ON THE STABLE EQUIVALENCE PROBLEM

Robert Dryło

PREPRINT IMUJ 2005/14

Abstract

In this paper we deal with the stable equivalence problem in a "strong version" over an algebraically closed field 𝕂 of characteristic 0. For this purpose we introduce a notion of stabilizing hypersurfaces, i.e., a hypersurface H in an affine variety X is called stabilizing if every isomorphism f : X × 𝕂^{m} → Y × 𝕂^{m} such that f(H × 𝕂^{m}) = H′ × 𝕂^{m}, where H′ is a hypersurface in a variety Y, satisfies the condition: for each x ∈ X there exists y ∈ Y such that f({x} × 𝕂^{m}) = {y} × 𝕂^{m} (obviously, then f induces the isomorphism f̃ : X → Y such that f̃(H) = H′). We prove that a non-uniruled hypersurface in a smooth affine variety and a non-ℂ-uniruled hypersurface in a smooth affine and dominated by ℂ^{n} variety are examples of stabilizing hypersurfaces. In particular, the stable equivalence problem has an affirmative solution for any non-ℂ-uniruled hypersurface in ℂ^{n}.

1 Introduction

Throughout this paper 𝕂 denotes an algebraically closed field of characteristic 0. Let X and Y be algebraic sets in 𝕂^{n}. Following [17], we say that X and Y are equivalent if there exists an automorphism of 𝕂^{n} that takes X onto Y. Furthermore, we say that X and Y are stably equivalent if for some m ≥ 0 the cylinders X × 𝕂^{m} and Y × 𝕂^{m} are equivalent in 𝕂^{n+m}.

STABLE EQUIVALENCE PROBLEM. Are any two stably equivalent hypersurfaces in 𝕂^{n} equivalent?

This problem for curves in 𝕂^{2} was solved affirmatively by Makar-Limanov, van Rossum, Shpilrain and Yu in [17]. Furthermore, it was proved by Shpilrain and Yu in [18] that two stably equivalent hypersurfaces in ℂ^{n} are equivalent if one of them is the set of zeros of a so-called test polynomial in the class of monomorphisms. The same was proved in [3] provided that one hypersurface is non-uniruled.

In this paper we will generalize the last result. We will consider the stable equivalence problem in a stronger version. For this purpose we introduce a notion of stabilizing hypersurfaces. (Let us note that we will mean by a variety an irreducible algebraic variety, and by a hypersurface in a variety a closed subset of pure codimension one.)

A hypersurface H in an affine variety X is called stabilizing if it satisfies the condition: if f : X × 𝕂^{m} → Y × 𝕂^{m} is an isomorphism such that f(H × 𝕂^{m}) = H′ × 𝕂^{m}, where H′ is a hypersurface in a variety Y, then for each x ∈ X there exists y ∈ Y such that f({x} × 𝕂^{m}) = {y} × 𝕂^{m} (obviously, then f induces the isomorphism between X and Y that takes H onto H′).

We will give some sufficient geometric condition for a hypesurface to be stabilizing, namely, a non-uniruled hypersurface in a smooth affine variety and a non-ℂ-uniruled hypersurface in a smooth affine and dominated by ℂ^{n} variety are stabilizing. In particular, the stable equivalence problem has an affirmative solution for non-ℂ-uniruled hypersurfaces in ℂ^{n}.

This article is divided into five section. In section 2 we give some examples of uniruled varieties. Section 3 contains a brief summary of properties of the Jelonek set. In sections 4 and 5 are stated and proved our main results mentioned above. Furthermore, we prove there that the set of zeros of a test polynomial in the class of monomorphisms is a stabilizing hypersurface in ℂ^{n} (this is a sharpened version of the result from [18]).

2 Uniruledness

A variety X of dimension n > 0 is said to be uniruled (𝕂-uniruled) if there exists a variety W of dimension n − 1 and a dominant rational map W × ℙ^{1}(𝕂) −−→ X (a dominant morphism W × 𝕂 → X). A reducible variety is said to be uniruled (𝕂-uniruled) if all its irreducible components are uniruled (𝕂-uniruled).

