Étale endomorphisms of algebraic surfaces with G_{m}-actions

Kayo Masuda and Masayoshi Miyanishi

Abstract

Let X be an algebraic surface defined over the complex field C and endowed with an A_{∗}^{1}-fibration. As such a surface we have a Platonic A_{∗}^{1}-fiber space, a weighted hypersurface with its singular point deleted off and, more generally, an affine algebraic surface with an unmixed G_{m}-action and its fixpoint deleted off. We consider an étale endomorphism φ : X → X and show that φ is an automorphism in most cases. Of particular interest is the case of a Platonic A_{∗}^{1}-fiber space, for which φ being an automorphism is closely related to the Jacobian Problem for the affine plane A^{2}. We observe also the automorphism group of such surfaces.

1 Introduction

Throughout the present article we work over the complex field C. Let X be a smooth algebraic surface with a surjective morphism ρ : X → B, where B is a smooth projective algebraic curve. We call a pair (X, ρ) (or simply X when the morphism ρ is tacitly given) an A_{∗}^{1}-fiber space (or a C^{∗}-fiber space) if the general fibers of ρ are isomorphic to A_{∗}^{1}, which is the affine line A^{1} with one point deleted off. We assume that there exists an open set U of B such that ρ^{−1}(U) is a trivial A_{∗}^{1}-bundle over U. The morphism ρ is called an A_{∗}^{1}-fibration (or C^{∗}-fibration). As such surfaces, we consider in the present article the following three surfaces:

(1) Platonic A_{∗}^{1}-fiber spaces, (2) A weighted hypersurface x_{1}^{m_{1}} + x_{2}^{m_{2}} + x_{3}^{m_{3}} = 0 in the affine 3-space A^{3}

with its singular point (0, 0, 0) deleted off,

(3) An algebraic surface with an unmixed G_{m}-action and its fixpoint deleted off.

The famous Jacobian problem asks whether or not an étale endomorphism φ of the affine plane A^{2} is an automorphism. We can generalize this problem as follows:

Generalized Jacobian Problem Let X be a smooth algebraic surface and let φ : X → X be an étale endomorphism. Is φ an automorphism ?

We consider the generalized Jacobian Problem for the above three surfaces. Of particular interest is the case of Platonic A_{∗}^{1}-fiber spaces. A Platonic A_{∗}^{1}-fiber space is written as A^{2}/G with the unique singular point deleted off, where G is a non-cyclic small finite subgroup of GL(2, C), and the generalized Jacobian problem is equivalent to asking whether or not an étale endomorphism ψ : A^{2} → A^{2} is an automorphism whenever ψ commutes with a linear action of the finite group G on A^{2} (cf. Lemma 2.2). Though we cannot solve the problem in the Platonic case, this became an introduction to consider the other surfaces with A_{∗}^{1}-fibrations. The generalized Jacobian problem does not necessarily have a positive answer (cf. Lemmas 5.4 and 5.8). The presentation of the present article is organized in such a way that we first treat the cases of Platonic A_{∗}^{1}-fiber spaces and weighted hypersurfaces as the motivations to the problem and then treat the general case of algebraic surfaces with G_{m}-actions. In the last section, we consider when a given A_{∗}^{1}-fibration comes from a G_{m}-action and also observe the automorphism group of algebraic surfaces with G_{m}-actions.

The first author would like to express her hearty thanks to the Max-Planck Institute for its hospitality and the splendid research environments which were given to her during the preparation of the present article.

2 Platonic A_{∗}^{1}-fiber spaces

Let ρ : X → B be an algebraic surface with an A_{∗}^{1}-fibration. We call X a Platonic A_{∗}^{1}-fiber space if the following conditions are satisfied:

(1) X is a smooth surface defined over C and the base curve B is isomorphic to the projective line P^{1}.

(2) Every fiber of ρ is irreducible and isomorphic to A_{∗}^{1} when taken with the reduced structure.

(3) There are three multiple fibers m_{i}Γ_{i} (i = 1, 2, 3) such that a triplet {m_{1}, m_{2}, m_{3}} is a Platonic triplet. Namely, {m_{1}, m_{2}, m_{3}} is one of the following triplets up to permutations:

{2, 2, n} (n ≥ 2), {2, 3, 3}, {2, 3, 4}, {2, 3, 5}

(4) There exists a smooth projective surface V with a P^{1}-fibration p : V → B such that

(i) V contains X as a Zariski open set and V − X is a divisor with simple normal crossings. The restriction of p onto X coincides with the given A_{∗}^{1}-fibration ρ. We call V a normal completion of the Platonic A_{∗}^{1}-fiber space X.

(ii) There exists two cross-sections S_{0} and S_{1} of p such that S_{0}, S_{1} ⊂ V − X and S_{0} ∩ S_{1} = ∅, and other irreducible components of V − X are contained in the fibers of p. Hence the A_{∗}^{1}-fibration ρ is untwisted.

It is known by [5] that a Platonic A_{∗}^{1}-fiber space X is isomorphic to A^{2}/G − (0), where G is a non-cyclic small finite subgroup of GL(2, C) and (0) is the image of the point of origin on A^{2}. The Platonic A_{∗}^{1}-fiber spaces represent an important class of almost minimal open algebraic surfaces with logarithmic Kodaira dimension \overline{κ}(X) = −∞ which are not affine ruled (cf. [7]). Here we call a finite subgroup G of GL(2, C) small if G contains no pseudo-reflections. Let us start with recalling the next result (cf. [5, Theorem 2]).

