotent derivations:

∂x = 0 ∂u = 0
∂y = x ∂t = u
∂z = 2y ∂z = 2t
∂t = 3z − 2 ∂y = 3z − 2
∂u = 4t ∂x = 4y

Therefore, AK(S_{1}) = ℂ.

Example 2.

S_{2} = {x′(x′y′ + 1) = z′^{2}} ⊂ ℂ^{3}.

It has one action ∂x′ = 0, ∂y′ = 2z′, ∂z′ = x^{′2}.

Let us show that it has just one class of lnds up to equivalence (i.e all other lnds define the same fibration). The ring O(S_{2}) = R can be embedded in ℂ[a, b] by mapping x′ to a^{2}, z′ to a(1 + a^{3}b), and y′ to ab(2 + a^{3}b). One lnd on R is defined by ∂(a) = 0 and ∂(b) = 1. Assume that there is another nonzero lnd ϵ which does not send a to zero. Let f be from its kernel. Let us take lexicographic order b >> a on ℂ[a, b] and the induced order on R. We

NON-STABILITY OF AK-INVARIANT 7

denote by \overline{u} the leading form of an element u and by \overline{R} the ring, generated by the leading forms of elements of R.

Clearly \overline{f} is a^{i}b^{j} where j > 0. Since R = ℂ[x′, y′] + z′ℂ[x′, y′] we see that \overline{R} is generated by a^{2}, a^{4}b^{2}, and a^{4}b.

So ϵ defines a nonzero lnd \overline{ϵ} on \overline{R} and \overline{f} ∈ ker(\overline{ϵ}) (see property 1). Replacing f by f^{2} if necessary, we may assume that j is even and so \overline{f} = (a^{4}b^{2})^{k} (since \overline{f} is \overline{ϵ}-constant, if a factorization of \overline{f} contains two different generators of \overline{R} then \overline{ϵ} is the zero derivation, due to property 3)).

Let us consider the degree function which is induced by \overline{ϵ} on \overline{R}. Since deg(a^{4}b^{2}) = 0 we may assume that deg(a) = d and deg(b) = −2d. Therefore deg(a^{2}) = 2d, deg(a^{4}b) = 2d and elements of \overline{R} have only even degrees. Since application of \overline{ϵ} decrees degree by 1 it is impossible. So ϵ must be the zero derivation.

It follows, that AK(S_{2}) = ℂ[x].

The same conclusion we get if we consider the “minimal” complement of this surface : the divisor in the complement is not a “zigzag” i.e. is not a linear chain (see [Giz], [Ber] and [BML2]).

Example 3. S_{3} ⊂ ℂ^{5} is defined by

vu = z^{2},
(u^{2} − 1)u = vt,
(v^{2} − 1)v = uw,
(v^{2} − 1)(u^{2} − 1) = tw.

Again we denote by R the ring O(S_{3}).

First of all, R = R_{0} + zR_{0} where R_{0} is generated by u, v, t, w. Next, R_{0} is the linear span of monomials u^{i}v^{j}, u^{i}t^{j}, j > 0, and v^{i}w^{j}, j > 0 since vt, uw, and tw can be replaced by a combination of monomials of smaller degree relative to t and w.

Since t = (u^{2} − 1)uv^{−1}, w = (v^{2} − 1)vu^{−1}, if we consider weights ω(u) = 1 and ω(v) = 1, then \overline{t} = u^{3}v^{−1}, \overline{w} = v^{3}u^{−1}. Therefore \overline{u^{i}t^{j}} = u^{3j+i}v^{−j}, \overline{v^{i}w^{j}} = v^{3j+i}u^{−j}. Since in these monomials j > 0 we see that the leading forms of monomials u^{i}v^{j}, u^{i}t^{j}, j > 0, and v^{i}w^{j}, j > 0 are linearly independent, which means that \overline{R_{0}} is generated by u, v, \overline{t}, \overline{w}. We can substitute u = z^{2}v^{−1} in the monomials u^{i}v^{j}, u^{i}\overline{t}^{j}, j > 0, and v^{i}\overline{w}^{j}, j > 0. The resulting monomials stay

8 TATIANA BANDMAN , LEONID MAKAR-LIMANOV

linearly independent. If we add to them their products with z we will get all \overline{R} and see that \overline{R} is generated by v, z, \overline{u} = z^{2}v^{−1}, \overline{t} = z^{6}v^{−4}, \overline{w} = z^{−2}v^{4}.

To get rid of the negative degrees, which are confusing, substitute z = a^{2}b^{2} and v = ab^{3}. Then \overline{R} = span\{a^{8}, a^{3}b, a^{2}b^{2}, ab^{3}, b^{8}\}.

