ON POLYNOMIALS IN THREE VARIABLES

ANNIHILATED BY TWO LOCALLY NILPOTENT DERIVATIONS

DANIEL DAIGLE

Abstract. Let B be a polynomial ring in three variables over an algebraically closed field k of characteristic zero. We are interested in irreducible polynomials f ∈ B satisfying the following condition: there exist nonzero locally nilpotent derivations D_{1}, D_{2} : B → B such that ker(D_{1}) ≠ ker(D_{2}) and D_{1}(f) = 0 = D_{2}(f). The main result asserts that a nonconstant polynomial f ∈ B satisfies the above requirement if and only if its “generic fiber” k(f) ⊗_{k[f]} B is isomorphic, as an algebra over the field K = k(f), to K[X, Y, Z]/(XY − φ(Z)) for some nonconstant φ(Z) ∈ K[Z].

1. Introduction

The main objective of this paper is to prove 1.10, below. Before stating that result, we introduce some notations and concepts.

Conventions. All rings and algebras are commutative, associative and unital. If R ⊆ A are rings, “ A = R^{[n]} ” means that A is R-isomorphic to the polynomial algebra in n variables over R. We write A^{∗} for the units of A. If a ∈ A then A_{a} denotes the localized ring S^{−1}A with respect to the multiplicative set S = {1, a, a^{2}, … }. If E is any subset of a ring R then V_{R}(E) = V (E) = { 𝔭 ∈ Spec R | 𝔭 ⊇ E }. All curves and surfaces are irreducible and reduced, unless the contrary is stated explicitely.

1.1. If R is a ring of characteristic zero, a derivation D : R → R is said to be locally nilpotent if for each r ∈ R there exists n ∈ ℕ (depending on r) such that D^{n}(r) = 0. We use the following notations:

lnd(R) = set of locally nilpotent derivations D : R → R such that D ≠ 0
klnd(R) = { ker D | D ∈ lnd(R) } (observe that R ∉ klnd(R))
ML(R) = intersection of all elements of klnd(R),

where we understand that ML(R) = R when klnd(R) = ∅.

1.2. If K is a field of characteristic zero, let 𝔇(K) be the set of K-algebras isomorphic to

K[X, Y, Z]/(XY − φ(Z))

2000 Mathematics Subject Classification. Primary: 14R10. Secondary: 14R20, 14R25, 13N15.

Key words and phrases. Locally nilpotent derivations, group actions, Danielewski surfaces, affine surfaces.

Research supported by a grant from NSERC Canada.

for some nonconstant polynomial φ(Z) ∈ K[Z] \ K, where X, Y, Z are indeterminates over K. This class of algebras was studied in [18], [6] and [7], in particular. It is well-known (see e.g. [7, 2.3]) that if K is a field of characteristic zero and R ∈ 𝔇(K) then R is a normal domain and ML(R) = K.

In [7], one defines a “Danielewski surface” to be a pair (R, K) such that R ∈ 𝔇(K), but we will not use that terminology here. In this paper we adopt the following definition, which is equivalent to the one given in [12]:

1.3. Definition. Let k be an algebraically closed field of characteristic zero. A Danielewski surface over k is an integral affine surface S over k together with a surjective morphism π : S → 𝔸_{k}^{1} with reduced fibers such that there exists a closed point x ∈ 𝔸_{k}^{1} satisfying:

(i) π^{−1}(𝔸_{k}^{1} \ {x}) is isomorphic to 𝔸_{k}^{1} × (𝔸_{k}^{1} \ {x}) over 𝔸_{k}^{1} \ {x} (ii) π^{−1}(x) is a disjoint union of affine lines 𝔸_{k}^{1}.

Basic elements. Until the end of this section, let k be an algebraically closed field of characteristic zero and let B = k^{[3]}. We propose the idea that the following elements of B are interesting:

1.4. Definition. An element of B is said to be basic if it is irreducible and belongs to at least two elements of klnd(B).

It is easy to see that all variables of B are basic (recall that an element f ∈ B is called a variable if there exist g, h such that B = k[f, g, h]). Moreover, the converse is true if we think fieldwise, i.e., each basic element is a good field generator:

1.5. If f is a basic element of B then there exist g, h ∈ B such that k(f, g, h) is the field of fractions of B.

Proof. Choose A ∈ klnd(B) such that f ∈ A, then (4.1) A = k[f, g] for some g. Let S = A \ {0} then by the well-known 2.3 we have S^{−1}B = k(f, g)^{[1]}, so for some h ∈ B we have k[f, g, h] ⊆ B ⊂ k(f, g)[h]. □

As the set of basic elements includes all variables and is included in the set of good field generators, it seems legitimate to state that the concept of basic element is a natural generalization of that of variable. Consequently, basic elements should be appealing even to those who are not primarily interested in locally nilpotent derivations.

Let us now look at concrete examples.

1.6. Example. Choose X, Y, Z such that B = k[X, Y, Z], choose a nonconstant φ(Z) ∈ k[Z] and let f = XY −φ(Z); then f is an irreducible element of B and k[X, f], k[Y, f] are two elements of klnd(B); thus f is a basic element of B.

1.7. Definition. A basic element f of B is trivial if there exist X, Y, Z satisfying B = k[X, Y, Z] and f = XY − φ(Z) for some nonconstant φ(Z) ∈ k[Z].

Note that all variables of B are trivial basic elements. To prove the existence of nontrivial basic elements, we make the following obvious observation:

1.8. If f is a trivial basic element of B then B/(f − λ) ∈ 𝔇(k) for every λ ∈ k.

1.9. Example. Consider the following homogeneous polynomial of degree 5:^{1}

f = X^{5} + 2X^{3}Y Z − 2X^{2}Y^{3} + X^{2}Z^{3} − 2XY^{2}Z^{2} + Y^{4}Z ∈ B = k[X, Y, Z].

In the notation of [14] we have f = H_{3}, and k[H_{2}, H_{3}], k[H_{3}, H_{4}] are distinct elements of klnd(B); so f is a basic element of B. As the domain B/fB is not normal, B/fB ∉ 𝔇(k). By 1.8, f is a nontrivial basic element of B.

The main result 1.10 characterizes basic elements in terms of their generic fiber; in a sense, this result asserts that all basic elements have properties in common with the trivial ones:

1.10. Main theorem. Let B be a polynomial ring in three variables over an algebraically closed field k of characteristic zero. For a nonconstant f ∈ B, the following conditions are equivalent:

(a) f is irreducible and f ∈ A ∩ A′ for some distinct elements A, A′ ∈ klnd(B);

(b) ML(ℬ) = k(f), where we define ℬ = k(f) ⊗_{k[f]} B;

(c) ℬ ∈ 𝔇(k(f)), where ℬ is defined as in (b).

Note that implications (a) ⇔ (b) ⇐ (c) are straightforward; we shall give a complete proof of 1.10 but the main point is to prove that (a) implies (c). The fact that (b) implies (c) is noteworthy because it is well-known that, for a two-dimensional K-algegra R, ML(R) = K is striclty weaker than R ∈ 𝔇(K). Also note the following consequence of our results (compare with 1.8):

1.11. If f is a basic element of B then B/(f − λ) ∈ 𝔇(k) for general λ ∈ k. Consequently, for general λ the hypersurface “f = λ” of 𝔸_{k}^{3} is isomorphic to a hypersurface with equation “xy = φ(z)”. (Note, however, that there does not necessarely exist an automorphism of 𝔸_{k}^{3} which maps one hypersurface onto the other, as the existence of such an automorphism is equivalent to f being a trivial basic element.)

Result 1.11 is part of 4.4, but one can also derive it from condition (c) of 1.10. Indeed, it is not difficult to show that if the generic fiber of f belongs to the class 𝔇, then its general fibers also belong to that class.

1.12. Example. Consider again the nontrivial basic element f = X^{5} + 2X^{3}Y Z − 2X^{2}Y^{3} + X^{2}Z^{3} − 2XY^{2}Z^{2} + Y^{4}Z of B = k[X, Y, Z] (cf. 1.9). Then one can show that k(f) ⊗_{k[f]} B is k(f)-isomorphic to k(f)[X_{1}, X_{2}, X_{3}]/(X_{1}X_{2} − X_{3}^{5} − f^{3}), where X_{1}, X_{2}, X_{3} are indeterminates over k(f). Moreover, if λ ∈ k^{∗} then the hypersurface f = λ in 𝔸_{k}^{3} is isomorphic to the hypersurface xy = z^{5} + 1, but no automorphism of 𝔸_{k}^{3} maps one hypersurface onto the other.

We think that the study of basic elements may lead to significant progress in the classification of locally nilpotent derivations of B. To support this claim we give the following result, which is a consequence of 1.10 and results 5.3 and 4.8 of [7].

1The zero-set of f in ℙ^{2} is “Yoshihara’s rational quintic” [25].

1.13. Corollary. Let f be a basic element of B and let A and A′ be elements of klnd(B) which contain f. Then there exists a finite sequence of local slice constructions which transforms A into A′.

Proof. In the notation of [7], the Corollary is equivalent to the following claim: Let f be a basic element of B and let R = k[f]; then the graph \underline{klnd}_{R}(B) is connected. So let us prove this claim.