Lemma 2.1 (1) Let X be a Platonic A_{∗}^{1}-fiber space. Then A^{2} − (0) is the universal covering space of X, G := π_{1}(X) is a small finite subgroup of GL(2, C) which is not a cyclic group, and X is isomorphic to the quotient variety A^{2}/G with the unique singular point deleted off.

(2) Conversely, let G be a small finite subgroup of GL(2, C) which is not a cyclic group. Then X := A^{2}/G − (0) is a Platonic A_{∗}^{1}-fiber space with respect to the G_{m}-action induced by the G_{m}-action on A^{2} via the center of GL(2, C), where (0) represents the image of the point of origin of A^{2}. Furthermore, the quotient morphism π : A^{2} → A^{2}/G induces an étale finite Galois covering A^{2} − (0) → X with group G.

Lemma 2.2 Let X be a Platonic A_{∗}^{1}-fiber space and let φ : X → X be an étale endomorphism. Then there is an étale endomorphism φ̃ : A^{2} → A^{2} such that φ ∘ π = π ∘ φ̃, φ̃(0) = (0) and φ̃ commutes with the G := π_{1}(X)- action on A^{2}, where π : A^{2} − (0) → A^{2}/G − (0) is the quotient morphism. Conversely, given an étale endomorphism φ̃ of A^{2} such that φ̃(0) = (0) and φ̃ commutes with an action of a small finite subgroup G of GL(2, C), then the induced endomorphism \overline{φ} : A^{2}/G → A^{2}/G gives rise to an étale endomorphism φ : X → X such that φ ∘ π = π ∘ φ̃.

Proof. Since π : A^{2}−(0) → X is the universal covering, the endomorphism φ : X → X induces an étale endomorphism φ̂ : A^{2} − (0) → A^{2} − (0) which commutes with the action of π_{1}(X). Then φ̂ extends to an endomorphism φ̃ : A^{2} → A^{2} such that φ̃(0) = (0). Note that A^{2} − φ̃(A^{2}) is a finite set of points. For otherwise, there is an irreducible curve C defined by an equation f = 0 on A^{2} such that C ⊂ A^{2} − φ̃(A^{2}). Then φ̃^{∗}(f) is an invertible element on the upper A^{2}, which is not the case. Write A^{2} = Spec k[x, y]. Let φ̃ is given by two polynomials f := φ̃^{∗}(x) and g := φ̃^{∗}(y). Then the Jacobian determinant J = |∂(f, g)/∂(x, y)| is invertible on A^{2} − (0) since φ̃ is etale on A^{2} − (0). Hence it is invertible on A^{2}. This implies that φ̃ is an étale endomorphism of A^{2}. It is clear that φ̃ commutes with the linear G-action. Conversely, given an endomorphism φ̃ of A^{2} such that φ̃(0) = (0) and φ̃ commutes with the G-action, φ̃ induces an endomorphism \overline{φ} on the quotient variety A^{2}/G such that \overline{φ}(0) = (0). It is clear that φ := \overline{φ} |_{X} is an étale endomorphism if so is φ̃. Q.E.D.

Lemma 2.3 Let ρ : X → B be the A_{∗}^{1}-fibration of a Platonic A_{∗}^{1}-fiber space and let φ : X → X be an étale endomorphism. Suppose that φ preserves the A_{∗}^{1}-fibration ρ, i.e., there exists an endomorphism β : B → B such that β ∘ ρ = ρ ∘ φ. If β is an automorphism then φ is an automorphism.

Proof. Write X = A^{2}/G − (0) as in Lemma 2.1. Let φ̃ : A^{2} → A^{2} be an étale endomorphism which lifts φ. It suffices to show that φ̃ is an automorphism of A^{2}. Since β induces a permutation of the set of points {p_{1}, p_{2}, p_{3}} with p_{i} := ρ(Γ_{i}) we may assume by replacing φ by a suitable iteration of φ that β fixes these three points. Then β is the identity automorphism of B := P^{1}. Note that each fiber of ρ complemented by the singular point on A^{2}/G is the image of a line on A^{2} passing through the point of origin. Furthermore, for each fiber of ρ there are finitely many such lines. For i = 1, 2, 3, let {ℓ_{i}^{(j)}; 1 ≤ j ≤ r_{i}} be the set of all such lines that q(ℓ_{i}^{(j)}) = Γ_{i} ∪ (0). Then φ induces an endomorphism on the set {ℓ_{i}^{(j)}; 1 ≤ j ≤ r_{i}}. Hence we may assume after replacing φ by a suitable iteration of φ that there are three lines ℓ_{1}, ℓ_{2}, ℓ_{3} which are mapped by q to the fibers Γ_{1}, Γ_{2}, Γ_{3} respectively and fixed by φ. We can choose a system of coordinates {x, y} on A^{2} such that ℓ_{1}, ℓ_{2}, ℓ_{3} are defined by x = 0, y = 0, x = y respectively. Note that the set defined by the equation φ̃^{∗}(x) = 0 is a union of lines among {ℓ_{1}^{(j)}; 1 ≤ j ≤ r_{1}}. Hence φ̃^{∗}(x) is a homogeneous polynomial in x, y. Similarly, φ̃^{∗}(y) is a homogeneous polynomial in x, y. Since φ̃ is étale, the Jacobian determinant J := |∂(φ̃^{∗}(x), φ̃^{∗}(y))/∂(x, y)| is a nonzero constant. Hence φ̃^{∗}(x) and φ̃^{∗}(y) are linear polynomials in x, y. Then φ is an automorphism. Q.E.D.