Example 2.1. (i) Let X be an affine variety of dimension n. Suppose that F_{1}, …, F_{n} are algebraically independent regular functions on X. Then the variety V = X \ {x ∈ X : F_{1}(x) … F_{n}(x) = 0} is non-𝕂-uniruled. (ii) A smooth hypersurface in ℙ^{n} of degree greater than n is non-uniruled.

Proof. (i) Since we have a dominant morphism (F_{1}, …, F_{n}) : V →(𝕂^{∗})^{n}, it suffices to show that (𝕂^{∗})^{n} is non-𝕂-uniruled. Let g : W × 𝕂 →(𝕂^{∗})^{n} be a morphism, where W is affine of dimension n−1. If g = (G_{1}, …, G_{n}) then all function G_{i} are invertible in the coordinate ring 𝕂[W × 𝕂], and hence they are in 𝕂[W]. Thus g is not dominant. (ii) see [12].

We say that a variety X has the strong cancellation property if every isomorphism f : X × 𝕂^{m} → Y × 𝕂^{m} satisfies the condition: for each x ∈ Y there exists y ∈ X such that f({x} × 𝕂^{m}) = {y} × 𝕂^{m} (clearly, f induces the isomorphism between X and Y). For example, a variety of non-negative logarithmic Kodaira dimension has this property, which was proved by Iitaka and Fujita in [10]. Other examples are given by

Proposition 2.2. ([4]) A non-𝕂-uniruled affine variety has the strong cancellation property.

3 The Jelonek set

Let f : X → Y be a morphism between varieties. We say that f is proper at y ∈ Y if there exists a neighborhood U of y such that res f : f^{−1}(U) → U is a proper morphism. It is well-known that over ℂ a morphism f is proper at y if and only if f is proper at y in the ℂ-topology, i.e., there exists an open (in the ℂ-topology) neighborhood U of y such that for each compact set T ⊂ U the inverse image f^{−1}(T) is compact.

A very important role in our consideration will play the Jelonek set

S_{f} := {y ∈ Y : f is not proper at y}.

Theorem 3.1. (Jelonek, [13].) Let f : X → Y be a dominant morphism between complex affine varieties of equal dimension. Suppose that X is dominated by ℂ^{n}. Then the set S_{f} is either empty or it is a ℂ-uniruled hypersurface.

The first version of this theorem was stated in [11], where the author proved that for a dominant morphism f : ℂ^{n} → ℂ^{n} the set S_{f} is either empty or it is a uniruled hypersurface. We will use the same idea to establish a little more general result below, which will be needed in the proof of Theorem 4.3. Note, that Theorem 3.1 was partially carried over an arbitrary algebraically closed field by Stasica [19]. Other generalization was given by Jelonek and Karaś in [15].

Now we give two well-known facts.

(1) By theorem of Hironaka [8] for a smooth affine variety there exists a smooth compactification (we mean by a compactification of a variety X any projective variety which contains X as an open subset).

(2) If X is an affine variety and a variety X′ contains X as an open subset then the set X′ \ X is either empty or it is of pure dimension dim X − 1.

Theorem 3.2. Let f : X → Y be a dominant morphism between affine varieties of dimension n. Suppose that X is smooth. Let \overline{X} be a smooth compactification of X and \overline{Y} a compactification of Y. Let

r := number of non-uniruled irreducible components of the set \overline{X} \ X, s := number of non-uniruled irreducible components of the set \overline{Y} \ Y.

Then

(i) r ≥ s; (ii) if r = s then every (n − 1)-dimensional irreducible component of the set S_{f} is uniruled.

In particular, if X and Y are smooth affine varieties of dimension n such that X dominates Y and conversely, and if f : X → Y is a dominant morphism, then every (n − 1)-dimensional irreducible component of the set S_{f} is uniruled.

Proof. The idea of this proof is due to [11,12]. By [8] for the rational map f : \overline{X} −−→ \overline{Y} there exists a commutative diagram