If A is any element of klnd(B) which contains f then by 4.1 we have A = R^{[1]}, so R is factorially closed in B; so we have R ∈ ℛ^{in}(B) (cf. [7, 5.2]) and consequently (cf. [7, 5.3]) we have an isomorphism of graphs \underline{klnd}_{R}(B) ≅ \underline{klnd}_{K}(ℬ) where we define ℬ = k(f) ⊗_{k[f]} B and K = k(f). As 1.10 asserts that ℬ ∈ 𝔇(K), we may apply [7, 4.8] and conclude that \underline{klnd}_{K}(ℬ) (and hence \underline{klnd}_{R}(B)) is connected. □

Let us also recall that it was once hoped that every element of klnd(B) contained a variable of B, because that would have enabled one to solve the classification problem; however Freudenburg [15] exhibited a kernel which did not contain a variable. In this regard, we propose:

1.14. Conjecture. Each element of klnd(B) contains a basic element of B.

Note that this conjecture is true in the homogeneous case (a proof will appear elsewhere). We think that if 1.14 is true in general, we are in a good position for developing a classification.

2. Preliminaries on locally nilpotent derivations

2.1. Definition. A subring A of an integral domain B is said to be factorially closed in B if the conditions x, y ∈ B and xy ∈ A \ {0} imply x, y ∈ A. Note that if A is factorially closed in B then the following hold: B^{∗} = A^{∗}; A is algebraically closed in B; an element of A is irreducible in A ⇔ it is irreducible in B; for x, y ∈ A, x | y in A ⇔ x | y in B; if B is a UFD then so is A.

2.2. Let B be an integral domain of characteristic zero. If A ∈ klnd(B), then A is factorially closed in B.

2.3. Let B be a ℚ-domain, D ∈ lnd(B) and A = ker D. There exists s ∈ B satisfying D(s) ∈ A \ {0} and, given any such s, we have B_{a} = A_{a}[s] = A_{a}^{[1]} where a = D(s). In particular, trdeg_{A}(B) = 1 for all A ∈ klnd(B).

Proofs of the above two facts can be found in [23].

2.4. Definition. A derivation D : B → B (where B is any ring) is said to be irreducible if the only principal ideal of B which contains D(B) is B itself. Now assume that B is a UFD of characteristic zero. It is easy to see that if A ∈ klnd(B) then there exists an irreducible Δ_{A} ∈ lnd(B) such that ker(Δ_{A}) = A, and such a Δ_{A} is unique up to multiplication by a unit of A. For each A ∈ klnd(B), define

I_{A} = A ∩ Δ_{A}(B).

Note that I_{A} is a nonzero ideal of A and is uniquely determined by A and B.

2.5. Let R be a k-algebra. If D : R → R is a locally nilpotent derivation and T is an indeterminate over R then we may define a ring homomorphism ξ : R → R[T] by ξ(r) = ∑_{i=0}^{∞} \frac{1}{n!}D^{n}(r)T^{n} (r ∈ R). Applying the “Spec” functor yields a morphism Spec(ξ) : Spec k[T] × Spec R → Spec R which is in fact a G_{a}-action on Spec R. Let us denote it by α_{D} : G_{a} × Spec R → Spec R, where α_{D} = Spec(ξ). It is well-known that D ↦ α_{D} is a bijection from the set of locally nilpotent derivations of R to the set of G_{a}-actions on Spec R. Also well-known is the fact that the set of fixed points of α_{D} is Fix(α_{D}) = V (DR) ⊆ Spec R. This justifies the following:

2.6. Definition. If D is a locally nilpotent derivation of a k-algebra R, we define the set of fixed points of D to be:

Fix(D) = V (DR) ⊆ Spec R.

In other words, we define Fix(D) = Fix(α_{D}).

Locally nilpotent derivations of k^{[3]}

From now-on, let B = k^{[3]} where k is a field of characteristic zero.

2.7. If A ∈ klnd(B) then:

(1) A = k^{[2]}

(2) B is faithfully flat as an A-module.

(3) The ideal I_{A} of A is principal (cf. 2.4).

In the case k = ℂ, assertion (1) was proved by Miyanishi [20] and (2) and (3) by Bonnet [3]. See [9] for the generalizations to all fields of characteristic zero.

2.8. Notation. Given A ∈ klnd(B), let us write ψ_{A} for a generator of the principal ideal I_{A} of A. So ψ_{A} is a nonzero element of A determined up to multiplication by an element of k^{∗}.

2.9 ([4]-2.6). Given A ∈ klnd(B), the derivation Δ_{A} (cf. 2.4) can be obtained as follows: choose X, Y, Z such that B = k[X, Y, Z] and f, g such that A = k[f, g], then

Δ_{A}(h) = |\frac{∂(f, g, h)}{∂(X, Y, Z)}| (for all h ∈ B).

2.10. Fix A ∈ klnd(B) and consider the morphism 𝒬 : Spec B → Spec A determined by the inclusion A ↪ B. As B is faithfully flat over A (cf. 2.7), 𝒬 is surjective and every fiber of 𝒬 has pure dimension 1. Moreover, 2.3 implies that the general fibers of 𝒬 are affine lines. Now consider the unique irreducible locally nilpotent derivation D : B → B with kernel A (i.e., D = Δ_{A}, cf. 2.4). We make two observations regarding the fixed points of D (cf. 2.6) in relation with the morphism 𝒬:

(a) Spec B \ Fix(D) = Smooth(𝒬)

(b) 𝒬 maps Fix(D) to a finite set of closed points in Spec A.

Proof. To prove (a), choose X, Y, Z such that B = k[X, Y, Z] and f, g such that A = k[f, g]. Then Fix(D) = V (J) where J is the ideal of B generated by D(B), i.e., J = (DX, DY, DZ). Using 2.9, we find that DX, DY, DZ are the 2 × 2 minors of the 2 × 3 jacobian matrix M = \frac{∂(f,g)}{∂(X,Y,Z)}, so the complement of Fix(D) is the set of points in Spec B = 𝔸^{3} at which rank M = 2. As 𝒬 is a flat morphism by 2.7, the set of points at which rank M = 2 is exactly the set of points at which the morphism 𝒬 is smooth.

To prove (b), recall that Fix(D) = Fix(α_{D}) where α_{D} is the G_{a}-action on Spec B determined by D. Consider a closed point P ∈ Fix(α_{D}); then P belongs to the fiber F_{P} = 𝒬^{−1}(𝒬(P)), which is a finite union of curves since every fiber of 𝒬 has pure dimension 1. The G_{a}-action α_{D} restricts to a G_{a}-action on F_{P}, and hence to a G_{a}- action on each irreducible component of F_{P}; let Γ_{P} be an irreducible component of F_{P} such that P ∈ Γ_{P}, then the action on Γ_{P} is trivial (any G_{a}-action on a curve and having a fixed point is trivial), so Γ_{P} ⊆ Fix(α_{D}). This shows that Fix(α_{D}) is a union of curves Γ_{P}, where each Γ_{P} is mapped to a point by 𝒬. Since D is irreducible, Fix(α_{D}) has codimension at least 2 in Spec B = 𝔸^{3} and consequently Fix(α_{D}) is a finite union of curves Γ_{P}, which proves (b). □

2.11. Let A ∈ klnd(B) and consider the fibers of 𝒬 : Spec B → Spec A. If 𝔭 ∈ Spec A, consider the residue field κ(𝔭) = A_{𝔭}/𝔭A_{𝔭} and the “fiber ring” B ⊗_{A} κ(𝔭). Define the following subset of Spec A:

𝒳_{A} = { 𝔭 ∈ Spec A | B ⊗_{A} κ(𝔭) ≠ κ(𝔭)^{[1]} }

By 2.3 the general fibers of 𝒬 are affine lines, i.e., 𝒳_{A} is not dense in Spec A. The following fact is needed in the proof of assertion (b-iii) of 3.3.

2.12. Lemma. If A ∈ klnd(B), then 𝒳_{A} is a constructible subset of Spec A.

2.12.1. Some comments are in order. Given a morphism of schemes φ : X → Y one may ask whether 𝒜(φ) = { y ∈ Y | X_{y} ≅ 𝔸_{κ(y)}^{1} } is a constructible subset of Y, where κ(y) = 𝒪_{Y,y}/𝔪_{Y,y} is the residue field of Y at y and X_{y} = Spec κ(y) ×_{Y} X is the fiber of φ over y. Apparently, the answer is affirmative whenever φ is a morphism of integral schemes of finite type over a field of characteristic zero. We don’t know a reference for this fact, so we will sketch the proof for the special case φ = 𝒬 (see the proof of 2.12). We shall need the following facts:

(i) (Verification left to the reader.) Let Q ⊂ R be integral domains and K the field of fractions of Q. If R is a finitely generated Q-algebra and K ⊗_{Q} R = K^{[n]}, then R_{s} = Q_{s}^{[n]} for some s ∈ Q \ {0}.

(ii) (Verification left to the reader.) Consider a field extension k′/k, a scheme Y of finite type over Spec k and the canonical morphism ε : Y′ → Y where Y′ = Spec k′ ×_{Spec k} Y. For a subset E of Y, E is constructible in Y ⟺ ε^{−1}(E) is constructible in Y′.

(iii) (Verification left to the reader.) Let φ : X → Y and ε : Y′ → Y be morphisms of schemes over a field of characteristic zero, and consider the morphism φ′ : Y′ ×_{Y} X → Y′ obtained by base extension. Then 𝒜(φ′) = ε^{−1}(𝒜(φ)).

(iv) Let φ : X → Y be a morphism of complex irreducible quasi-projective algebraic varieties. Then there exists a Zariski-dense open subset Y_{0} of Y so that the fibers of φ over any two closed points of Y_{0} are homeomorphic in the standard topology. (Special case of the Varchenko equisingularity theorem [24, Thm 5.2].)