3 Étale endomorphisms preserving the A_{∗}^{1}- fibration

Let ρ : X → B be an A_{∗}^{1}-fibration as given in the introduction. In this section we assume that an étale endomorphism φ of X preserves the A_{∗}^{1}-fibration ρ : X → B. Let β : B → B be the induced endomorphism of the base curve B. To avoid confusion of the notation, we write φ : X_{1} → X_{2} to denote the endomorphism φ, where X_{1}, X_{2} are isomorphic to X. We also denote the A_{∗}^{1}-fibrations for X_{1}, X_{2} by ρ_{1} : X_{1} → B_{1} and ρ_{2} : X_{2} → B_{2} respectively. Hence we have ρ_{2} ∘ φ = β ∘ ρ_{1}. We denote by g the genus of the curve B. The next result is useful.

Lemma 3.1 With the above notation and assumptions, let mΓ be an irreducible fiber of ρ_{2} lying over a point p ∈ B_{2} with m ≥ 1 and Γ reduced, and let q ∈ B_{1} be a point such that β(q) = p. Suppose ρ_{1}^{∗}(q) = ℓΔ, where Δ is reduced and irreducible and ℓ is its multiplicity. If the ramification index for β at q is e then ℓe = m. In particular, if m = 1 then ℓ = e = 1.

Proof. Let Q ∈ Δ and P ∈ Γ be smooth points such that φ(Q) = P. Let η = 0 and ξ = 0 be the local equations of Δ and Γ at Q and P in X_{1} and X_{2} respectively. Choose local parameters u and t for q ∈ B_{1} and p ∈ B_{2} respectively. Then β^{∗}(t) ∼ u^{e}, where ∼ means that t and u^{e} differ by an invertible element at q. We have ρ_{1}^{∗}(u) ∼ η^{ℓ} near Q and ρ_{2}^{∗}(t) ∼ ξ^{m} near P. Since φ is étale at Q, η ∼ φ^{∗}(ξ). Hence we obtain

η^{eℓ} ∼ ρ_{1}^{∗}(u)^{e} = ρ_{1}^{∗}(u^{e}) ∼ ρ_{1}^{∗}β^{∗}(t) = φ^{∗}ρ_{2}^{∗}(t) ∼ φ^{∗}(ξ^{m}) ∼ η^{m} .

So, the equality ℓe = m holds. Q.E.D.

Let m_{1}Γ_{1}, …, m_{r}Γ_{r} exhaust all multiple fibers of ρ, where we assume without loss of generality that 2 ≤ m_{1} ≤ ⋯ ≤ m_{r}. We denote by {r; m_{1}, …, m_{r}} the number of multiple fibers and the multiplicities. Let p_{i} = ρ(Γ_{i}) for 1 ≤ i ≤ r.

Theorem 3.2 With the above notation and assumptions, suppose r ≥ 3. Then the induced endomorphism β : B_{1} → B_{2} is an automorphism except when B is isomorphic to P^{1} and {r; m_{1}, …, m_{r}} is one of the following collections:

{4; 2, 2, 2, 2}, {3; 2, 3, 6}, {3; 2, 4, 4}, {3; 3, 3, 3} .

The proof consists of the subsequent lemmas 3.3 ∼ 3.6. Note that {3; m_{1}, m_{2}, m_{3}} is one of the above collections if and only if

1 = \frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} .

Lemma 3.3 Suppose that either r ≥ 5 or r > 0 and g > 0. Then β : B_{1} → B_{2} is an automorphism. If r = 4 and g = 0 then β is an automorphism except for the case {4; 2, 2, 2, 2}.

Proof. Let d = deg β. By Lemma 3.1 the endomorphism β induces an endomorphism of the set {p_{1}, …, p_{r}}. Let q_{1}, …, q_{s} exhaust all points of B_{1} which are contained in B_{1} − {p_{1}, …, p_{r}} and mapped to the points in {p_{1}, …, p_{r}} by β : B_{1} → B_{2}. In fact, we have s_{i} points of them mapped to the point p_{i} and s = s_{1} + ⋯ + s_{r}. First of all, since β induces an étale finite morphism

β : B_{1} − {p_{1}, …, p_{r}, q_{1}, …, q_{s}} ⟶ B_{2} − {p_{1}, …, p_{r}} ,

the comparison of Euler numbers gives rise to an equality

2 − 2g − (r + s) = d(2 − 2g − r) .  (1)

Hence we have

s = (d − 1){2g − 2 + r} .

Let e_{j} be the ramification index of β at the point q_{j}. Since e_{j} ≥ 2 by Lemma 3.1 we have the following inequality

2s + r ≤ dr .  (2)

By (1) and (2) we obtain an inequality

2(d − 1){2g − 2 + r} ≤ (d − 1)r .  (3)

Suppose d > 1. Then 4g − 4 + r ≤ 0 by (3). This is impossible if either r ≥ 5 or r > 0 and g > 0.