If we prove that \overline{R} does not have nonzero lnds then R itself also does not have nonzero lnds (see property 1))

Assume that ∂ is a nonzero lnd on \overline{R}. Let us take the lexicographic ordering a >> b on \overline{R} and let \overline{∂} be the corresponding lnd induced on R̂ which is obtained from \overline{R} by this ordering. If f ∈ ker(\overline{∂)} then f is a monomial. If this monomial contains both a and b then f^{8} is also a \overline{∂}-constant which is divisible by a^{8} and b^{8} in R̂ which means that both a^{8} and b^{8} are constants of \overline{∂} and \overline{∂} = 0.

Thus we may assume that either f = a^{8} or f = b^{8}. Both cases are similar. Let \overline{∂}(a^{8}) = 0. Then deg\overline{∂}(b) = 1 since otherwise there are no elements in R̂ with \overline{∂}-degree equal to 1. So \overline{∂}(a^{3}b) = λa^{8k} where λ ∈ ℂ and \overline{∂}(b) = λa^{8k−3}. But then \overline{∂}(a^{2}b^{2}) = 2λa^{8k−1}b which is not in R̂ if λ ≠ 0.

Thus, AK(S_{3}) = O(S_{3}).

The surfaces S_{1} and S_{2} both have fibration over ℂ (with fibers {x = const} and {x′ = const} respectively) with only one singular fiber {x = 0} (respectively {x′ = 0}), which is isomorphic to ℂ and has multiplicity 2. The surface S_{3} has fibering over ℙ^{1}, defined by the rational function u/z = z/v. There are two singular fibers, both isomorphic to ℂ and both of multiplicity 2. Thus, (see Section 2) π_{1}(S_{i}) = ℤ/2ℤ, i = 1, 2, 3.

Next, we want to build a surface S, such that 1. π_{1}(S) = ℤ/2ℤ,

2. AK(S) = ℂ[x],

3. AK(S × ℂ) = ℂ. We start with a description of the surfaces with only two singular fibers: one fiber of

multiplicity μ = ∞ and one fiber of multiplicity m, which consists of the single curve C ≅ ℂ^{1}. These surfaces are ℚ−planes ( see, for example, [Mi3]). It was proved in [MiMa1] that any ℚ−plane S with AK(S) = ℂ is a quotient of Danielewski surface S_{1,m} by an action of the group ℤ/mℤ = π_{1}(S). We want to use a similar description for ℚ−planes S with AK(S) = ℂ[x].

NON-STABILITY OF AK-INVARIANT 9

Proposition 2. Let S be a ℚ-plane admitting a ℂ^{+}-action. Let π_{1}(S) = ℤ/mℤ, where m is prime. Then S ≅ V/G, where

- V is a hypersurface in ℂ^{3} with coordinates (u, y, v) : V = {u^{s}y = v^{m} − v_{1}^{m} +

u^{m}q(u^{m}, u^{α}v)}, where q(u, v) is a polynomial of degree less then m relative to v;

- G is a group of transformations, generated by g(u, y, v) = (uε, yε^{−s}, vε^{−α}) for some

α ∈ ℕ and ε = e^{\frac{2iπ}{m}} .

Proof of Proposition 2. Let ∂ be an lnd, corresponding to a ℂ^{+}-action on S. From the general properties of ℚ-planes (see, for example, [Mi3] and Prop.1 of [BML2]), it follows that there are two functions x ∈ O(S) and z ∈ O(S) such that:

a) a fiber {x = c ≠ 0} = F_{c} ∼ ℂ is reduced;

b) the fiber {x = 0} = F_{0} ∼ ℂ has multiplicity m;

c) for c ≠ 0 the function z |_{F_{c}} is linear (i.e., takes any finite value exactly at one point);

d) z |_{F_{0}}= 0;

e) ∂x = 0, ∂z = x^{n}, and n is minimal possible for z with these properties. Consider a surface U which is defined as the normalization of S ×_{ℂ} Γ, where Γ stands

for the m-ramified cover of ℂ with the only branch point at γ = 0 over x = 0 (similar to Theorem 3.2 of [MiMa1]). Let π : U → S and p : U → Γ be the natural projections.

The surface U has the following properties:

a) there is a function u ∈ O(U) such that π^{∗}x = u^{m};

b) U is an unramified m-sheeted cover of S;

c) fiber E_{0} = {u = 0} = ⋃_{i=1}^{m} E_{i} is a union of m reduced components E_{i} ≃ ℂ^{1};

d) U is affine ([Sh], Ch.II, Th.4);

e) the general fiber E_{c} = {u = c} ≃ ℂ. Indeed,

– In property a) u = p^{∗}γ.

– The properties b) and c) are proved in [BPV], chapter III, §9 for proper maps. Since both properties are local, the same proof is valid in our case as well. The only modification one needs to make in the proof is to use π^{∗}ϕ_{f}, where ϕ_{f} = 0 is a local equation of F_{0} in an appropriate neighborhood of a point f ∈ F_{0} instead of function g (the global equation of the multiple fiber there).

– e) is valid because E_{c} ≃ F_{c^{m}} by construction .