(v) Let S be a geometrically integral scheme of finite type over a field of characteristic zero and let φ : X → S be an affine, flat morphism of finite type. Assume that there is a nonempty open subset S_{0} of S such that every closed point of S_{0} belongs to 𝒜(φ). Then the generic point of S belongs to 𝒜(φ). (Special case of [17, Lemma 2.2].)

Proof of 2.12. Let k be a field of characteristic zero and B = k^{[3]}. We say that “2.12 is valid over k” if for every A ∈ klnd(B) the set 𝒳_{A} is constructible in Spec A, or equivalently, for every A ∈ klnd(B) the set 𝒜(𝒬) is constructible in Spec A (where 𝒬 : Spec B → Spec A is determined by A ↪ B). We have to show that 2.12 is valid over each field of characteristic zero. We first show that it is valid over ℂ.

So let B = ℂ^{[3]}, A ∈ klnd(B) and 𝒬 : X → Y where X = Spec B and Y = Spec A. We want to show that 𝒜(𝒬) is constructible. Let C be an irreducible closed subset of Y such that 𝒜(𝒬) ∩ C is dense in C; in view of (6.C) of [19], it suffices to show that 𝒜(𝒬)∩C contains a nonempty Zariski-open subset of C. This is obvious if C is a closed point; this is also easy if C = Y, because 2.3 implies that 𝒜(𝒬) contains a nonempty open subset of Y; so we may assume that C is a curve. Consider the irreducible element p ∈ A such that C = V (p). Give C the reduced induced structure of a closed subscheme of Y (i.e., C ≅ Spec(A/pA)) and consider the morphism 𝒬′ : X ×_{Y} C → C obtained by base extension. As A is factorially closed in B, it follows that p is irreducible in B, so X ×_{Y} C = Spec(B/pB) is an integral scheme, i.e., an integral surface. The fibers of 𝒬 and 𝒬′ over any point of C are the same, so 𝒜(𝒬) ∩ C = 𝒜(𝒬′) (this also follows from (iii)). This means that the proof in the complex case reduces to proving:

(∗)

Let φ : S → C be a morphism of affine complex algebraic varieties, where S is a surface and C is a curve. If 𝒜(φ) is dense in C, then it contains a nonempty Zariski-open subset of C.

Note that if U is a nonempty affine open subset of C then the restriction φ′ : φ^{−1}(U) → U of φ satisfies the hypothesis of (∗), and it suffices to prove (∗) for φ′. Using this observation, we see that it suffices to prove (∗) in the special case where the following additional conditions are satisfied: (a) φ is surjective; (b) C is a smooth curve; (c) φ is a smooth morphism; (d) (using (iv)) the fibers of φ over any two closed points of C are homeomorphic in the standard topology. Moreover, as 𝒜(φ) is dense in C, at least one of the following conditions is satisfied:

(e) 𝒜(φ) contains infinitely many closed points of C;

(f) the generic point of C belongs to 𝒜(φ). Suppose that (e) holds. If P is any closed point of C then the fiber φ^{−1}(P) is smooth and homeomorphic to ℂ in the standard topology, so it is isomorphic to 𝔸_{ℂ}^{1} as an algebraic variety. This shows that all closed points of C belong to 𝒜(φ), so applying (v) to φ implies that (f) holds. So (f) must hold in all cases and by (i) we obtain that some nonempty open subset of C is included in 𝒜(φ), which proves (∗). This also proves that 2.12 is valid over ℂ.

Using (ii), (iii) and a straightforward “Lefschetz Principle” argument, we now extend validity to all fields of characteristic zero. We begin by proving:

(∗∗)

If k′/k is an extension of fields of characteristic zero and if 2.12 is valid over k′, then it is valid over k.

Indeed, consider B = k^{[3]}, A ∈ klnd(B) and 𝒬 : Spec B → Spec A. Define B′ = k′⊗_{k}B = k^{′[3]} and A′ = k′⊗_{k}A; note that A′ ∈ klnd(B′) and let 𝒬′ : Spec B′ → Spec A′ be the morphism determined by the inclusion A′ ↪ B′. Then in the diagram

Spec B′ ⟶^{𝒬′} Spec A′
↓  ↓^{ε}
Spec B ⟶^{𝒬} Spec A

𝒬′ is obtained from 𝒬 by base extension via ε : Spec A′ → Spec A. Since we are assuming that 2.12 is valid over k′, 𝒜(𝒬′) is constructible in Spec A′. We have 𝒜(𝒬′) = ε^{−1}(𝒜(𝒬)) by (iii) and consequently 𝒜(𝒬) is constructible by (ii). This proves (∗∗).

Now let k be any field of characteristic zero and consider B = k[X, Y, Z] = k^{[3]}, A ∈ klnd(B) and 𝒬 : Spec B → Spec A. We leave it to the reader to show that there exist a subfield k_{0} of k finitely generated over ℚ and an element A_{0} ∈ klnd(B_{0}) (where B_{0} = k_{0}[X, Y, Z]) such that B = k ⊗_{k_{0}} B_{0} and A = k ⊗_{k_{0}} A_{0}; so if 𝒬_{0} : Spec B_{0} → Spec A_{0} is the morphism determined by A_{0} ↪ B_{0} then 𝒬 is obtained from 𝒬_{0} by base extension via ε : Spec A → Spec A_{0}. Since k_{0} is isomorphic to a subfield of ℂ, (∗∗) implies that 2.12 is valid over k_{0} and hence that 𝒜(𝒬_{0}) is constructible. Since 𝒜(𝒬) = ε^{−1}(𝒜(𝒬_{0})) by (iii), 𝒜(𝒬) is constructible by (ii). □

2.13. Lemma. Let A ∈ klnd(B). For a multiplicative subset S of A, the following are equivalent:

(1) S^{−1}B = (S^{−1}A)^{[1]}

(2) S ∩ I_{A} ≠ ∅.

Proof. It follows from 2.3 that if a ∈ I_{A} \ {0} then B_{a} = (A_{a})^{[1]}, so (2) implies (1). The converse is Lemma 1.4 of [5]. □

2.14. Lemma. If A ∈ klnd(B) then V (I_{A}) is the closure of 𝒳_{A} in Spec A.

Remark. In fact, 2.13 and 2.14 are valid (with the same proofs) under weaker hypotheses: for 2.13, it suffices to assume that B is a factorial ℚ-algebra; for 2.14, assume that B is a factorial ℚ-algebra and that A ∈ klnd(B) satisfies:

(∗)

A is noetherian and B is flat as an A-module and finitely generated as an A-algebra.

Proof of 2.14. Let \overline{𝒳_{A}} be the closure of 𝒳_{A} in Spec A. Then \overline{𝒳_{A}} ⊆ V (I_{A}) easily follows from 2.13.

For the reverse inclusion, consider a basic affine open set ∅ ≠ 𝒟(f) ⊆ Spec(A) such that 𝒟(f) ∩ 𝒳_{A} = ∅ (where f ∈ A and 𝒟(f) = Spec(A) \ V (f)); it suffices to show that 𝒟(f) ∩ V (I_{A}) = ∅. Now (∗) implies that A_{f} is noetherian and that B_{f} is flat as an A_{f}-module and finitely generated as an A_{f}-algebra. Moreover, the condition 𝒟(f) ∩ 𝒳_{A} = ∅ implies that B_{f} ⊗_{A_{f}} κ(𝔭) = κ(𝔭)^{[1]} holds for every 𝔭 ∈ Spec(A_{f}). Thus, by Theorem 3.4 of [1], A_{f} ⊂ B_{f} ⊂ (A_{f})^{[n]} for some n > 0. Together with the fact that A_{f} and B_{f} are UFD’s such that trdeg_{A_{f}} (B_{f}) = 1, the last condition implies that B_{f} = (A_{f})^{[1]}. Then 2.13 gives f^{m} ∈ I_{A} for some m, and consequently 𝒟(f) ∩ V (I_{A}) = ∅. □

2.15. Corollary. Let A ∈ klnd(B) and consider 𝒬 : Spec B → Spec A determined by A ↪ B. Let U = Spec A \ \overline{𝒳}_{A}, where \overline{𝒳}_{A} is the closure of 𝒳_{A} in Spec A. Then 𝒬^{−1}(U) is isomorphic to 𝔸_{k}^{1} × U over U.

Proof. By 2.14, and since I_{A} = (ψ_{A}), it suffices to show that B_{ψ_{A}} = A_{ψ_{A}}^{[1]}. This follows from 2.13, because ψ_{A} ∈ I_{A}. □

2.16. Proposition. Let B = k^{[3]}, where k is an algebraically closed field of characteristic zero. Let A ∈ klnd(B) and let p be a prime factor of ψ_{A}. Then A/pA is a subalgebra of k^{[1]}.

Proof. By a straightforward Lefschetz Principle argument which is left to the reader, it suffices to prove the case k = ℂ. Assume that k = ℂ, and let us obtain this result as a corollary to results 2.2 and 3.2 of Kaliman’s [16]. The “Γ ” which occurs in the statement of those results is a closed subset of Spec A and satisfies the following conditions: Γ is a finite union of curves in Spec A, and if we define E = 𝒬^{−1}(Γ) ⊂ Spec(B) then the variety Spec(B)\E is naturally isomorphic to (Spec(A)\Γ)×ℂ over Spec(A) \ Γ (as before, 𝒬 : Spec B → Spec A is the morphism determined by A ↪ B). It follows that Γ ⊇ 𝒳_{A}, and consequently that Γ ⊇ \overline{𝒳_{A}} = V_{A}(I_{A}) = V_{A}(ψ_{A}) ⊇ V_{A}(p) by 2.14 and 2.8; so if we define Γ^{1} = V_{A}(p) then

Γ^{1} ⊆ \overline{𝒳_{A}} (1)

(2) Γ^{1} is an irreducible component of the set Γ of [16].