\xymatrix{Z \ar[d]_{g} \ar[dr]^{h} \\ \overline{X} \ar@{-->}[r]_{f} & \overline{Y}}

where g : Z → \overline{X} is a composition of blowing ups with smooth centers and h is a morphism. Let E ⊂ Z be the largest exceptional divisor for g. By properties of blowing up E is ruled, i.e., is birational to V × ℙ^{1}. Consequently every (n − 1)-dimensional irreducible component of the set h(E) is uniruled. Clearly, \overline{Y} \ Y is contained in h(g^{−1}(\overline{X} \ X)) = h((\overline{X} \ X)′) ∪ h(E), where (\overline{X} \ X)′ denotes the proper transform of \overline{X} \ X by g. Therefore every nonuniruled component of \overline{Y} \ Y coincides with some component of h((\overline{X} \ X)′). This proves (i). To prove (ii) it suffices to check that S_{f} ⊂ h(g^{−1}(\overline{X} \ X)). This is equivalent that res f : f^{−1}(U) → U is a proper morphism, where U = \overline{Y} \ h(g^{−1}(\overline{X} \ X)). It is clear that f^{−1}(U) is contained in X and the following diagram commutes

\xymatrix{g^{−1}(f^{−1}(U)) = h^{−1}(U) \ar[d]_{res g} \ar[dr]^{res h} \\ f^{−1}(U) \ar[r]_{res f} & U}

Since the vertical arrow is surjective and the wedge arrow is proper, the horizontal arrow is proper too. This completes the proof.

4 Stabilizing hypersurfaces

We shall begin with an easy necessary condition for a hypersurface to be stabilizing.

Proposition 4.1. Let X be an affine variety. Suppose that F is a regular function on X such that the set F^{−1}(0) is a stabilizing hypersurface. Then D(F) ≠ 0 for every non-zero locally nilpotent 𝕂-derivation D on the coordinate ring 𝕂[X].

Proof. A locally nilpotent 𝕂-derivation D on 𝕂[X] induces the automorphism exp T D : 𝕂[X][T] ∋ G ↦ ∑_{i=0}^{∞} \frac{1}{i!}D^{i}(G)T^{i} ∈ 𝕂[X][T], where D is extended on 𝕂[X][T] such that D(T) = 0 (see [5, p.17]). Therefore if D(F) = 0 and a hypersurface F^{−1}(0) is stabilizing then exp T D(F) = F, and consequently exp T D(𝕂[X]) ⊂ 𝕂[X], so D = 0.

Note the following

Proposition 4.2. Let H be a hypersurface in an affine variety X. Suppose that the variety X \ H has the strong cancellation property (in particular, if X \ H is either non-𝕂-uniruled or if \overline{κ}(X \ H) ≥ 0). Then H is stabilizing.

For example, if n = dim X and F_{1}, …, F_{n} are algebraically independent regular functions on X then {x ∈ X : F_{1}(x) … F_{n}(x) = 0} is a stabilizing hypersurface.

Proof. The assertion follows immediately from section 2 and an easy observation that for a morphism f : Y × 𝕂^{m} → Z the set {y ∈ Y : dim f({y} × 𝕂^{m}) = 0} is closed in Y.

Now we give the main result of this section.

Theorem 4.3. A non-uniruled hypersurface contained in a smooth affine variety is stabilizing.

We will need three lemmas.

Lemma 4.4. Let f : X → Y be a dominant morphism between smooth varieties of equal dimension. Suppose that f is étale at a ∈ X, proper at b = f(a), and f^{−1}(b) = {a}. Then there exists a neighborhood U of b such that res f : f^{−1}(U) → U is an isomorphism.

Proof. We may assume that f is proper. Let R_{f} be the set of points at which f is not étale. Then f(R_{f}) is closed in Y and b ∈ Y \ f(R_{f}). Replacing Y \ f(R_{f}) by Y and X \ f^{−1}(f(R_{f})) by X we may assume that f is étale. Hence f is an étale covering, so it is a finite morphism (see [9, p.248]). Now we may assume that X and Y are affine. Consider the corresponding finite extension of coordinate rings 𝕂[Y] ⊂ 𝕂[X]. Let (ϑ_{x}, 𝔪_{x}) and (ϑ_{y}, 𝔪_{y}) be local rings of points x and y, respectively. Put M = 𝔪_{y} ∩ 𝕂[Y]. Clearly, the extension ϑ_{y} = 𝕂[Y]_{M} ⊂ 𝕂[X]_{𝕂[Y]\M} is finite and the ring 𝕂[X]_{𝕂[Y]\M} has a unique maximal ideal. Thus 𝕂[X]_{𝕂[Y]\M} = ϑ_{x} and consequently ϑ_{x} is a finitely generated ϑ_{y}-module. Moreover, 𝔪_{x} = 𝔪_{y}ϑ_{x}, since f is étale at x. Because ϑ_{x}/𝔪_{x} ≅ ϑ_{y}/𝔪_{y} ≅ 𝕂, so ϑ_{x} = ϑ_{y} + 𝔪_{x} = ϑ_{y} + 𝔪_{y}ϑ_{x}. Hence, by Nakayama’s lemma, ϑ_{y} = ϑ_{x}. From this the assertion follows at once.