Suppose r = 4, g = 0 and d > 1. Then 4g − 4 + r = 0. Hence the equalities occur in (2) and (3). Hence the ramification index at the point p_{i} ∈ B_{1} is one for 1 ≤ i ≤ 4 and e_{j} = 2 for 1 ≤ j ≤ s. Since s = 2(d − 1) > 0 it follows from Lemma 3.1 that m_{1} = 2. Since d ≥ 2 by the hypothesis we have 2s > d, which implies that one of the q_{i} is mapped to a point in {p_{2}, p_{3}, p_{4}}. Hence we may assume that m_{2} = 2. Suppose m_{3} > 2. Then s_{3} = s_{4} = 0, and 2d ≥ 2s which implies that d = 2 and s = 2. This easily leads to a contradiction. Hence m_{3} = 2. Suppose m_{4} > 2. Then 3d ≥ 2s, whence d ≤ 4. If d = 2 then s = 2. It is easily seen that this is impossible. If d = 3 then s = 4. In this case, no two of q_{1}, q_{2}, q_{3}, q_{4} are mapped to one of {p_{1}, p_{2}, p_{3}, p_{4}}. But this leads to a contradiction becasuse s_{4} = 0. Hence d = 4 and s = 6. Then the six points q_{1}, …, q_{6} form three couples which are mapped to p_{1}, p_{2}, p_{3}. Then p_{1}, p_{2}, p_{3}, p_{4} are mapped to p_{4} and d ≥ 2 + 2 + 2 + 1 = 7. This is a contradiction. Thus we have shown that if d > 1 then {r; m_{1}, …, m_{r}} = {4; 2, 2, 2, 2}. Q.E.D.

In the subsequent arguments we assume that g = 0 and r = 3.

Lemma 3.4 Suppose that β induces a bijection on the set {p_{1}, p_{2}, p_{3}}. Then β is an automorphism except when {m_{1}, m_{2}, m_{3}} is one of {2, 3, 6}, {2, 4, 4}, {3, 3, 3}.

Proof. We consider only the case β(p_{i}) = p_{i} for 1 ≤ i ≤ 3. The other cases are treated in the same fashion. We have

d = m_{1}s_{1} + 1 = m_{2}s_{2} + 1 = m_{3}s_{3} + 1 .

By the Riemann-Hurwitz formula, we have

−2 = −2d + (m_{1} − 1)s_{1} + (m_{2} − 1)s_{2} + (m_{3} − 1)s_{3} .

By plugging into this equality the data

s_{1} = \frac{d − 1}{m_{1}}, s_{2} = \frac{d − 1}{m_{2}}, s_{3} = \frac{d − 1}{m_{3}},

we obtain

2(d − 1) = (d − 1) \{3 − \frac{1}{m_{1}} − \frac{1}{m_{2}} − \frac{1}{m_{3}} \} .

Hence if d > 1 we have

1 = \frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} .

Q.E.D.

Lemma 3.5 Suppose that the image by β of {p_{1}, p_{2}, p_{3}} consists of two points. Then β is an automorphism except when {m_{1}, m_{2}, m_{3}} is one of {2, 3, 6}, {2, 4, 4}, {3, 3, 3}.

Proof. We consider only the case β(p_{1}) = β(p_{2}) = p_{2} and β(p_{3}) = p_{3}. The other cases are treated in the same fashion. The assumption implies that

d = m_{1}s_{1} = \frac{m_{2}}{m_{1}} + m_{2}s_{2} + 1 = m_{3}s_{3} + 1.

The Riemann-Hurwitz formula is then written as

−2 = −2d + (m_{1} − 1)s_{1} + (\frac{m_{2}}{m_{1}} − 1) + (m_{2} − 1)s_{2} + (m_{3} − 1)s_{3}.

Since

s_{1} = \frac{d}{m_{1}} , s_{2} = \frac{d − 1}{m_{2}} − \frac{1}{m_{1}} , s_{3} = \frac{d − 1}{m_{3}},

we obtain

2(d − 1) = (d − 1) \{3 − \frac{1}{m_{1}} − \frac{1}{m_{2}} − \frac{1}{m_{3}} \} .

Hence if d > 1 we have

1 = \frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} .

Q.E.D.

Lemma 3.6 Suppose that the image by β of {p_{1}, p_{2}, p_{3}} consists of one point. Then β is an automorphism except when {m_{1}, m_{2}, m_{3}} is one of {2, 3, 6}, {2, 4, 4}, {3, 3, 3}.

Proof. Upto the renumbering of the points p_{1}, p_{2}, p_{3} we may assume that β(p_{1}) = β(p_{2}) = β(p_{3}) = p_{3}. Then we have

d = m_{1}s_{1} = m_{2}s_{2} = \frac{m_{3}}{m_{1}} + \frac{m_{3}}{m_{2}} + m_{3}s_{3} + 1.

The Riemann-Hurwitz formula is written as

−2 =

−2d + (m_{1} − 1)s_{1} + (m_{2} − 1)s_{2} + (\frac{m_{3}}{m_{1}} − 1) + (\frac{m_{3}}{m_{2}} − 1) + (m_{3} − 1)s_{3}.

Then we obtain also an equality

2(d − 1) = (d − 1) \{3 − \frac{1}{m_{1}} − \frac{1}{m_{2}} − \frac{1}{m_{3}} \} .

Hence if d > 1 we have

1 = \frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} .

Q.E.D.

Theorem 3.2 implies the next result.