By [16, 2.2, 2.3], there exist a (not necessarely irreducible) curve Z, a finite morphism θ : Z → Γ and a surjective morphism κ : E → Z so that 𝒬|_{E} = θ ∘ κ : E → Γ and Z contains a Zariski dense subset Z^{∗} for which

(3)  κ^{−1}(Z^{∗}) is isomorphic to Z^{∗} × ℂ over Z^{∗}.

Define Z^{1} = θ^{−1}(Γ^{1}) and E^{1} = 𝒬^{−1}(Γ^{1}). Note that E^{1} is irreducible, since E^{1} = V_{B}(p) and p is irreducible in B. As κ is surjective, we have Z^{1} = κ(κ^{−1}(Z^{1})) = κ(E^{1}), so Z^{1} is irreducible. As θ is surjective, Γ^{1} = θ(θ^{−1}(Γ^{1})) = θ(Z^{1}). Hence,

(4)  Z^{1} is an irreducible component of Z and θ(Z^{1}) = Γ^{1} = V_{A}(p).

We claim:

(5)  θ|_{Z^{1}} : Z^{1} → Γ^{1} is not bijective.

Indeed, if (5) is false then θ|_{Z^{1}} is a birational morphism, so there is a Zariski open subset U ⊆ Γ^{1} such that (θ|_{Z^{1}})^{−1}(U) → U is an isomorphism. By (3), it follows that for a general point P ∈ Γ^{1} we have 𝒬^{−1}(P) = 𝔸^{1} (scheme-theoretically), i.e., P ∉ 𝒳_{A}. This contradicts (1), so (5) is true.

Now (2), (4) and (5) allow us to make use of [16, 3.2] and conclude that Γ^{1} is a “polynomial curve” (i.e., its normalization is 𝔸_{ℂ}^{1}). This implies that A/pA is a subalgebra of ℂ^{[1]}, which completes the proof. □

3. Irreducible polynomials belonging to at least one kernel

In this section, k is an algebraically closed field of characteristic zero. We say that a condition is satisfied for general λ ∈ k if there are at most finitely many λ ∈ k for which that condition is false.

3.1. Global Assumptions. Throughout the section, the following assumptions and notations are in effect. Let B = k^{[3]} and fix a pair (f, A) such that A ∈ klnd(B), f ∈ A and f − λ_{0} is irreducible^{2} for at least one λ_{0} ∈ k.

The following objects are determined by the pair (f, A):

• Let 𝒬 : Spec B → Spec A be the morphism determined by the inclusion A ↪ B. Let D : B → B be the essentially unique (cf. 2.4) irreducible locally nilpotent derivation such that ker D = A.

• For each λ ∈ k, define the k-algebra B_{λ} = B/(f − λ) and consider the locally nilpotent derivation D_{λ} : B_{λ} → B_{λ} determined by the commutative diagram

B ⟶^{D} B
π_{λ}↓  ↓^{π_{λ}}
B_{λ} ⟶_{D_{λ}} B_{λ}

where π_{λ} is the canonical epimorphism. Let 𝒬_{λ} : Spec B_{λ} → Spec ( ker D_{λ}) be the morphism determined by the inclusion ker(D_{λ}) ↪ B_{λ}.

Warning. Several results in this section begin with the phrase “Let the setup be as in 3.1.” Each such result gives information about the pair (f, A) or the objects (B_{λ}, 𝒬_{λ}, etc) which are determined by that pair, and this is the case even when A does not appear in the statement of the result. For instance, 3.12 appears to be asserting that if f ∈ B is such that ML(B/(f − λ)) = k for infinitely many λ ∈ k, then k(f) ⊗_{k[f]} B belongs to 𝔇(k(f)); however, one also requires as a hypothesis that f belongs to some A ∈ klnd(B), and that f − λ_{0} is irreducible for some λ_{0}.

2Note that an element of A is irreducible in A if and only if it is irreducible in B, because A is factorially closed in B (2.1, 2.2).

3.2. Notation. Let φ : X → Y be a morphism of schemes such that Y \ 𝒜(φ) is a finite set (cf. 2.12.1). Then the cardinality of the set Y \ 𝒜(φ) is denoted N(φ).

3.3. Lemma. Let the setup be as in 3.1. For general λ ∈ k, the following hold:

(a) B_{λ} is an integral domain and Spec(B_{λ}) is isomorphic to a smooth hypersurface of 𝔸^{3}

(b) Fix(D_{λ}) = ∅, ker(D_{λ}) = π_{λ}(A) ≅ A/(f − λ) and the morphism

𝒬_{λ} : Spec B_{λ} → Spec ( ker D_{λ})

has the following properties:

(i) 𝒬_{λ} is surjective and smooth; in particular every fiber of 𝒬_{λ} is reduced (ii) every fiber of 𝒬_{λ} which is not an affine line is a disjoint union of (reduced)

affine lines; (iii) N(𝒬_{λ}) = |V_{A}(ψ_{A}, f − λ)| (cf. 3.2, 2.8).

Proof. We have A = k^{[2]} by 2.7, and for polynomials in two variables it is well-known (cf. [22]) that the set Λ = { λ ∈ k | f − λ is reducible in A } is either finite or equal to k. As λ_{0} ∉ Λ, Λ is finite and consequently B_{λ} is an integral domain for general λ. So the general fibers of the morphism 𝔸^{3} = Spec B → Spec k[f] are integral schemes. Since in characteristic zero the general fibers are smooth, (a) is proved.

The same argument applied to Spec A → Spec k[f] shows that

(6) for general λ ∈ k, A/(f − λ) is an integral domain and is smooth over k.

Let E = 𝒬( Fix(D)) ⊂ Spec A and recall from 2.10 that E is a finite set of closed points of Spec A. Consequently, V_{A}(f − λ) is disjoint from E for general λ ∈ k. If Fix(D_{λ}) ≠ ∅ then there exists 𝔪 ∈ Spec B_{λ} such that D_{λ}(B_{λ}) ⊆ 𝔪; let 𝔐 = π_{λ}^{−1}(𝔪), then 𝔐 ∈ Fix(D) and f − λ ∈ 𝔐, so 𝔐 ∩ A ∈ E ∩ V_{A}(f − λ). As E ∩ V_{A}(f − λ) = ∅ for general λ, Fix(D_{λ}) = ∅ for general λ.

Define the k-subalgebra A_{λ} = π_{λ}(A) of B_{λ} and note that A_{λ} ⊆ ker(D_{λ}). We claim that equality holds (for general λ). Before proving this, first note that π_{λ} restricts to a surjective homomorphism A → A_{λ} with kernel A ∩ (f − λ)B; as A is factorially closed in B we get A ∩ (f − λ)B = (f − λ)A, so A_{λ} ≅ A/(f − λ)A. By (6), it follows that Spec(A_{λ}) is a smooth integral curve.

As B_{λ} is an integral domain, it follows that ker(D_{λ}) is factorially closed in B_{λ}; in particular, ker(D_{λ}) = K ∩ B_{λ} where K is the field of fractions of ker(D_{λ}). By (a), B_{λ} is in particular normal; so it follows from a theorem of Zariski [26] that ker(D_{λ}) is finitely generated as a k-algebra. By 2.3, ker(D_{λ}) has transcendence degree 1 over k. Consequently, C_{λ} = Spec(ker D_{λ}) is an algebraic curve. Since B_{λ} is a normal domain and ker D_{λ} is integrally closed in B_{λ}, ker D_{λ} is a normal domain; hence C_{λ} is a smooth integral curve. Consider the dominant morphisms

Spec(B_{λ}) ⟶^{u} C_{λ} ⟶^{v} Spec A_{λ}

determined by the inclusions A_{λ} ⊆ ker(D_{λ}) ⊂ B_{λ} (i.e., u = 𝒬_{λ}). To complete the proof that ker(D_{λ}) = A_{λ}, it suffices to show that v is an isomorphism. As C_{λ} and Spec A_{λ} are smooth curves, it suffices to show that v is surjective and birational.

We know from 2.10 that the map 𝒬 : Spec B → Spec A is surjective, so the restriction ρ_{λ} : 𝒬^{−1}(L_{λ}) → L_{λ} of 𝒬 is also surjective, where we define L_{λ} to be the closed subscheme of Spec A determined by the principal ideal (f − λ)A of A. Now it is clear that we have a commutative diagram:

(7)  Spec(B_{λ}) ⟶^{≅} 𝒬^{−1}(L_{λ}) ^{v∘u}↓  ↓^{ρ_{λ}} Spec(A_{λ}) ⟶^{≅} L_{λ} (for general λ)

so the fact that ρ_{λ} is surjective implies that v is surjective.

As the general fibers of 𝒬 are 𝔸^{1}, a general λ ∈ k satisfies the condition:

𝒬^{−1}(P) = 𝔸^{1} for general P ∈ L_{λ}

or equivalently:

(8)  (v ∘ u)^{−1}(P) = 𝔸^{1} for general P ∈ Spec A_{λ}.

As u is dominant, the complement of im(u) is a finite set; thus for a general point P ∈ Spec A_{λ} we have v^{−1}(P) ⊂ im(u), and consequently

|v^{−1}(P)| ≤ number of connected components of (v ∘ u)^{−1}(P).