Note that for complex varieties we may argue as follows. Choose an open (in the ℂ-topology) neighborhood V of a such that f : V → f(V) is a biholomorphism. By properness of f at b there is an open neighborhood U of b such that f^{−1}(U) is contained in V. This means that the generic fiber of f consists of one point, so f is birational. By Zariski’s Main Theorem f^{−1} is regular at b.

In the sequel π_{X} denotes the projection X × 𝕂^{m} ∋ (x, y) ↦ x ∈ X.

Lemma 4.5. Let X be a smooth affine variety and W be a subvariety of X × 𝕂^{m}. Suppose that W is non-singular at points a_{1}, …, a_{s} ∈ W and π_{X}(a_{i}) ≠ π_{X}(a_{j}) for i ≠ j. Then there exists a morphism p : X → 𝕂^{m} such that its graph meets W transversally at a_{1}, …, a_{s}.

Proof. It suffices to give a proof for X = 𝕂^{n}. (Indeed, if X is contained in 𝕂^{n} and a morphism p : 𝕂^{n} → 𝕂^{m} satisfies the above condition then res_{X} p satisfies this condition too.) Choose n-dimensional affine subspaces L_{1}, …, L_{s} in 𝕂^{n} × 𝕂^{m} such that L_{i} and W are transversal at a_{i} and the projection L_{i} → 𝕂^{n} is an isomorphism for all i. Using an easy interpolation we can find a polynomial mapping p : 𝕂^{n} → 𝕂^{m} the graph of which pass through a_{1}, …, a_{s} and has L_{i} as a tangent space at a_{i} for all i. This concludes the proof.

Lemma 4.6. Let f : X → Y be a birational morphism between smooth varieties. If a fiber of f contains at least two points then its all irreducible components have positive dimension.

Proof. Let f^{−1}(y) satisfy the assumptions. Since f^{−1} is not regular at y, by Zariski’s Main Theorem there exists an exceptional divisor E containing f^{−1}(y). Hence all irreducible components of f^{−1}(y) = (f|_{E})^{−1}(y) have positive dimension.

Proof of Theorem 4.3. Let X be a smooth affine variety and H a nonuniruled hypersurface in X. Let f : X × 𝕂^{m} → Y × 𝕂^{m} be an isomorphism such that f(H × 𝕂^{m}) = H′ × 𝕂^{m}, where H′ is a hypersurface in Y. We may assume that H is irreducible.

For any morphism p : X → 𝕂^{m} define the morphism f_{p} : X ∋ x ↦ π_{Y} (f(x, p(x))) ∈ Y. We shall show that f_{p} is birational.

Let a ∈ X, A := (a, p(a)) and Γ_{p} denote the graph of p. By Proposition 2.2, f({a} × 𝕂^{m}) = {b} × 𝕂^{m} for some b ∈ H′. Clearly, Γ_{p} and {a} × 𝕂^{m} are transversal at A, hence f(Γ_{p}) and {b} × 𝕂^{m} are transversal at f(A) too. Consequently res π_{Y} : f(Γ_{p}) → Y is étale at f(A). This implies that f_{p} is étale at a as a composition of étale morphisms. It is clear that f_{p} : H → H′ is an isomorphism and f_{p}^{−1}(f_{p}(x)) = {x} for each x ∈ H′. Furthermore, X dominates Y and conversely, so by Theorem 3.2, f_{p} is proper at some point of H′. It implies that f_{p} is birational by Lemma 4.4.

Suppose now that π_{X}(f^{−1}({y} × 𝕂^{m})) has positive dimension for some y ∈ Y. Choose two points a, b ∈ f^{−1}({y} × 𝕂^{m}) such that π_{X}(a) ≠ π_{X}(b). By Lemma 4.5 there exists a morphism p : X → 𝕂^{m} such that its graph meets transversally f^{−1}({y} × 𝕂^{m}) at a, b. Hence {π_{X}(a)} and {π_{X}(b)} are components of the fiber f_{p}^{−1}(y), which is impossible by Lemma 4.6. The proof is complete.