10 TATIANA BANDMAN , LEONID MAKAR-LIMANOV

The property e) provides the existence of a ℂ^{+}-action φ on U with general orbit {u = const}. Let ∂_{1} be a corresponding lnd: ∂_{1}(π^{∗}(f)) = π^{∗}(∂f) for f ∈ O(S); ∂_{1}(u) = 0. Since all the fibers {u = const} are reduced, we have π_{1}(U) = 0. It follows that U is the universal covering of S and S = U/G, where G is the group of deck transformations. Any transformation g ∈ G sends the fiber E_{u_{0}} = {u = u_{0}} to the fiber E_{u_{0}ε^{k}} = {u = u_{0}ε^{k}}, where k ∈ N and ε = e^{\frac{2iπ}{m}} . In particular, it moves a component E_{i} of E_{0} to the component E_{j} ⊂ E_{0}.

Since z |_{F_{0}}= 0, we have π^{∗}z |_{E_{0}}= 0. But all the components E_{i} are reduced; hence π^{∗}z = u^{α} · v for some α ∈ ℕ and v ∈ O(U) such that v |_{E_{0}}≢ 0. Further on we shall use z and x instead of π^{∗}z and π^{∗}x when no confusion may arise.

Lemma 3.

a) (α, m) = 1.

b) v is linear along each fiber E_{u_{0}} = {u = u_{0} ≠ 0}.
c) v |_{E_{i}}= v_{i} is a constant, and v_{i} ≠ v_{j} if i ≠ j.
d) ∂_{1}v = u^{mn−α}.
e) g^{∗}(∂_{1}h) = ∂_{1}(g^{∗}h) for any h ∈ O(U)

Proof of Lemma 3. a) Since z |_{F_{0}}= 0, there are coprime numbers r and k and a function ψ ∈ O(S) such that z^{k} = x^{r} · ψ and ψ |_{F_{0}}≢ 0. It follows that u^{αk}v^{k} = u^{mr}π^{∗}(ψ), i.e., αk = mr. Since m is prime, the following two cases are possible.

(1) α = mr, k = 1, and z = x^{r} · ψ, ∂z = x^{r} · ∂ψ = x^{n}, ∂ψ = x^{n−r}. Then ψ is linear along a general fiber and constant along F_{0} ([BML2], Prop.1), i.e. n is not minimal;

(2) α = r and k = m. But then (α, m) = (k, r) = 1.

b) v = z/u^{α} is linear along each fiber {u = c ≠ 0} since z is linear along a fiber {x = const}.

c) Take now E_{1} and g a generator of G such that g^{∗}(u) = uε and denote: E_{2} = g(E_{1}), E_{3} = g(E_{2}) … E_{m} = g(E_{m−1}). Then

g^{∗}(u) = u · ε, g^{∗}(v) = g^{∗}(z) · g^{∗}(u)^{−α} = z · u^{−α}ε^{−α} = vε^{−α}.

If v were not constant along a component E_{1} ⊂ E_{0}, it would be non-constant along each E_{i}, i = 1, …, m. This means that v would take any value v_{0} ∈ ℂ at least m times in E_{0}, which contradicts the fact that it is linear along the general fiber. (Indeed, {v = v_{0}} intersects each fiber at most at one point).

Thus, v |_{E_{i}}= v_{i} = const . Then v_{2} = ε^{−α}v_{1}, v_{3} = ε^{−2α}v_{1}, …, v_{m} = ε^{−(m−1)α}v_{1}, i.e., v_{i} ≠ v_{j}.

d) ∂_{1}v = ∂_{1}(z/u^{α}) = x^{n}/u^{α} = u^{mn−α}.

e) Any function h ∈ O(U) may be represented as h = q(u, v)/u^{l} for some polynomial q, since U is rational. The property may be obtained by a direct computation. □

Lemma 4. The surface U admits a fixed-point-free ℂ^{+}-action ψ, equivalent to φ.

Proof of Lemma 4. We have to prove that ∂_{1} = u^{β} ˜∂, where β ≥ 0 and ˜∂ is an lnd of O(U) such that for any i = 1, …, m if h ∈ O(U) and h |_{E_{i}}≠ const, then ˜∂h |_{E_{i}}≠ 0.

Let ∂_{1}h |_{E_{i}}= 0 for all h ∈ O(U), i.e., ∂_{1}h = u^{n_{i}(h)}ψ, where ψ is rational on U and ψ |_{E_{i}}≢ 0, ∞. Let n_{i} be the minimal value of n_{i}(h) for h ∈ O(U).

Since the action of group G permutes the components,

n_{1} = n_{2} ⋯ = n_{m} and ˜∂ = \frac{∂_{1}}{u^{n_{1}}}

is defined and corresponds to a fixed-point-free ℂ^{+}-action. □

Lemma 5. Let ˜∂v = u^{k}. Then the there exists a polynomial p(u, v) = p_{0}(v) + up_{1}(u, v) such that the function y = \frac{p(u,v)}{u^{k}} ∈ O(U) is linear along each E_{i}, i = 1, …, m.