By (8) we obtain |v^{−1}(P)| ≤ 1, so v is birational and we have shown that

ker(D_{λ}) = π_{λ}(A) ≅ A/(f − λ)A

for general λ ∈ k. In fact v is the identity map and u = 𝒬_{λ}, so (7) is really:

(9)  Spec(B_{λ}) ⟶^{≅} 𝒬^{−1}(L_{λ}) ^{𝒬_{λ}} ↓  ↓^{ρ_{λ}} Spec(ker D_{λ}) ⟶^{≅} L_{λ} (for general λ).

As ρ_{λ} is surjective, it follows that 𝒬_{λ} is surjective.

Consider the open subset W = Spec A \ E of Spec A where, as before, E is the finite set 𝒬( Fix(D)); using 2.10 again, we see that the restriction 𝒬^{−1}(W) → W of 𝒬 is a smooth morphism. As L_{λ} ⊂ W (for general λ), it follows that 𝒬^{−1}(L_{λ}) ⟶^{ρ_{λ}} L_{λ} is a smooth morphism and consequently (by (9)) that 𝒬_{λ} : Spec B_{λ} → Spec(ker D_{λ}) is a smooth morphism. It follows that all fibers of 𝒬_{λ} are reduced.

Consider x ∈ Spec(ker D_{λ}) such that 𝒬_{λ}^{−1}(x) is not an affine line (over the suitable residue field). Then x is a closed point of Spec(ker D_{λ}). Moreover, the fixed point free G_{a}-action on Spec B_{λ} determined by D_{λ} restricts to a fixed point free G_{a}-action on each irreducible component of 𝒬_{λ}^{−1}(x); thus 𝒬_{λ}^{−1}(x) is a disjoint union of (reduced) affine lines.

Only (b–iii) remains to be proved. By (9) we have N(𝒬_{λ}) = N(ρ_{λ}), and since we have an isomorphism of schemes ρ_{λ}^{−1}(P) ≅ 𝒬^{−1}(P) for each P ∈ L_{λ}, we also have N(ρ_{λ}) = |L_{λ} ∩ 𝒳_{A}|. Since 𝒳_{A} is constructible (2.12) and not dense in Spec A, 𝒳̄_{A} \ 𝒳_{A} is a finite set of closed points and consequently L_{λ} ∩ 𝒳_{A} = L_{λ} ∩ 𝒳̄_{A} for general λ. We have 𝒳̄_{A} = V_{A}(ψ_{A}) (2.14), so N(𝒬_{λ}) = |L_{λ} ∩ V_{A}(ψ_{A})| for general λ, from which we obtain the last assertion. □

3.4. Lemma. Let the setup be as in 3.1 and consider an element λ ∈ k for which all conclusions of 3.3 are valid. The following conditions are equivalent:

(a) ker(D_{λ}) = k^{[1]} and N(𝒬_{λ}) ≤ 1

(b) 𝒬_{λ} : Spec B_{λ} → Spec ( ker D_{λ}) defines a Danielewski surface.

Proof. It is obvious that (b) implies (a). Assume that (a) holds then, as ker(D_{λ}) = k^{[1]}, the codomain of 𝒬_{λ} is 𝔸_{k}^{1}; Spec B_{λ} is a smooth affine surface and 𝒬_{λ} is a surjective morphism with reduced fibers, and such that each fiber is a disjoint union of affine lines; so there only remains to check condition (i) of 1.3. Consider ρ_{λ} as in (9); N(𝒬_{λ}) ≤ 1 and N(𝒬_{λ}) = |V_{A}(ψ_{A}, f −λ)| imply that the curve L_{λ} = V_{A}(f −λ) ⊂ Spec A meets \overline{𝒳}_{A} in at most one point x, and 2.15 implies that ρ_{λ}^{−1}(L_{λ} \ {x}) is isomorphic to 𝔸_{k}^{1} × (L_{λ} \ {x}) over L_{λ}\{x}; via (9), this implies that 𝒬_{λ} satisfies condition (i) of 1.3, so (b) holds. □

3.5. Lemma. Let Spec R be a smooth affine surface such that ML(R) = k, let R_{0} ∈ klnd(R) and let π : Spec R → Spec R_{0} be the morphism determined by R_{0} ↪ R. Then R_{0} = k^{[1]} and N(π) ≤ 1.

Proof. This is well-known, see for instance [10, 2.3], [11, 2.15]. □

3.6. Proposition. Let the setup be as in 3.1. If ML(B_{λ}) = k for infinitely many λ ∈ k (resp. for general λ ∈ k) then:

(a) A = k[f]^{[1]}

(b) B_{λ} ∈ 𝔇(k) for infinitely many λ ∈ k (resp. for general λ ∈ k).

Proof. There is a cofinite subset Λ ⊆ k such that all conclusions of 3.3 are valid for each λ ∈ Λ; define Λ_{∞} = { λ ∈ Λ | ML(B_{λ}) = k } and note that Λ_{∞} is an infinite (resp. a cofinite) subset of k included in Λ.

Let λ ∈ Λ_{∞}. As Spec(B_{λ}) is a smooth affine surface satisfying ML(B_{λ}) = k, 3.5 gives

(10)  ker(D_{λ}) = k^{[1]} and N(𝒬_{λ}) ≤ 1

so 3.4 implies that 𝒬_{λ} defines a Danielewski surface. Since Spec(B_{λ}) is isomorphic to a smooth hypersurface of 𝔸^{3}, its canonical sheaf is trivial and 𝒬_{λ} therefore satisfies condition (a) of [12, 5.6]; by that result,^{3} it follows that Spec(B_{λ}) is isomorphic to a hypersurface of 𝔸^{3} with equation xy = p(z), for some nonconstant univariate polynomial p(z) with distinct roots; so B_{λ} ∈ 𝔇(k) (for each λ ∈ Λ_{∞}), which proves (b).

3In the case k = ℂ we may use instead the result of Bandman and Makar-Limanov [2].

If λ ∈ Λ_{∞} then (10) and 3.3 give A/(f − λ) ≅ ker(D_{λ}) = k^{[1]}, so the Abhyankar-Moh-Suzuki Theorem implies that A = k[f]^{[1]}, which proves (a). □

3.7. Lemma. Let the setup be as in 3.1 and suppose that the condition

(11)  ker(D_{λ}) = k^{[1]} and N(𝒬_{λ}) ≤ 1

holds for infinitely many λ ∈ k. Then there exists g such that A = k[f, g] and ψ_{A} = a(f)g^{n} for some a(f) ∈ k[f] \ {0} and n ∈ ℕ.

Proof. There is a cofinite subset Λ ⊆ k such that, for each λ ∈ Λ, all conclusions of 3.3 are valid, and an infinite subset Λ_{∞} ⊆ Λ such that (11) holds for every λ ∈ Λ_{∞}.

Let λ ∈ Λ_{∞}. Since A/(f −λ) ≅ ker(D_{λ}) = k^{[1]} by 3.3 and (11), the Abhyankar-Moh-Suzuki Theorem gives A = k[f, h] for some h. So ψ_{A} is a polynomial in two variables f, h and (by 3.3 and (11)) satisfies

|V_{A}(ψ_{A}, f − λ)| = N(𝒬_{λ}) ≤ 1

for all λ ∈ Λ_{∞}. This implies that ψ_{A} = a(f)(u(f)h + v(f))^{n}, for some n ∈ ℕ and a, u, v ∈ k[f] such that a ≠ 0 and gcd(u, v) = 1.

Note that u ∈ k. Indeed, if u ∉ k then v ≠ 0, because gcd(u, v) = 1; so p = uh + v is a prime factor of ψ_{A} which violates 2.16 (A/pA has at least two places at infinity, so cannot be embedded in k[t]). Moreover, if u = 0 then ψ_{A} ∈ k[f] and we are done. So we may assume from now-on that u ∈ k^{∗}. Define g = uh + v, then A = k[f, g] and ψ_{A} = a(f)g^{n}, where a ∈ k[f] \ {0} and n ∈ ℕ. □

3.8. Lemma. Suppose that R ∈ 𝔇(k) and let D : R → R be an irreducible locally nilpotent derivation. Then there exists s ∈ R such that ker D = k[Ds] = k^{[1]}.

Proof. Write R = k[X, Y, Z]/(XY − P (Z)) where P (Z) ∈ k[Z] \ k and let x, y, z be the images in R of X, Y, Z respectively. The main result of [6] implies, in particular, that the group of k-automorphisms of R acts transitively on the set klnd(R); so we may assume that ker D = k[x]. Then it is easy to see that D(z) = ax for some a ∈ k^{∗}, which proves the claim. □

3.9. Lemma. Let x, y be elements of an affine k-domain R such that (1) x is transcendental over k

(2) for each λ in an infinite subset of k, there exists c_{λ} ∈ k such that y − c_{λ} ∈

(x − λ)R. Then there exist α, β ∈ k[x] such that α ≠ 0 and αy + β = 0.

Proof. Consider the subalgebras k[x] and k[y] of R. If y ∈ k then the conclusion of the lemma holds trivially, so we may assume throughout that k[y] = k^{[1]} (and we have k[x] = k^{[1]} by assumption (1)). Let R_{0} = k[x, y] ⊆ R and consider the commutative diagram

\xymatrix{Spec k[x] & Spec R \ar[l]_{u} \ar[d]^{w} \ar[r]^{v} & Spec k[y] \\  & Spec R_{0} \ar@{-->}[ul]^{u_{0}} \ar@{-->}[ur]_{v_{0}}}

where the morphisms are determined by the corresponding inclusions. For a closed point P ∈ Spec k[x], write F_{P} = u_{0}^{−1}(P) ⊆ Spec R_{0}. The reader may verify that

w(u^{−1}(P)) is a dense subset of F_{P}

holds for a general closed point P ∈ Spec k[x]; consequently, v_{0}(w(u^{−1}(P))) is dense in v_{0}(F_{P} ); as v_{0}(w(u^{−1}(P))) = v(u^{−1}(P)), we conclude that

(12)  for a general closed point P ∈ Spec k[x], v(u^{−1}(P)) is dense in v_{0}(F_{P} ).