Corollary 3.7 Let ρ : X → B be a Platonic A_{∗}^{1}-fiber space. Suppose that an étale endomorphism φ of X preserves the A_{∗}^{1}-fibration ρ. Let β : B → B be the induced endomorphism. Then β is an automorphism. Hence φ is an automorphism by Lemma 2.3.

4 Weighted hypersurfaces

In this section we consider a weighted hypersurface Σ in A^{3} defined by

x_{1}^{m_{1}} + x_{2}^{m_{2}} + x_{3}^{m_{3}} = 0,

where m_{1}, m_{2}, m_{3} are integers larger than 1 and gcd(m_{1}, m_{2}, m_{3}) = 1. Let d be the least common multiple of m_{1}, m_{2}, m_{3} and let q_{i} = d/m_{i} for i = 1, 2, 3. Then the one-dimensional algebraic torus G_{m} acts on X by

t · (x_{1}, x_{2}, x_{3}) = (t^{q_{1}}x_{1}, t^{q_{2}}x_{2}, t^{q_{3}}x_{3})

for t ∈ G_{m}. Let X = Σ − (0, 0, 0), let B = X/G_{m} and let π : X → B be the quotient morphism. The following result is known (cf. [10]), where we abbreviate gcd(m, n) as (m, n) :

Lemma 4.1 The curve B is a nonsingular projective curve of genus given by

g = \frac{d^{2}}{2q_{1}q_{2}q_{3}} − \frac{d}{2} \{\frac{(q_{1}, q_{2})}{q_{1}q_{2}} + \frac{(q_{2}, q_{3})}{q_{2}q_{3}} + \frac{(q_{3}, q_{1})}{q_{3}q_{1}}\} + 1 .

The morphism π : X → B is an A_{∗}^{1}-fibration whose fibers are all isomorphic to A_{∗}^{1} if taken with the reduced structure. If (q_{1}, q_{2}) > 1 (resp. (q_{2}, q_{3}) > 1, (q_{3}, q_{1}) > 1) there are d(q_{1}, q_{2})/q_{1}q_{2} (resp. d(q_{2}, q_{3})/q_{2}q_{3}, d(q_{3}, q_{1})/q_{3}q_{1}) multiple fibers of multiplicity (q_{1}, q_{2}) (resp. (q_{2}, q_{3}), (q_{3}, q_{1})). Except for these fibers there are no multiple fibers.

We consider a special case where m_{1}, m_{2}, m_{3} are pairwise coprime. Then d = m_{1}m_{2}m_{3}, q_{1} = m_{2}m_{3}, q_{2} = m_{1}m_{3}, q_{3} = m_{1}m_{2}. Hence Lemma 4.1 implies the next result:

Corollary 4.2 Suppose m_{1}, m_{2}, m_{3} are pairwise coprime. Then the following assertions hold:

(1) The curve B is a rational curve.

(2) The A_{∗}^{1}-fibration π : X → B has three multiple fibers of respective multiplicities m_{1}, m_{2}, m_{3}.

We consider the case where m_{1}, m_{2}, m_{3} are pairwise coprime. The next result is due to Bundagaard-Nielsen [1] and Fox [2] (cf. [9, Theorem 1.2.15]).

Lemma 4.3 Let B be a smooth projective curve of genus g. Let r ≥ 1 and let p_{1}, p_{2}, …, p_{r} be distinct points in B and m_{1}, m_{2}, …, m_{r} arbitrary integers ≥ 2. Then there exists a finite Galois covering τ : B̃ → B such that the ramification index at any point over p_{i} is m_{i} for i = 1, 2, …, r except when (i) g = 0 and r = 1 and (ii) g = 0, r = 2 and m_{1} ≠ m_{2}. If B ≅ P^{1}, r = 2 and gcd(m_{1}, m_{2}) = d > 1 then we can construct a finite cyclic covering τ : B̃ → P^{1} with the ramification index d at p_{1}, p_{2}.

Corollary 4.4 Suppose g = 0 and r = 3. Then B̃ in Lemma 4.3 has genus 0 (resp. 1) if and only if {m_{1}, m_{2}, m_{3}} = {2, 2, n} (n ≥ 2), {2, 3, 3}, {2, 3, 4}, {2, 3, 5} (resp. {m_{1}, m_{2}, m_{3}} = {2, 3, 6}, {2, 4, 4}, {3, 3, 3}), where we assume m_{1} ≤ m_{2} ≤ m_{3}. The genus of B̃ is greater than 1 otherwise.

Proof. Let d be the degree of τ : B̃ → P^{1} and let g̃ be the genus of B̃. Let s_{i} be the number of points on B̃ lying over the point p_{i} for i = 1, 2, 3. Then d = m_{1}s_{1} = m_{2}s_{2} = m_{3}s_{3}. By the Riemann-Hurwitz formula, we have

2g̃ − 2 = −2d + (m_{1} − 1)s_{1} + (m_{2} − 1)s_{2} + (m_{3} − 1)s_{3} .

Hence we have

g̃ = \frac{d}{2} \{1 − (\frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} )\} + 1 .

This implies g̃ = 0 (resp. 1, > 1) if and only if

1 − (\frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} ) < 0 (resp. = 0, > 0) .

This yields the stated result. Q.E.D.

We consider an étale endomorphism φ : X → X when m_{1}, m_{2}, m_{3} are pairwise coprime integers such that

\frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} < 1  (1)

Lemma 4.5 Under the assumption (1) above, X has log Kodaira dimension one. If φ : X → X is an étale endomorphism then φ preserves the A_{∗}^{1}- fibration. Namely, there exists an endomorphism β : P^{1} → P^{1} such that π ∘ φ = β ∘ π.