5 Stabilizing hypersurfaces over ℂ

Theorem 5.1. Let X be a complex affine and smooth variety which is dominated by ℂ^{n}. Then any non-ℂ-uniruled hypersurface in X is stabilizing.

Proof. The proof of Theorem 4.3 works also in this case, we need only use Theorem 3.1 instead of Theorem 3.2.

Corollary 5.2. Two stably equivalent hypersurfaces in ℂ^{n} are equivalent if one of them is non-ℂ-uniruled.

Corollary 5.3. Let X be an irreducible curve in ℂ^{2} which is either non-ℂ-uniruled or it is simply connected. Then any curve in ℂ^{2} stably equivalent to X is equivalent to it.

Proof. It remains to prove the case when X is simply connected, but this follows at once from two remarks. (1) By theorems of Abhyankar-Moh [2] and Lin-Zaidenberg [16] (see also [7]) we know that two simply connected curves in ℂ^{2} are isomorphic if and only if they are equivalent. (2) Affine curves have the cancellation property, i.e., if C_{1} and C_{2} are affine curves and C_{1} × 𝕂^{m} ≅ C_{2} × 𝕂^{m} then C_{1} ≅ C_{2} (this follows, for example, from a more general theorem proved by Abhyankar, Eakin and Heinzer in [1]).

Our next purpose is to give a sharpened version of the result of Shpilrain and Yu from [18]. Recall that a polynomial P ∈ ℂ[T_{1}, …, T_{n}] is said to be a test polynomial in the class of monomorphisms if every ℂ-monomorphism Φ : ℂ[T_{1}, …, T_{n}] → ℂ[T_{1}, …, T_{n}] such that Φ(P) = cP for some c ∈ ℂ^{∗} is an automorphism. A question about the existence and some properties of such polynomials were stated by van den Essen and Shpilrain in [6]. Later it was showed by Jelonek in [14] that a generic polynomial in ℂ[T_{1}, …, T_{n}] of degree greater than n is a test polynomial in the class of monomorphisms. (Note that the above definition is due to [14] and it is slightly different than that introduced in [6].)

Proposition 5.4. Let P ∈ ℂ[T_{1}, …, T_{n}] be a test polynomial in the class of monomorphisms. Then the set P^{−1}(0) is a stabilizing hypersurface in ℂ^{n}.

Proof. Let P = P_{1}^{r_{1}} … P_{s}^{r_{s}}, where P_{1}, …, P_{s} are pairwise different irreducible factors of P. Observe that P′ = P_{1} … P_{s} is a test polynomial in the class of monomorphisms too. Indeed, if Φ : ℂ[T_{1}, …, T_{n}] → ℂ[T_{1}, …, T_{n}] is a monomorphism such that Φ(P′) = cP′ for some c ∈ ℂ^{∗} then Φ(P_{i}) = c_{i}P_{σ(i)}, where σ is a permutation of the set {1, …, s} and c_{i} ∈ ℂ^{∗}. Hence Φ^{s!}(P_{i}) = c′_{i}P_{i}, so Φ^{s!}(P) = c′P, where c′_{i}, c′ ∈ ℂ^{∗}. This implies that Φ^{s!} is an automorphism, and so is Φ. Therefore for this proof we may assume that r_{1} = … = r_{s} = 1.