On the other hand, the assumptions of the lemma imply:

(13) For infinitely many closed points P ∈ Spec k[x],

v maps u^{−1}(P) to a closed point of Spec k[y].

By (12) and (13), it follows that v_{0}(F_{P} ) is one closed point, i.e.,

(14) For infinitely many closed points P ∈ Spec k[x],

v_{0} maps u_{0}^{−1}(P) to a closed point of Spec k[y].

From (14) it is clear that x, y must be algebraically dependent. Indeed, if they are not then Spec R_{0} = 𝔸^{2} and u_{0}, v_{0} are the standard projections

Spec k[x] ⟵^{u_{0}} 𝔸^{2} ⟶^{v_{0}} Spec k[y],

but obviously these maps do not satisfy (14).

So there exists an irreducible Φ(X, Y) ∈ k[X, Y] = k^{[2]} such that Φ(x, y) = 0. Moreover, assumption (1) implies that deg_{Y} Φ(X, Y) > 0. As R_{0} ≅ k[X, Y]/Φk[X, Y], we may identify Spec R_{0} with the curve V (Φ) ⊂ 𝔸^{2}; then u_{0}, v_{0} are the projections 𝔸^{1} ⟵^{u_{0}} V (Φ) ⟶^{v_{0}} 𝔸^{1}, i.e., u_{0}(x, y) = x and v_{0}(x, y) = y. Then (14) states that, for infinitely many λ ∈ k, the equation Φ(λ, Y) = 0 has only one solution Y in k. This implies that deg_{Y} (Φ) = 1, which proves the lemma. □

3.10. Proposition. Let the setup be as in 3.1 and suppose that ML(B_{λ}) = k for infinitely many λ ∈ k. Then there exists g such that A = k[f, g] and ψ_{A} = a(f)g^{n} for some a(f) ∈ k[f] \ {0} and n ∈ {0, 1}.

Proof. There is a cofinite subset Λ ⊆ k such that, for each λ ∈ Λ, all conclusions of 3.3 are valid; by 3.6, there is an infinite subset Λ_{∞} ⊆ Λ such that B_{λ} ∈ 𝔇(k) (and hence ML(B_{λ}) = k) holds for every λ ∈ Λ_{∞}.

Let λ ∈ Λ_{∞}. Then Spec(B_{λ}) is a smooth affine surface and ML(B_{λ}) = k, so 3.5 implies that ker(D_{λ}) = k^{[1]} and N(𝒬_{λ}) ≤ 1; since this holds for infinitely many λ, 3.7 implies that there exists g such that A = k[f, g] and ψ_{A} = a(f)g^{n} for some a(f) ∈ k[f] \ {0} and n ∈ ℕ. There only remains to show that n ≤ 1. By contradiction, we assume:

(15)  n > 1.

If λ ∈ Λ_{∞} then B_{λ} ∈ 𝔇(k) and, by 3.3, Fix(D_{λ}) = ∅; as the last condition implies that D_{λ} is irreducible, we may apply 3.8 to B_{λ} and D_{λ} and obtain

(16) for each λ ∈ Λ_{∞}, there exists σ_{λ} ∈ B_{λ} such that ker D_{λ} = k[D_{λ}(σ_{λ})].

For λ ∈ Λ_{∞}, define g_{λ} = π_{λ}(g) ∈ B_{λ}; then 3.3 gives ker(D_{λ}) = π_{λ}(A) = k[g_{λ}] = k^{[1]}. Define the ideal I_{λ} = D_{λ}(B_{λ}) ∩ k[g_{λ}] of k[g_{λ}].

Note that if λ is such that I_{λ} = k[g_{λ}] then 2.3 implies that B_{λ} = (ker D_{λ})^{[1]} = k[g_{λ}]^{[1]}, i.e., B/(f −λ)B = k^{[2]}. If this is the case for infinitely many λ, then [9, Thm 3] implies that f is a variable of B and in this case it is well-known (see for instance part (5) of [8, 3.2]) that ψ_{A} ∈ k[f], which contradicts (15). So there exists an infinite subset Λ′_{∞} of Λ_{∞} for which:

I_{λ} ≠ k[g_{λ}] for all λ ∈ Λ′_{∞}.

From now-on, let λ ∈ Λ′_{∞}.

Choose a maximal ideal 𝔪_{λ} of k[g_{λ}] such that I_{λ} ⊆ 𝔪_{λ}. By definition of ψ_{A} there exists s ∈ B such that D(s) = ψ_{A} = a(f)g^{n}, so D_{λ}(π_{λ}(s)) = π_{λ}(Ds) = a(λ)g_{λ}^{n} where a(λ) ∈ k^{∗}. This shows that g_{λ}^{n} ∈ I_{λ} ⊆ 𝔪_{λ} and consequently

𝔪_{λ} = g_{λ} k[g_{λ}].

The element D_{λ}(σ_{λ}) of condition (16) must belong to I_{λ} ⊆ 𝔪_{λ} and must generate k[g_{λ}] as a k-algebra; so D_{λ}(σ_{λ}) = α_{λ}g_{λ} for some α_{λ} ∈ k^{∗}, and we may therefore choose σ_{λ} so that

D_{λ}(σ_{λ}) = g_{λ}.

Choose t_{λ} ∈ B such that π_{λ}(t_{λ}) = σ_{λ} and consider s − ag^{n−1}t_{λ} ∈ B. Then

π_{λ}(D(s − ag^{n−1}t_{λ})) = π_{λ}(ag^{n} − ag^{n−1}D(t_{λ})) = a(λ)g_{λ}^{n} − a(λ)g_{λ}^{n−1}π_{λ}(D(t_{λ}))

= a(λ)g_{λ}^{n} − a(λ)g_{λ}^{n−1}D_{λ}(σ_{λ}) = 0.

Since (for x ∈ B) the condition π_{λ}(D(x)) = 0 is equivalent to x ∈ (f − λ)B + k[g], there exists P_{λ}(g) ∈ k[g] such that s − ag^{n−1}t_{λ} − P_{λ}(g) ∈ (f − λ)B. If c_{λ} ∈ k is the constant coefficient of P_{λ} then (using n > 1) we conclude that s − c_{λ} ∈ (g, f − λ)B. We record:

(17)  For each λ ∈ Λ′_{∞}, ∃ c_{λ} ∈ k such that s − c_{λ} ∈ (g, f − λ)B.

Applying 3.9 to the elements x = f + gB and y = s + gB of R = B/gB, we obtain that there exist α(f), β(f) ∈ k[f] such that α(f) ≠ 0 and α(f)s + β(f) ∈ gB. Define s′ = (α(f)s + β(f))/g ∈ B; then D(s′) = α(f)a(f)g^{n−1}, so α(f)a(f)g^{n−1} ∈ I_{A} \ {0}. This is absurd, because I_{A} is the principal ideal of k[f, g] generated by ψ_{A} = a(f)g^{n}. So (15) is false and the proof is complete. □

The following is a corollary to [7, 2.6.2]:

3.11. Let ℬ be a factorial domain containing a field K of characteristic zero, and suppose that 𝒟 ∈ lnd(ℬ) and s ∈ ℬ satisfy:

ker 𝒟 = K[𝒟s] = K^{[1]}.

Then ℬ ∈ 𝔇(K). Moreover, if 𝒟 is irreducible then there exist h ∈ ℬ and a nonconstant polynomial P (T) ∈ K[T] such that ℬ = K[𝒟(s), h, s] and 𝒟(s)h = P (s).

3.12. Proposition. Let the setup be as in 3.1 and suppose that ML(B_{λ}) = k for infinitely many λ ∈ k. Then ℬ ∈ 𝔇(K), where we define ℬ = k(f) ⊗_{k[f]} B and K = k(f).

Proof. By definition of ψ_{A}, there exists s_{0} ∈ B such that D(s_{0}) = ψ_{A}. By 3.10, there exists g such that A = k[f, g] and ψ_{A} = ag^{n} for some a ∈ k[f] \ {0} and n ∈ {0, 1}. Define

s = \begin{cases}s_{0}, & if n = 1 \\ gs_{0}, & if n = 0, \end{cases}

so that D(s) = ag. Consider the locally nilpotent derivation 𝒟 = S^{−1}D : ℬ → ℬ obtained by localizing D with respect to S = k[f] \ {0} and note that

ker 𝒟 = S^{−1}k[f, g] = K[g] = K[𝒟(s)] = K^{[1]}.

Then 3.11 asserts that ℬ ∈ 𝔇(K). □

4. Irreducible polynomials belonging to at least two kernels

In this section we set B = k^{[3]}, where k is an algebraically closed field of characteristic zero.

The following fact is due to Freudenburg and this author, and is in fact valid without assuming that k is algebraically closed; a proof should appear in the forthcoming [13].

4.1. Suppose that A, A′ are distinct elements of klnd(B) such that A ∩ A′ ≠ k. Then A ∩ A′ = k^{[1]}, A = (A ∩ A′)^{[1]} and A′ = (A ∩ A′)^{[1]}.