Proof. Let X̃ be the normalization of a fiber product X ×_{B} B̃ and let q : X̃ → X be the natural finite morphism. Then the A_{∗}^{1}-fibration π : X → B induces an A_{∗}^{1}-fibration π̃ : X̃ → B̃. Since B̃ has genus > 1 by the assumption (1), the addition formula of log Kodaira dimension yields

\overline{κ}(X̃) ≥ \overline{κ}(B̃) + \overline{κ}(A_{∗}^{1}) = 1 .

On the other hand, since q : X̃ → X is an étale finite morphism (cf. Lemma 4.7 below), we have \overline{κ}(X̃) = \overline{κ}(X) and \overline{κ}(X) ≤ 1 by the weak addition formula. Hence \overline{κ}(X̃) = \overline{κ}(X) = 1.

The second assertion follows from [3, Lemma 2.4]. Q.E.D.

Corollary 4.6 With the notations and assumptions in Lemma 4.5, the endomorphism β is the identity.

Proof. We make use of Lemma 3.1. Let m_{i}Γ_{i} (i = 1, 2, 3) be the multiple fibers of π : X → B, where Γ_{i} ≅ A_{∗}^{1}. Since m_{1}, m_{2}, m_{3} are pairwise coprime, φ : X → X maps the fiber m_{i}Γ_{i} to itself. Furthermore, assume that there are s_{i} smooth fibers of π mapped to the fiber m_{i}Γ_{i}. Let those fibers lie over the points q_{i}^{(j)} of B, where 1 ≤ j ≤ s_{i}. Then β(q_{i}^{(j)}) = p_{i}. The ramification index of β at q_{i}^{(j)} is m_{i}. By the Riemann-Hurwitz formula, we have

−2 = −2d + (m_{1} − 1)s_{1} + (m_{2} − 1)s_{2} + (m_{3} − 1)s_{3} ,

where d := deg β = 1 + m_{1}s_{1} = 1 + m_{2}s_{2} = 1 + m_{3}s_{3}. Since m_{1}, m_{2}, m_{3} are pairwise coprime integers, we can write s_{1} = q_{1}s, s_{2} = q_{2}s, s_{3} = q_{3}s for some integer s. From these relations one can deduce

m_{1}m_{2}m_{3}s = (m_{1}m_{2} + m_{2}m_{3} + m_{3}m_{1})s ,

where m_{1}m_{2}m_{3} ≠ m_{1}m_{2} + m_{2}m_{3} + m_{3}m_{1}. So, s = 0 and d = 1. Namely β is an automorphism. Meanwhile, since β fixes three points p_{1}, p_{2}, p_{3} it is the identity. Q.E.D.

Before going to make further arguments we need the following result:

Lemma 4.7 Let ρ : X → B be an A_{∗}^{1}-fibration as given in the introduction. Let m_{1}Γ_{1}, …, m_{r}Γ_{r} exhaust all multiple fibers of ρ and let p_{i} = ρ(Γ_{i}). Suppose r ≥ 3. Let τ : B̃ → B be a finite Galois covering which ramifies over the points p_{1}, …, p_{r} with respective multiplicities m_{1}, …, m_{r} (cf. Lemma 4.3). Let X̃ be the normalization of the fiber product X ×_{B} B̃, let ρ̃ : X̃ → B̃ be the A_{∗}^{1}-fibration induced by ρ and let μ : X̃ → X be a composite of the normalization morphism ν : X̃ → X ×_{B} B̃ and the projection X ×_{B} B̃ → X. Then the following assertions hold:

(1) The A_{∗}^{1}-fibration ρ̃: X̃ → B̃ is an A_{∗}^{1}-fiber bundle over B̃. In particular, there are no multiple fibers in ρ̃.

(2) The morphism μ : X̃ → X is a finite étale morphism.

Proof. Let p be a point of B. Suppose first that p is none of p_{1}, …, p_{r}. Then τ is unramified over p. So, if q is a point of B̃ lying over p then B̃ and B are (analytically) locally isomorphic to each other near q and p. Hence the fibers ρ̃^{−1}(q) and ρ^{−1}(p) are isomorphic to each other. Suppose p coincides with one of p_{1}, …, p_{r}, say p = p_{1}. Write m = m_{1}. Let t be a local parameter of B at p. Let P be a point on Γ = Γ_{1}. Then there exists a local system of parameters {ξ, η} at P such that ξ = 0 defines locally the curve Γ, t = aξ^{m} and η is a fiber coordinate of Γ, i.e., a local parameter of the curve Γ. Here a is a regular function such that a(P) ≠ 0. Then we can write analytically a = b^{m} with b holomorphic and nonzero near P. Replacing ξ by bξ, we can write t = ξ^{m}. Let q be a point of B̃ lying above p and let u be a local parameter at q. Since the ramification index of τ is m at q, we may assume that t = u^{m}. Then the fiber product X ×_{B} B̃ is identified near the point (q, P) with a hypersurface u^{m} = ξ^{m} in the (u, ξ, η)-space. This hypersurface is singular because it is isomorphic to a union of m-planes meeting along the line u = ξ = 0. Taking the normalization of the fiber product X ×_{B} B̃ is equivalent to separating these m-planes to disjoint m-planes. Recall here that we are looking an analytic local chart. Anyway, the fiber of ρ̃ over q consists of m-copies of a curve locally given by u = 0 in the (u, η)-plane. Hence it is smooth and has multiplicity one. Furthermore, the morphism μ : X̃ → X is étale, while μ is finite because the normalization morphism ν is finite and τ : B̃ → B is finite. In order to show that ρ̃ : X̃ → B̃ is an A_{∗}^{1}-fiber bundle it suffices to show that the fiber ρ̃^{−1}(q) is irreducible.