Put H = P^{−1}(0) and H_{i} = P_{i}^{−1}(0). Suppose that H is not stabilizing. Let f : ℂ^{n} × ℂ^{m} → X × ℂ^{m} be an isomorphism such that f(H × ℂ^{m}) = H′ × ℂ^{m} and for some x ∈ X the dimension of π_{ℂ}^{n}(f^{−1}({x} × ℂ^{m})) is positive. Choose points a_{i} ∈ H_{i} \ ⋃_{j≠i} H_{j} such that H_{i} is non-singular at a_{i} for all i. Assume that f(a_{i}) lies on {b_{i}} × ℂ^{m} for some b_{i} ∈ X (we identify ℂ^{n} with ℂ^{n} ×{0}). By Lemma 4.5 there exists a morphism p : ℂ^{n} → ℂ^{m} such that its graph meets transversally f^{−1}({b_{i}} × ℂ^{m}) at a_{i} and cuts f^{−1}({x} × ℂ^{m}) in at least two points. Similarly, we can find a morphism q : X → ℂ^{m} with the graph meeting transversally f({a_{i}} × ℂ^{m}) at f(a_{i}) for all i. Consider the morphism g := f^{−1}_{q} ∘ f_{p} : ℂ^{n} → ℂ^{n}, where f_{p} : ℂ^{n} ∋ x ↦ π_{X}(f(x, p(x))) ∈ X and f^{−1}_{q} : X ∋ x ↦ π_{ℂ}^{n}(f^{−1}(x, q(x))) ∈ ℂ^{n}. By definition, g is not injective, g(a_{i}) = a_{i} and g is a local biholomorphism at a_{i}. In particular, g is dominant. Obviously, g^{−1}(H) = H. Hence for the induced monomorphism g^{∗} : ℂ[T_{1}, …, T_{n}] → ℂ[T_{1}, …, T_{n}] we have g^{∗}(P) = cP_{1}^{d_{1}} … P_{s}^{d_{s}} for some c ∈ ℂ^{∗} and d_{1}, …, d_{s} > 0. But the tangent space to H at a_{i} is given by two equations d_{a_{i}}P = 0 and d_{a_{i}}g^{∗}(P) = 0, so d_{1} = … = d_{s} = 1. Consequently g is an automorphism, despite g is not injective. This concludes the proof.

Acknowledgements. I would like to express my gratitude to Professor Kamil Rusek for introducing me to the subject and helpful discussion. I thank also to Grzegorz and Michał Kapustka for indicated me how to simplify the proof of Theorem 4.3.

References

[1] S. S. Abhyankar, P. Eakin, and W. J. Heinzer, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310-342.

[2] S. S. Abhyankar and T. T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148-166.

[3] R. Dryło, Non-uniruledness and the cancellation problem, Ann. Polon. Math. (to appear).

[4] R. Dryło, Cancellation theorem for algebraic line bundles, Preprint IMUJ, 2005/13.

[5] A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progr. Math. 190, Birkhäuser, 2000.

[6] A. van den Essen and V. Shpilrain, Some combinatorial questions about polynomial mappings, J. Pure and Appl. Alg. 119 (1997), 47-52.

[7] R. V. Gurjar and M. Miyanishi, On contractible curves in the complex affine plane, Tahôku Math. J. 48 (1996), 459-469.

[8] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79 (1964), 109-326.

[9] S. Iitaka, An Introduction to Birational Geometry of Algebraic Varieties, Springer, Berlin, 1982.

[10] S. Iitaka and T. Fujita, Cancellation theorem for algebraic varieties, J. Fac. Sci. Univ. Tokyo 24 (1977), 123-127.

[11] Z. Jelonek, The set of points at which a polynomial map is not proper, Ann. Polon. Math. 58 (1993), 259-266.

[12] Z. Jelonek, Irreducible identity sets for polynomial automorphisms, Math. Z. 212 (1993), 601-617.

[13] Z. Jelonek, Testing sets for properness of polynomial mappings, Math. Ann. 315 (1999), 1-35.

[14] Z. Jelonek, Test polynomials, J. Pure and Appl. Alg. 147 (2000), 125-132.

[15] Z. Jelonek and M. Karaś, The set of points at which a morphism of affine schemes is not finite, Colloq. Math. 92 (2002), 59-66.

[16] M. Zeidenberg and V. Lin, An irreducible simply connected algebraic curves in ℂ^{2} is equivatent to a quasihomogeneous curve, Soviet Math. Dokl. 28 (1983), 200-204.

[17] L. Makar-Limanov, P. van Rossum, V. Shpilrain and J. T. Yu, The stable equivalence and cancellation problems, Comment. Math. Helv. 79 (2004), 341-349.

[18] V. Shpilrain and Y. T. Yu, Affine varieties with equivalent cylinders, J. Algebra 251 (2002), 295-307.

[19] A. Stasica, Geometry of the Jelonek set, J. Pure and Appl. Alg. 198 (2005), 317-327.

Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland E-mail: Robert.Drylo@im.uj.edu.pl