We also need the following:

4.2. Lemma. Suppose that R is an integral domain and a finitely generated k-algebra, and that f, g, h ∈ R are algebraically independent over k. For each λ ∈ k, let π_{λ} : R → R/(f − λ)R be the canonical epimorphism. Then, for general λ ∈ k, the elements π_{λ}(g), π_{λ}(h) ∈ R/(f − λ)R are algebraically independent over k.

Proof. Let φ : k[X, Y, Z] → R be the k-homomorphism which maps X, Y, Z to f, g, h respectively, and let φ^{∗} : Spec R → 𝔸^{3} be the morphism determined by φ. The hypothesis implies that φ is injective, so φ^{∗} is dominant and consequently there exists an open subset U ≠ ∅ of 𝔸^{3} such that U ⊆ im(φ^{∗}).

Let λ ∈ k be such that π_{λ}(g), π_{λ}(h) are algebraically dependent over k. Then there exists P (Y, Z) ∈ k[Y, Z] \ {0} such that P (g, h) ∈ (f − λ)R. We claim that

(18)  V (X − λ) ∩ im(φ^{∗}) ⊆ V (X − λ, P).

Indeed, if 𝔭 ∈ V (X − λ) ∩ im(φ^{∗}) then φ^{−1}(𝔮) = 𝔭 for some 𝔮 ∈ Spec R; then X − λ ∈ 𝔭 implies that f −λ = φ(X −λ) ∈ 𝔮, so φ(P) = P (g, h) ∈ (f −λ)R ⊆ 𝔮 and consequently P ∈ 𝔭, proving (18). It follows that V (X −λ)∩U ⊆ V (X −λ, P); as V (X −λ, P (Y, Z)) has dimension at most 1, whereas V (X − λ) ∩ U is either empty or two-dimensional, we obtain that V (X − λ) ∩ U = ∅, a condition which can hold for at most finitely many λ. □

4.3. Lemma. Suppose that f is an irreducible element of B which belongs to at least two elements of klnd(B). Define B_{λ} = B/(f − λ)B for each λ ∈ k.

(a) For each A ∈ klnd(B) satisfying f ∈ A, we have A = k[f]^{[1]}.

(b) B_{λ} is an integral domain, for every λ ∈ k.

(c) ML(B_{λ}) = k for general λ ∈ k.

Proof. Let A, A′ be distinct elements of klnd(B) such that f ∈ A ∩ A′. By 4.1 we have A = k[f]^{[1]} and A′ = k[f]^{[1]}, which proves (a). Note that A = k[f]^{[1]} implies that, for every λ ∈ k, f − λ is an irreducible element of A, hence an irreducible element of B; so (b) is proved. Choose g, h ∈ B such that A = k[f, g] and A′ = k[f, h]. Then h ∉ A; it follows that h is not algebraic over A and hence that f, g, h are algebraically independent over k; so, by 4.2, the condition

π_{λ}(g), π_{λ}(h) are algebraically independent over k

is satisfied for general λ ∈ k (where π_{λ} : B → B_{λ} is the canonical epimorphism). Let D (resp. D′) be an irreducible locally nilpotent derivation of B such that ker D = A (resp. ker D′ = A′). Consider D_{λ}, D′_{λ} : B_{λ} → B_{λ}, the locally nilpotent derivations determined by D and D′ respectively. Then D_{λ}, D′_{λ} are nonzero, because D and D′ are irreducible, so each one of R = ker(D_{λ}), R′ = ker(D′_{λ}) has transcendence degree 1 over k. Clearly, D_{λ}(π_{λ}(g)) = 0 and D′_{λ}(π_{λ}(h)) = 0. If D_{λ}(π_{λ}(h)) = 0 then π_{λ}(g), π_{λ}(h) must be algebraically dependent over k; so for general λ we have D_{λ}(π_{λ}(h)) ≠ 0 and consequently R, R′ are distinct elements of klnd(B_{λ}). Then R ∩ R′ has transcendence degree 0 over k; as k is algebraically closed, R ∩ R′ = k and consequently ML(B_{λ}) = k. □

4.4. Corollary. Let f be an irreducible element of B which belongs to at least two elements of klnd(B). Then the following hold:

(a) B/(f − λ) ∈ 𝔇(k) for general λ ∈ k

(b) ℬ ∈ 𝔇(K), where we define ℬ = k(f) ⊗_{k[f]} B and K = k(f)

(c) For any A ∈ klnd(B) satisfying f ∈ A, there exists g such that A = k[f, g] and ψ_{A} = a(f)g^{n} for some a(f) ∈ k[f] \ {0} and n ∈ {0, 1}.

Proof. Choose A ∈ klnd(B) which contains f. Then the pair (f, A) satisfies the global assumptions 3.1 and we may apply the results of Section 3. By 4.3, the hypotheses of 3.6, 3.12 and 3.10 are satisfied; so we are done. □

We may now obtain the Main Theorem 1.10 as an easy consequence of 4.4.

Proof of 1.10. Let f ∈ B \ k, S = k[f] \ {0}, ℬ = S^{−1}B and K = k(f).

By 4.4, we obtain that 1.10(a) ⇒ 1.10(c). Implication 1.10(c) ⇒ 1.10(b) is wellknown (see for instance [7, 2.3(d)]). Implication 1.10(b) ⇒ 1.10(a) is also quite wellknown, but let us recall the argument. Suppose that f is such that ML(ℬ) = K. Let us first show that if 𝒜 ∈ klnd(ℬ) then B ∩ 𝒜 ∈ klnd(B). Choose 𝒟 ∈ lnd(ℬ) such that ker 𝒟 = 𝒜. Let X_{1}, X_{2}, X_{3} be such that B = k[X_{1}, X_{2}, X_{3}]. For each i ∈ {1, 2, 3} we have 𝒟(X_{i}) ∈ S^{−1}B, so there exists s ∈ S such that (for all i = 1, 2, 3) s𝒟(X_{i}) ∈ B. Then s𝒟(B) ⊆ B and (since S ⊂ ker 𝒟) s𝒟 ∈ lnd(ℬ); so if we let D : B → B be the restriction of 𝒟 then D ∈ lnd(B) and ker D = B ∩ 𝒜. So we have a well-defined map

klnd(ℬ) ⟶ { A ∈ klnd(B) | f ∈ A }, 𝒜 ↦ B ∩ 𝒜

which is easily seen to be injective. As trdeg_{K}(ℬ) = 2 and ML(ℬ) = K, it follows that klnd(ℬ) has at least 2 elements, so f ∈ A ∩ A′ for some distinct elements A, A′ ∈ klnd(B). By 4.1 we have A ∩ A′ = k[f_{0}] = k^{[1]} for some f_{0}, and this implies that k[f] ⊆ k[f_{0}]. As k(f_{0}) is algebraic over k(f) and k(f) = ML(ℬ) is algebraically closed in k(X_{1}, X_{2}, X_{3}), we must have k(f) = k(f_{0}) and hence k[f] ⊆ k[f_{0}] ⊂ k(f); thus k[f] = k[f_{0}] = A ∩ A′, so k[f] is factorially closed in B. As f is irreducible in k[f], it therefore follows that it is irreducible in B. So 1.10(b) implies 1.10(a). □

4.5. Corollary. Suppose that f is an element of B satisfying:

(i) f belongs to at least one element of klnd(B)

(ii) for infinitely many λ ∈ k, B/(f − λ) is a domain and ML (B/(f − λ)) = k. Then f belongs to at least two elements of klnd(B).

Proof. By (i), we may consider A ∈ klnd(B) such that f ∈ A. As (f, A) satisfies the global assumptions 3.1, we can use 3.12 and obtain that f satisfies condition (c) of 1.10; so f satisfies condition (a) of the same result. □

5. Families of Danielewski surfaces in 𝔸^{3}

Let k be an algebraically closed field of characteristic zero and let B = k^{[3]}. Given a nonconstant polynomial f ∈ B, define the family of surfaces

S_{λ} = V_{B}(f − λ) ⊂ Spec B = 𝔸_{k}^{3} (for each λ ∈ k).

We are interested in polynomials f satisfying:

(∗)  For general λ, S_{λ} is a Danielewski surface.

To be precise, “S_{λ} is a Danielewski surface” means that there exists a morphism π_{λ} : S_{λ} → 𝔸_{k}^{1} which satisfies definition 1.3. Example 5.1 presents a method for obtaining such polynomials f, but the method produces polynomials which satisfy a condition stronger than (∗); result 5.2 identifies exactly that condition. Note that we don’t know any example showing that the “stronger” condition of 5.2 is indeed stronger than (∗).

5.1. Example. We show that if A = k[f, g] is any element of klnd(B) satisfying:^{4}

(19)  ψ_{A} = a(f)g^{n} (a(f) ∈ k[f] \ {0}, n ≥ 0),

then f satisfies (∗). Note that (f, A) satisfies the global assumptions of 3.1, so all results of Section 3 can be applied. Let B_{λ} = B/(f − λ) and consider

𝒬_{λ} : Spec B_{λ} → Spec ( ker D_{λ})

defined as in 3.1. By 3.3, we have (for general λ ∈ k) ker D_{λ} ≅ A/(f − λ) = k^{[1]} and N(𝒬_{λ}) = |V_{A}(ψ_{A}, f − λ)| = |V_{A}(a(f)g^{n}, f − λ)| ≤ 1, so by 3.4:

for general λ, 𝒬_{λ} defines a Danielewski surface.

So Spec B_{λ} is a Danielewski surface; since S_{λ} = V_{B}(f − λ) ≅ Spec B_{λ}, f satisfies (∗). (Note that f also satisfies condition (a) of 5.2, below.)