For this purpose we embed the given A_{∗}^{1}-fibration ρ : X → B into a P^{1}-fibration π : V → B in such a way that the restriction of π onto the open set X is ρ and that if G_{i} is the fiber of π containing the multiple fiber m_{i}Γ_{i} the dual graph of G_{i} is a linear chain and the closure \overline{Γ} of Γ in V is a unique (−1) curve in G_{i}, where V is a smooth projective surface. Then there exist two cross-sections S_{0}, S_{1} of π lying in V − X. Let Ṽ be the normalization of the fiber product V ×_{B} B̃. Then Ṽ is a normal projective surface with the induced P^{1}-fibration π̃ : Ṽ → B̃ such that X̃ is an open set of Ṽ and π̃|_{X} = ρ̃. Let h : V̂ → Ṽ be a minimal resolution of singularities of Ṽ. Then there is an induced P^{1}-fibration π̂ : V̂ → B and every connected component of the exceptional locus of h is contained in a fiber of π̂. It is known that every singular point obtained by contracting a connected union of irreducible components of a reducible fiber of a P^{1}-fibration is a rational singularity. The surface X̃ is an open set of V̂ and the proper transforms Ŝ_{0}, Ŝ_{1} of S_{0}, S_{1} by h are the cross-sections of π̂.

Let Γ̃ = ρ̃^{−1}(q) with the above notation. Since μ : Γ̃ → Γ is a finite étale morphism, Γ̃ is a disjoint union of Δ_{1}, …, Δ_{n}, where Δ_{i} ≅ A_{∗}^{1} for each i. We have only to show that n = 1. Let \overline{Δ}_{i} be the closure of Δ_{i} in V̂. Then \overline{Δ}_{1}, …, \overline{Δ}_{n} are contained in the fiber \overline{G} = π̂^{−1}(q). These components together with the components in \overline{G} ∩ (V̂ − X̃) are linked to the cross-sections Ŝ_{0}, Ŝ_{1}. Then it is easy to see that the dual graph of \overline{G} should contain loops provided n > 1. But this is impossible. Hence n = 1. Q.E.D.

Now we shall come back to the situation before Lemma 4.7 and look for a smooth completion of X̃ which is the normalization of X ×_{B} B̃. Note that every fiber of the A_{∗}^{1}-fibration π̃ : X̃ → B̃ is smooth and irreducible. Hence there is a P^{1}-bundle ρ : V → B̃ such that X̃ = V − (S_{0} ∪ S_{1}), where S_{0}, S_{1} are the cross-sections of ρ and that π̃ is the restriction of ρ onto X̃. Since S_{0}, S_{1} are disjoint from each other, the P^{1}-bundle V is written as P(S) := Proj (S), where S is the symmetric O_{B}-algebra generated by a rank 2 locally free module O_{B} ⊕ L with an invertible sheaf L. Let φ : X → X be an étale endomorphism. Then φ is a B-morphism by Corollary 4.6. Hence φ lifts to an étale endomorphism φ̃ : X̃ → X̃ which is a B̃-morphism. Let K̃ be the function field of B̃ and let X̃_{K} be the generic fiber of π̃. If K is the function field of B and X_{K} is the generic fiber of π then X̃_{K} = X_{K} ⊗_{K} K̃. Hence we may write X̃_{K} = X_{K}. By the above construction, X_{K} ≅ A_{∗,_{K}}^{1} :=

Spec K̃[t, t^{−1}]. Furthermore, X_{K} = P1 ; _{K} − {S_{0,_{K}}, S_{1,_{K}}}, where S_{0,_{K}}, S_{1,_{K}} are

the generic points of S_{0}, S_{1}. The B̃-morphism φ̃ induces a K̃-endomorphism φ̃_{K} : X_{K} → X_{K}. Replacing φ̃ by φ̃^{2} if necessary, we may assume that φ̃_{K} extends to an endomorphism from P1 ; _{K} into itself which ramifies totally over the points S_{0,_{K}}, S_{1,_{K}}. We denote this endomorphism by the same letter φ̃_{K}. This implies that there is a rational mapping φ̃ : V → V which is regular except for finitely many points lying on the cross-sections S_{0}, S_{1}. Then φ̃ is an endomorphism, that is to say, φ̃ is regular everywhere. Furthermore, φ̃ : V → V is a finite morphism and totally ramifying over S_{0}, S_{1}. It then follows that φ̃^{∗}(S_{0}) = NS_{0}, where N is the degree of φ̃. We may assume that the cross-section S_{0} contracts to the point (0, 0, 0) on \overline{Σ} by a morphism V → \overline{Σ} which is induced by the composite X̃ → X ×_{B} B̃ → X ↪ Σ, where \overline{Σ} is a suitable projective completion of Σ. Hence (S_{0}^{2}) < 0. We have N(S_{0}^{2}) = (S_{0}^{2}) by the projection formula. So, N = 1. Namely, φ̃ is an automorphism. Hence φ : X → X is also an automorphism. If m_{1}, m_{2}, m_{3} are pairwise coprime integers larger than one and if {m_{1}, m_{2}, m_{3}} ≠ {2, 3, 5} then

\frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} < 1.