Proof of Lemma 5. The polynomial p_{0}(v) = v^{m} − v_{1}^{m} vanishes along E_{0}, i.e., p_{0}(v) = u^{s_{1}} · y_{1}, where y_{1} ∈ O(U) and y_{1} |_{E_{i}}≢ 0 at least for one value of i. Since g^{∗}(y_{1}) = y_{1}ε^{−s_{1}}, y_{1} is constant or non-constant along all E_{i} simultaneously and y_{1} |_{E_{i}}≢ 0 for all i.

We have:

u^{s_{1}}y_{1} = p_{0}(v)
u^{s_{1}} ˜∂y_{1} = p′_{0}(v)u^{k}
˜∂y_{1} = p′_{0}(v)u^{k−s_{1}} ∈ O(U).

It follows, that k ≥ s_{1}.

Consider two cases.

I. y_{1} |_{E_{i}}≠ const. Then ˜∂y_{1} = p′(v)u^{k−s_{1}} ≢ 0, i.e. k − s_{1} = 0, ˜∂y_{1} |_{E_{i}}= const, and y_{1} is linear along each component E_{i}.

II. y_{1} |_{E_{1}}= c_{1} = const. Then y_{1} |_{E_{i}}= c_{1}ε^{−s_{1}i}. Let h < m be the natural number, such that s_{1} ≡ αh(mod m), and p_{1}(v) = c_{1}(\frac{v}{v_{1}} )^{h}. Then y_{1} − p_{1}(v) |_{E_{0}}≡ 0 and y_{1} − p_{1}(v) = u^{s_{2}}y_{2}, where y_{2} ∈ O(U) and y_{2} |_{E_{i}}≢ 0 at least for one value of i.

Since g^{∗}(y_{2}) = y_{1}ε^{−s_{1}−s_{2}}, y_{1} is constant or non-constant on all E_{i} simultaneously and y_{2} |_{E_{i}}≢ 0 for all i.

We have :

u^{s_{1}+s_{2}}y_{2} = p_{0}(v) + u^{s_{1}}p_{1}(v),
˜∂y_{2} = (p′_{0}(v) + u^{s_{1}}p′_{1}(v))u^{−(s_{1}+s_{2})+k}

It follows that k ≥ (s_{1} + s_{2}). If y_{2} |_{E_{i}}≠ const, it is linear along each E_{i}, if not, we continue the process. After r steps we get

u^{s_{1}+s_{2}+...+s_{r}}y_{r} = p_{0}(v) + u^{s_{1}}p_{1}(v) + ... + u^{s_{1}+s_{2}+...+s_{r−1}}p_{r−1}(v),

and either

s_{1} + s_{2} + ... + s_{r} < k, y_{r} |_{E_{i}}= const

or

s_{1} + s_{2} + ... + s_{r} = k, y_{r} |_{E_{i}}≠ const,
˜∂y_{r} = (p′_{0}(v) + u^{s_{1}}p′_{1}(v) + ... + u^{s_{1}+s_{2}+...+s_{r−1}}p′_{r−1}(v))

and y_{r} is linear along the components E_{i}.

So, after a finite number of steps we will get a function y, which is liner along each E_{i} and

u^{k}y = p_{0}(v) + uq̃(u, v).

Moreover, by construction, g^{∗}y = yε^{−k} and q(u, v) has degree less then m relative to v. □

We proceed with the proof of the Proposition 2. Let θ : U → ℂ^{3} be defined as θ(p) = (u, y, v) for a point p ∈ U.

This regular map is bijective: indeed, the function u distinguishes the fibers, v is linear along the general fiber and distiguishes components of the singular fiber, and y is linear along the components of the singular fiber.

NON-STABILITY OF AK-INVARIANT 13

The surface V = θ(U) is given by equation

{u^{k}y = v^{m} − v_{1}^{m} + uq̃(u, v)}.

This equation is invariant under the action of group G if only uq̃(u, v) = u^{m}q(u^{m}, u^{α}v). Since V is a smooth surface bijection θ is an isomorphism. □

Example 4. We use Proposition 2 for the case m = 2, s = 2. The Danielewski surface

S_{2,2} = {x^{2}y_{1} = z_{2}^{2} − 1}

is the universal cover of the surface S_{2} (see Example 2). Indeed, consider the following map:

(x, y_{2}, z_{2}) → (x′ = x^{2}, y′ = y_{2}, z′ = xz_{2}).

This map glues the points s = (x, y_{2}, z_{2}) and T s = (−x, y_{2}, −z_{2}) and is unramified double covering. Obviously,

x′^{2}y′ + x′ = z′^{2}.

The surface S_{2} has the fundamental group ℤ/2ℤ and precisely one singular fiber {x′ = 0}, which is a double straight line.