Let us also prove:

• if n = 0 then f is a variable of B, so B_{λ} = k^{[2]} for all λ;

• if n = 1 then f is a basic element of B (cf. 1.4) and, for general λ, B_{λ} ∈ 𝔇(k) and B_{λ} ≠ k^{[2]};

• if n > 1 then ML(B_{λ}) = k^{[1]} for general λ.

Proof. If n = 0 then ψ_{A} ∈ k[f], so I_{A} ∩ k[f] ≠ {0}; this implies that k(f) ⊗_{k[f]} B = (k(f) ⊗_{k[f]} A)^{[1]} = k(f)^{[2]}, so f is a variable of B by [9, Thm 3] and consequently B_{λ} = k^{[2]} for all λ.

Note that if B_{λ} = k^{[2]} for infinitely many λ then f is a variable of B by [9, Thm 3] and consequently ψ_{A} ∈ k[f] (part (5) of [8, 3.2]), so n = 0. Consequently, n = 1 implies that B_{λ} ≠ k^{[2]} for general λ. If n = 1 then 3.11 implies (as in the proof of 3.12) that k(f) ⊗_{k[f]} B belongs to 𝔇(k(f)), i.e., f satisfies condition (c) of 1.10. Then f belongs to at least two elements of klnd(B), i.e., is a basic element of B; by 4.4, B_{λ} ∈ 𝔇(k) for general λ.

If ML(B_{λ}) = k for infinitely many λ ∈ k then (4.5) f belongs to two elements of klnd(B) and consequently part (c) of 4.4 implies that n ≤ 1. In other words, if n > 1 then (for general λ) ML(B_{λ}) ≠ k, so ker(D_{λ}) must be the unique element of klnd(B_{λ}) and ML(B_{λ}) = ker D_{λ} ≅ A/(f − λ) = k^{[1]}, for general λ. □

5.2. Proposition. Let f ∈ B \ k and define S_{λ} = V_{B}(f − λ) ⊂ 𝔸_{k}^{3} for each λ ∈ k. For the polynomial f, conditions (a) and (b) are equivalent:

(a) The following two conditions hold: (i) For general λ ∈ k, S_{λ} is a Danielewski surface (ii) for some λ_{0} ∈ k satisfying condition (i), some nontrivial G_{a}-action on S_{λ_{0}}

can be extended to a G_{a}-action on 𝔸_{k}^{3}.

(b) There exists g ∈ B such that if we set A = k[f, g] then A ∈ klnd(B) and ψ_{A} = a(f)g^{n}, for some a(f) ∈ k[f] \ {0} and n ≥ 0.

4See 5.3 for examples of kernels satisfying (19).

Proof. We prove that (a) implies (b) (the converse is proved in 5.1). Suppose that f satisfies (a). There is a G_{a}-action α on 𝔸_{k}^{3} which restricts to a nontrivial G_{a}-action on S_{λ_{0}}. Let D′ : B → B be the locally nilpotent derivation such that α = α_{D′} (2.5). Using that α maps S_{λ_{0}} = V (f −λ_{0}) into itself, it is easy to see that D′(f −λ_{0}) = 0. So f ∈ A, where we define A = ker D′; hence the pair (f, A) satisfies the global assumptions 3.1 and the results of Section 3 can be applied. Choose an irreducible locally nilpotent derivation D : B → B with kernel A and for each λ define B_{λ} = B/(f − λ) and D_{λ} : B_{λ} → B_{λ} as in 3.1. If ML(B_{λ}) = k for infinitely many λ then 3.10 implies that f satisfies condition (b) with n ≤ 1. We may therefore assume that

(20)  ML(B_{λ}) ≠ k for general λ.

By assumption, for general λ there exists a morphism π_{λ} : Spec(B_{λ}) → 𝔸_{k}^{1} which makes Spec(B_{λ}) a Danielewski surface. Algebraically, π_{λ} corresponds to an injective homomorphism φ : k^{[1]} → B_{λ} whose image is an element of klnd(B_{λ}), but (20) implies that ker(D_{λ}) is the unique element of klnd(B_{λ}), so φ factors as

k^{[1]} ⟶^{≅} ker(D_{λ}) ↪ B_{λ}

and in particular ker(D_{λ}) = k^{[1]}. It follows that π_{λ} factors as

Spec B_{λ} ⟶^{𝒬_{λ}} Spec(ker D_{λ}) ⟶^{≅} 𝔸_{k}^{1},

so 𝒬_{λ} defines a Danielewski surface and consequently N(𝒬_{λ}) ≤ 1. As the conditions ker(D_{λ}) = k^{[1]} and N(𝒬_{λ}) ≤ 1 hold for general λ, it follows from 3.7 that f satisfies (b). □

5.3. Here are some examples of kernels A = k[f, g] ∈ klnd(B) satisfying (19).

(1) Let f = X^{n}Y + Z^{2} + Xq(X, Z) ∈ B = k[X, Y, Z], then A = k[f, X] belongs to klnd(B) and ψ_{A} = X^{n}. In [21], the Danielewski surfaces V_{B}(f − λ) in the case n = 2 are shown to have interesting pathologies (for instance automorphisms which do not extend to 𝔸^{3}).

(2) Let n ≥ 1 and let A = ker Δ_{n} with Δ_{n} as in [14, 4.1]. Then A = k[G_{n}, F] and ψ_{A} = G_{n}F^{n} (G_{n} and F are defined in [14] and are homogeneous polynomials of degrees 4n + 1 and 2 respectively).

(3) (Generalization of (2).) Let D : B → B be any nonzero locally nilpotent derivation which is homogeneous with respect to a positive grading of B. Then A = ker D satisfies (19). Indeed, by [5, 3.10], for a suitable choice of f, g we have A = k[f, g] and ψ_{A} = f^{i}g^{n} for some i, n ∈ ℕ.

Note that if Conjecture 1.14 is true then 4.4 implies that for every element A of klnd(B) there is at least one pair (f, g) which satisfies A = k[f, g] and (19). In particular, every known example of kernel satisfies (19).

References

1. T. Asanuma, Polynomial fibre rings of algebras over noetherian rings, Invent. math. 87 (1987), 101–127.

2. T. Bandman and L. Makar-Limanov, Affine surfaces with AK(S) = ℂ, Michigan Math. J. 49 (2001), 567–582.

3. P. Bonnet, Surjectivity of quotient maps for algebraic (ℂ, +)-actions and polynomial maps with contractible fibres, Transform. Groups 7 (2002), 3–14.

4. D. Daigle, On some properties of locally nilpotent derivations, J. Pure Appl. Algebra 114 (1997), 221–230.

5. , Homogeneous locally nilpotent derivations of k[x, y, z], J. Pure Appl. Algebra 128 (1998), 109–132.

6. , On locally nilpotent derivations of k[X_{1}, X_{2}, Y]/(φ(Y)−X_{1}X_{2}), J. Pure and Appl. Algebra 181 (2003), 181–208.

7. , Locally nilpotent derivations and Danielewski surfaces, Osaka J. Math. 41 (2004), 37–80.

8. D. Daigle and G. Freudenburg, Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations of k[X_{1}, …, X_{n}], J. of Algebra 204 (1998), 353–371.

9. D. Daigle and S. Kaliman, A note on locally nilpotent derivations and variables of k[x, y, z], to appear in Can. Bull. Math., 2006.

10. D. Daigle and P. Russell, On log ℚ-homology planes and weighted projective planes, Canadian J. of Math. 56 (2004), 1145–1189.

11. A. Dubouloz, Completions of normal affine surfaces with a trivial Makar-Limanov invariant, Michigan Math. J. 52 (2004), 289–308.

12. , Danielewski-fieseler surfaces, Transformation Groups 10 (2005), 139–162.

13. G. Freudenburg, Locally nilpotent derivations and G_{a}-actions, Invariant theory and algebraic transformation groups, Springer-Verlag, book in preparation.

14. , Local slice constructions in K[X, Y, Z], Osaka J. Math. 34 (1997), 757–767.

15. , Actions of G_{a} on A^{3} defined by homogeneous derivations, J. Pure and Appl. Algebra 126 (1998), 169–181.

16. S. Kaliman, Free ℂ_{+}-actions on ℂ^{3} are translations, Invent. Math. 156 (2004), 163–173.

17. T. Kambayashi and M. Miyanishi, On flat fibrations by the affine line, Illinois J. of Math. 22 (1978), 662–671.

18. L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel J. of Math. 69 (1990), 250–256.

19. H. Matsumura, Commutative Algebra, 2nd edition, Mathematics Lecture Note Series, Benjamin/Cummings, 1980.

20. M. Miyanishi, Normal affine subalgebras of a polynomial ring, Algebraic and Topological Theories—to the memory of Dr. Takehiko MIYATA, Kinokuniya, 1985, pp. 37–51.

21. L. Moser-Jauslin and P. M. Poloni, Embeddings of a family of Danielewski hypersurfaces and certain ℂ^{+}-actions on ℂ^{3}, preprint, 2006.

22. Y. Stein, The total reducibility order of a polynomial in two variables, Israel J. Math. 68 (1989), 109–122.

23. A. van den Essen, Polynomial automorphisms, Progress in Mathematics, vol. 190, Birkhäuser, 2000.

24. A. N. Varchenko, Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings, Math. USSR-Izv. 6 (1972), 949–1008.

25. H. Yoshihara, On Plane Rational Curves, Proc. Japan Acad. (Ser. A) 55 (1979), 152–155.

26. O. Zariski, Interprétations algébro-géométriques du quatorzième problème de Hilbert, Bull. Sci. Math. 78 (1954), 155–168.

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada K1N 6N5

E-mail address: ddaigle@uottawa.ca