All actions on this surface are equivalent: AK(S_{2}) = ℂ[x]

Now we want to compute the invariant of its cylinder Z = S_{2} × ℂ. The universal cover of this cylinder is Y = S_{2,2} × ℂ ≅ S_{1,2} × ℂ ([F],[D]), where S_{1,2} = {xy_{1} = z_{1}^{2} − 1}.

Lemma 6. (communicated by P. Russell).

Y = \{ \begin{matrix}xy_{1} = z_{1}^{2} − 1 \\ x^{2}y_{2} = z_{2}^{2} − 1 \\ xu = z_{1} − z_{2} \\ y_{1} − xy_{2} = u(z_{1} + z_{2}). \end{matrix}

The surface S_{2,2} is defined in it as u = const and the surface S_{1,2} as w = u^{3}x + 3u^{2}z_{2} + 3uxy_{2} + z_{2}y_{2}.

Proof of Lemma 6. Let A = ℂ[x, y_{1}, y_{2}, z_{1}, z_{2}, u] where xy_{1} = z_{1}^{2} − 1, x^{2}y_{2} = z_{2}^{2} − 1, and xu = z_{1} − z_{2}. Then A = ℂ[x, y_{2}, z_{2}][u] since z_{1} = xu + z_{2} and y_{1} = u^{2}x + 2uz_{2} + xy_{2}.

14 TATIANA BANDMAN , LEONID MAKAR-LIMANOV

Similarly, A = ℂ[x, y_{1}, z_{1}][w], where w = u^{3}x + 3u^{2}z_{2} + 3uxy_{2} + z_{2}y_{2} since xw = y_{1}z_{1} − 2u and wz_{1} = y_{2} − u^{2} + y_{1}^{2}. So 2u = −xw + z_{1}y_{1}, y_{2} = wz_{1} + u^{2} − y_{1}^{2} and 2z_{2} = x^{2}w − z_{1}^{3} + 3z_{1}.

□

Cylinder Z = S_{2} × ℂ is a factor of Y by the transformation T′, which preserves S_{2,2} ( i.e. the value of u) and coinsides on S_{2,2} with T:

T′ : (x, y_{2}, z_{2}, y_{1}, z_{1}, u) → (−x, y_{2}, −z_{2}, −y_{1}, −z_{1}, u).

On the other hand Z = {(x′, y′, z′, t) : x′^{2}y′ + x′ = z′^{2}} ⊂ ℂ^{4} and the map Y → Z is defined as

x′ = x^{2}, y′ = y_{2}, z′ = xz_{2}, t = u.

Due to Lemma 6

u = (1/2)(−xw + z_{1}y_{1}),
y_{2} = wz_{1} + u^{2} − y_{1}^{2},
z_{2} = (1/2)(x^{2}w − z_{1}^{3} + 3z_{1}).

Any lnd defined on O(S_{1,2}) may be extended to an lnd on O(Y). In particular, the lnd defined by:

∂y_{1} = 0, ∂z_{1} = y_{1}, ∂x = 2z_{1}, ∂w = 0.

Then

∂u = (1/2)(−2z_{1}w + y_{1}^{2}),
∂y_{2} = (wy_{1} + u(y_{1}^{2} − 2wz_{1})),
∂z_{2} = (1/2)(4xwz_{1} − 3z_{1}^{2}y_{1} + 3y_{1}),

and ∂ is invariant under T′, i.e.∂T′ = T′∂. Indeed, say,

T′(∂x) = T′(2z) = −2z = ∂T′(x),

and we can check it similarly for the remaining generators.

Therefore this lnd can be pushed down to O(Z), which is the factor of Y by T′. This means that in Z there is an lnd, for which x′ is not invariant. Hence AK(Z) = ℂ, though AK(S_{2}) = ℂ[x].

NON-STABILITY OF AK-INVARIANT 15

4. Cylinders over surfaces with reduced fibers and primitive Feiseler factor

We want to show in this section that the property 2) of Danielewski surfaces ( see Section 1) is not essential for non-stability of AK−invariant of a surface.

The Danielewski construction represents a surface with a ℂ^{+}action with reduced components of fibers as an affine bundle over the “Danielewsky-Fieseler” factor.

For a given c.l.s. U ⊂ S or a corresponding fibering ρ the “Danielewsky-Fieseler” (DF) factor X = S/_{U} = S/_{ρ} is a (non-separated) prevariety X, such that the points of X are in one-to-one correspondence with the connected components of ρ^{−1}(c) for all c ∈ C. The precise and detailed definition was given in [D], [F].

We describe this factor in the following way.

Definition 6. A factor X of a smooth surface S by a pencil ρ is an algebraic prevariety X included into the following commutative diagram

[figure: S; X; C; π; ρ; σ]

and having the following properties:

1) maps π, σ, ρ are regular;

2) σ is an isomorphism of X \ {x_{i,j}}, i = 1, ⋯ r onto C \ {c_{1}, …, c_{r}} where the set of points {x_{i,j}} ∈ X is finite;

3) let F_{c_{i}} = ρ^{−1}(c_{i}) = ⋃_{j=1}^{t} C_{ij} be the union of t connected (necessarily irreducible [Mi1],Lemma 4.4.1 ) components C_{ij}. Then π|_{C_{ij}} = x_{ij} and x_{ij} ≠ x_{ik} if j ≠ k. There is a one-to-one correspondence between points x_{ij} ∈ σ^{−1}(c_{i}) and components C_{ij} ⊂ ρ^{−1}(c_{i}).

We call X primitive, if C ≅ ℂ and there is only one singular fiber.

Theorem 1. Let S be a smooth affine rational surface and let α : S×ℂ → S be a C^{+}−action on S for which all the components of all the fibers are reduced. Assume that the DF factor X = S/α is primitive. Then AK(O(S × ℂ)) ≅ ℂ

The proof of the Theorem is based on the following Theorem of Sh. Kaliman and M. Zaidenberg ([KZ]).

Theorem KZ. Assume that there exists a dominant morphism f: X → S of a smooth quasiprojective variety X to a smooth quasiprojective variety S, and that a general fiber f^{−1}(s), s ∈ S, is isomorphic to ℂ^{2}. Then there exists a Zariski open subset S_{0} of S such that f^{−1}(S_{0}) ≅ S_{0} ×ℂ^{2}. Moreover, if we denote by ϕ the isomorphism f^{−1}(S_{0}) → S_{0} ×ℂ^{2}, and by p: S_{0} × ℂ^{2} → S_{0} the projection to the first factor, then p^{−1}(s) = ϕ(f^{−1}(s)) for any s ∈ S_{0}.

Proof of Theorem 1. Since X is primitive, the base of fibering C ≅ ℂ and we may assume that the multiple point is c = 0. Let σ^{−1}(0) = {x_{1}, ..., x_{k}}. Consider a surface S_{1} ⊂ ℂ^{3} defined by

xy = (z − 1)(z − 2)...(z − k) = p(z),

where {x, y, z} are coordinates in ℂ^{3}. S_{1} is a smooth affine surface with two C^{+}-actions β and δ. The orbits of β, δ are the curves R_{x} = {x = const}, R_{y} = {y = const} respectively, and the corresponding lnds of O(S_{1}) are defined as follows: I. β : ∂x = 0, ∂z = x, ∂y = p′(z);

II. δ : ∂y = 0, ∂z = y, ∂x = p′(z).

Note that the DF factors S_{1}/_{β} ≅ S_{1}/δ ≅ X.

Now consider a commutative diagram

[figure: S × ℂ; ≅; S ×_{X} S_{1}; =; W; S; S_{1}; X; X; ℂ; ℂ; p; p_{1}; π_{α}; π_{β}; π_{δ}; σ; σ_{1}]

NON-STABILITY OF AK-INVARIANT 17

All the fibers of π_{β} are reduced , and the action β is fixed-point-free. Following Danielewsky and Feiseler ([D],[F]) it means that W = S ×_{X} S_{1} has the following properties:

a) W is a ℂ^{+} bundle over S;

b) (W, p) and (W, p_{1}) are locally-trivial fiber-bundles over S and S_{1} respectively.

It follows, that:

a’) W ≅ S × ℂ, since any ℂ^{+}- bundle over affine surface S is trivial ;

b’)Consider a map μ = σ_{1} ∘ π_{δ} ∘ p_{1} : S × ℂ → ℂ. The general fiber P_{y} = μ^{−1}(y), y ∈ ℂ is a locally trivial fiber-bundle (restriction of (W, p_{1})) over a curve R_{y} = (σ_{1} ∘ π_{δ})^{−1}(y) ⊂ S_{1}. The fiber of this bundle is reduced and isomorphic to ℂ, and R_{y} ≅ ℂ for a general y. Thus, P_{y} ≅ ℂ^{2} ([MiSu]) for a general y.

According to Theorem KZ, there exist a Zariski open subset Z ⊂ S × ℂ and U ⊂ ℂ such that Z = μ^{−1}(U), Z ≅ U × ℂ^{2}, and μ is the projection on the first factor in this product. Due to [Mi2] Lemma 1.4, there exist two commuting lnds on the ring O(S × ℂ), such that the general orbit of the group generated by the corresponding ℂ^{+}-actions coincides with P_{y}. The image σ ∘ π_{β} ∘ p_{1}(P_{y}) = σ ∘ π_{β}(R_{y}) = ℂ. Thus, there is a ℂ^{+}-action ϵ on S × ℂ, whose orbit is not contained in a fiber Q_{z} = p_{1}^{−1} ∘ π_{β}^{−1} ∘ σ^{−1}(z), z ∈ ℂ.

We obtained three ℂ^{+} actions on S ×ℂ: α′, induced by α, ω, acting along the fibers of the map p : W → S, and ϵ. The general orbit of the group generated by α′ and ω is Q_{z} = p_{1}^{−1} ∘ π_{β}^{−1} ∘ σ^{−1}(z). The orbit of the third action ϵ is not contained in Q_{z}. Hence, the general orbit of the group generated by all three actions is three-dimensional, and AK(O(S×ℂ)) ≅ ℂ. □

Example 5. Consider a surface

S_{5} = \{ \begin{matrix}x^{2}y_{2} = z_{2}^{2} − 1 \\ (z_{2} − 1)y_{2} = xt \\ xy_{2}^{2} = (z_{2} + 1)t \end{matrix}

This surface is obtained from S_{2,2} = {x^{2}y_{2} = z_{2}^{2} − 1} by the following affine modification: the point x = 0, z_{2} = −1, y_{2} = 0 is blown up, and the curve x = 0, z_{2} = −1, y_{2} ≠ 0 is taken off. Since the graph of the complement of this surface is not linear ([Giz], [Ber]) surface S_{5} admits only one class of ℂ^{+}−actions, i.e. any lnd is proportional to the following one:

∂x = 0, ∂z_{2} = x^{3}, ∂y_{2} = 2z_{2}x, ∂t = 3z_{2}^{2} − 2z_{2} − 1.

The corresponding fibering has one singular fiber x = 0 with two connected reduced components x = 0, z_{2} = 1, t = 0 and x = 0, z_{2} = −1, y_{2} = 0.

18 TATIANA BANDMAN , LEONID MAKAR-LIMANOV

This lnd vanishes along the curve x = 0, z_{2} = 1, t = 0, i.e. there are fixed points for any ℂ^{+}−action.

According to Theorem 1, AK(ℂ × S_{5}) = ℂ. We will show it explicitly, using the same formulae as in Example 4 Let B = ℂ[x, y_{1}, y_{2}, z_{1}, z_{2}, u, t] where xy_{1} = z_{1}^{2} − 1, x^{2}y_{2} = z_{2}^{2} − 1, (z_{2} − 1)y_{2} = xt,

xy_{2}^{2} = (z_{2} + 1)t and xu = z_{1} − z_{2}. Then B = ℂ[x, y_{2}, z_{2}, t][u] since z_{1} = xu + z_{2} and y_{1} = u^{2}x + 2uz_{2} + xy_{2}.

Therefore B = O(S_{5} × ℂ) and ∂ can be extended on B by ∂u = 0. Substituting into the identity xt = (z_{2} − 1)y_{2} expressions for y_{2} and z_{2} trough x, y_{1}, z_{1}, w,

(see Example 4) we get:

xt = (z_{1} − xu − 1)y_{2} = −xuy_{2} + (z_{1} − 1)[wz_{1} − y_{1}^{2} + \frac{1}{4}(z_{1}y_{1} − xw)^{2}] =

−xuy_{2} + \frac{1}{4}(z_{1} − 1)(x^{2}w^{2} − 2xy_{1}z_{1}w) + (z_{1} − 1)[wz_{1} − y_{1}^{2} + \frac{1}{4}z_{1}^{2}y_{1}^{2}].

To simplify this expression take

s = t + uy_{2} − \frac{1}{4}(z_{1} − 1)(xw^{2} − 2y_{1}z_{1}w).

Then

xs = \frac{1}{4}(z_{1} − 1)(4wz_{1} − 4y_{1}^{2} + z_{1}^{2}y_{1}^{2}).

Next,

4xs = (z_{1} − 1)(4wz_{1} − 4y_{1}^{2} + z_{1}^{2}y_{1}^{2}) = (z_{1}^{2} − 1)(y_{1}^{2}(z_{1} − 1) + 4w)−
(z_{1} − 1)(4w + 3y_{1}^{2}) = xy_{1}[y_{1}^{2}(z_{1} − 1) + 4w] − (z_{1} − 1)(4w + 3y_{1}^{2}).

Introducing now

r = −4s + y_{1}[y_{1}^{2}(z_{1} − 1) + 4w]

we obtain xr = (z_{1} − 1)(4w + 3y_{1}^{2}). Since

(z_{1} + 1)xr = (z_{1}^{2} − 1)(4w + 3y_{1}^{2}) = xy_{1}(4w + 3y_{1}^{2})

we also have (z_{1} + 1)r = y_{1}(4w + 3y_{1}^{2}). Clearly, B = ℂ[x, y_{1}, y_{2}, z_{1}, z_{2}, u, r], and, as in Example 4, B = ℂ[x, y_{1}, z_{1}, r, w],

Let us define now an lnd of B by ˜∂y_{1} = ˜∂r = 0, ˜∂z_{1} = 4y_{1}, ˜∂x = 8z_{1}, and ˜∂w = r. It acts on the cylinder, and ˜∂x ≠ 0.

A third lnd, ∂′, can be obtained from the presentation B = ℂ[x, y_{2}, z_{2}, t][u] : ∂′u = 1, ∂′x = ∂′z_{2} = ∂′y_{2} = ∂′t = 0.

Using the notations of Theorem 1, the map μ(x, y_{2}, z_{2}, u, t) = y_{1} and the fiber of this map

is ℂ^{2} with coordinates z_{1}, r. Indeed, for y_{1} = c we have

x = \frac{z_{1}^{2} − 1}{c},

w = \frac{(z_{1} + 1)r − 3c^{2}}{4c} ,

u = 1/2(−xw + z_{1}c),

y_{2} = wz_{1} + u^{2} − c^{2},

z_{2} = z_{1} − xu.

Aknowledgements. Conversations and contacts with many colleagues were helpful to us.

We would like to single out two of them: Peter Russell whose help with Example 4 was

indispensable and Masayoshi Miyanishi for making accessible his preprints and his book

“Open Algebraic surfaces” long before they were published. This paper was finished while

both authors were visiting Max Planck Institute of Mathematics.

References

[BML1] T. Bandman, L. Makar-Limanov, Cylinders over affine surfaces, Japan. J. Math., 26 (2000), 207-217.

[BML2] T. Bandman, L. Makar-Limanov, Affine surfaces with AK(S) = ℂ actions, Michigan Journal of Mathematics, 49 (2001), 567-582.

[BPV] W. Barth, C. Peters, A. van de Ven, Compact complex surfaces, Springer Verlag, (1984).

[Ber] J.Bertin, Pinceaux de droites et automorphismes des surfaces affines, J. Reine und Angew. Math.,341(1983), 32-53.

[D] W. Danielewski, On the cancellation problem and automorphism groups of affine algebraic varieties, preprint, Warsaw (1989).

[FLN] M. Ferrero, Y. Lequain, A. Nowicki, A note on locally nilpotent derivations, J. of Pure and Appl. Algebra, 79(1992), 45–50.

[F] K.-H. Fieseler, On complex affine surfaces with ℂ^{+}-action , Comment. Math. Helvetici, 69(1994), 5-27.

[Fu] T. Fujita On the topology of non-complete algebraic surfaces , J. Fac.Sci. Univ. Tokyo, Sect.IA 29(1982 ), 503-566.

[Giz] M.H.Gizatullin, Quasihomogeneous affine surfaces, Math. USSR Izvestiya, 5(1971), 1057-81. [Ii-Fu] S.Iitaka, T.Fujita, Cancellation theorem for algebraic varieties, J.Fac.Sci.Univ.Tokyo, Sect.IA,

24(1977), 123-127.

[KML] S. Kaliman, L. Makar-Limanov, On the Russell-Koras contractible threefolds , Journ. of the Algebraic Geometry, 6(2)(1997), 247–268.

[KZ] S. Kaliman, M. Zaidenberg, Families of affine planes: the existence of a cylinder , Mich. Math. Journ, 49(2001), 353-367.

[ML1] L. Makar-Limanov, Locally nilpotent derivations, a new ring invariant, and applications, preprint.

[ML2] L. Makar-Limanov, On the hypersurface x + x^{2}y + z^{2} + t^{3} = 0 in ℂ^{4} or a ℂ^{3}-like threefold which is not ℂ^{3} , Israel Math. J. 96(1996), 419–429.

[Mi1] M.Miyanishi, Non-complete algebraic surfaces, Lecture Notes in Math. 857, Springer-Verlag, (1981).

[Mi2] M. Miyanishi On algebro-topological characterization of the affine space of dimension 3 , Amer. Math. Jour., 106(1984 ), 1469-1485.

[Mi3] M.Miyanishi, Open algebraic surfaces, CRM Monograph series, 12, AMS, RI, (2001).

[MiMa1] M. Miyanishi, K.Masuda, The additive group action on ℚ− homology planes, Ann. de l’Inst. Fourier, 53(2003), 429–464.

[MiMa2] M. Miyanishi, K.Masuda, Open algebraic surfaces with finite group actions, Transformation groups, 7(2002), 185-207.

[MiSu] M.Miyanishi, T. Sugie Affine surfaces containing cylinderlike open set, J. Math. Univ.Kyoto, 20, (1980), 11-42.

[S] M. Snow Unipotent actions on affine space, Topological methods in Algebraic Transformation groups Progress in Math. 80, (1989), 165-177.

[Sh] I. Safarevic, Algebraic surfaces, Proceedings Steclov Institute in Mathematics, 75,(1965) AMS, 1967.

[Su] T.Sugie Algebraic characterization of the affine plane and the affine 3-space, Topological methods in Algebraic Transformation groups Progress in Math. 80,(1989), 177-190. .

[Wi] J. Wilkens On the cancellation problem for surfaces, C.R.Acad.Sci.Paris, 326,(1998 ), 1111-1116.

T. Bandman, Department of Mathematics and Statistics, Bar-Ilan University, 52900, Ramat Gan, ISRAEL

  E-mail address: bandman@macs.biu.ac.il

L. Makar-Limanov, Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

  E-mail address: lml@math.wayne.edu