Hence, in view of Lemma 4.5 and Corollary 4.6, we have shown the following result.

Theorem 4.8 Let Σ be an affine hypersurface in A^{3} defined by x_{1}^{m_{1}} + x_{2}^{m_{2}} + x_{3}^{m_{3}} = 0, where m_{1}, m_{2}, m_{3} are pairwise coprime positive integers larger than one. Let X = Σ − (0, 0, 0). Then any étale endomorphism φ : X → X is an automorphism possibly except for the case {m_{1}, m_{2}, m_{3}} = {2, 3, 5}.

5 Algebraic surfaces with G_{m}-action

Let Σ be anew an affine normal algebraic surface defined over C. We suppose that Σ has an unmixed algebraic action of the one-dimensional torus G_{m} satisfying the following conditions:

(1) Σ has a unique fixpoint, say O. The point O is in the closure of every orbit.

(2) Let X := Σ − {O}. Then the algebraic quotient B := X/G_{m} is a smooth projective curve. We denote its genus by g.

Let π : X → B be the quotient morphism. Then X is a smooth surface and π gives rise to an A_{∗}^{1}-fibration on X. By abuse of notations, we use π in this section to denote the A_{∗}^{1}-fibration induced by the quotient morphism X → X/G_{m}.

Lemma 5.1 If g > 0 then an étale endmorphsim φ : X → X preserves the A_{∗}^{1}-fibration π. Hence there exists an endomorphism β : B → B such that π ∘ φ = β ∘ π. If either g > 1 or g = 1 and π has a multiple fiber then β is an automorphism.

Proof. Let F be a fiber of π. Since F_{red} ≅ A_{∗}^{1}, (π ∘ φ)(F_{red} ) is a point of B. Hence φ(F_{red}) is contained in a fiber of π. Namely φ preserves the A_{∗}^{1}-fibration. If g > 1 the Riemann-Hurwitz formula shows that β is an automorphism. Suppose g = 1. Let m_{1}Γ_{1}, …, m_{r}Γ_{r} exhaust all multiple fibers of π, where Γ_{i} ≅ A_{∗}^{1} and m_{1} ≤ ⋯ ≤ m_{r}. Let p_{i} = π(Γ_{i}). By the hypothesis, r ≥ 1. Suppose d := deg β > 1. If β(p) = p_{1}, it follows from Lemma 3.1 that either p = p_{i} with m_{i} = m_{1} or p is a ramification point of index m_{1}. Since β is surjective, one can show easily that there exists a ramification point for β. Then the Riemann-Hurwitz formula gives a contradiction. Q.E.D.

Note that any automorphism β of a smooth projective curve of genus g is of finite order if either g > 1 or g = 1 and β fixes one point.

Lemma 5.2 Let π : X → B be as in Lemma 5.1. Let φ : X → X be an étale endomorphism. If either g > 1 or g = 1 and π has a multiple fiber then φ is an automorphism.

Proof. Let β be the automorphism of B induced by φ. Then β permutes the points over which lie the multiple fibers of π. Hence, replacing β by its iteration, we may assume that β fixes those points on B. Then β has finite order, say n. Now, replacing φ by φ^{n}, we may and shall assume that β is the identity. Now the argument proceeds as in the last part of section 4. Let m_{1}Γ_{1}, …, m_{r}Γ_{r} exhaust all multiple fibers of π and let p_{i} = π(Γ_{i}). Let B̃ be a finite Galois covering of B ramified over the points p_{i} with ramification indices m_{i}. Such a covering exists by Lemma 4.3. Let X̃ be the normalization of X ×_{B} B̃. Then X̃ together with the induced A_{∗}^{1}-fibration π̃ : X̃ → B̃ is embedded into a P^{1}-bundle ρ : V → B̃ in such a way that X̃ = V − (S_{0} ∪ S_{1}) with disjoint cross-sections S_{0}, S_{1} and π̃ = ρ|_{X}. Here we may assume that (S_{0}^{2}) < 0. On the other hand, the given endomorphism φ : X → X extends to an étale endomorphism φ̃ : X̃ → X̃, which also extends to an endomorphism φ̃ : V → V. Then φ̃ is totally ramifying over S_{0}. Hence we have (S_{0}^{2}) = N(S_{0}^{2}) with N := deg φ̃. Hence N = 1 and this shows that φ is an automorphism. Q.E.D.

Lemma 5.3 Let π : X → B be as in Lemma 5.1. Assume that B is an elliptic curve and that π has no multiple fibers. Then there exists a nontrivial étale endomorphism φ : X → X if and only if there exist an endomorphism β of B, an invertible sheaf L on B of positive degree and a positive integer d > 1 satisfying the condition: L^{⊗d} ≅ β^{∗}(L). We have necessarily d = deg β and deg φ = d^{2}.

Proof. There exists an invertible sheaf L such that X = P(O_{B} ⊕L)−(S_{0} ∪ S_{1}), where S_{0}, S_{1} are the cross-sections of the P^{1}-bundle. The invertible sheaf L is determined uniquely up to isomorphisms by the A_{∗}^{1}-bundle X. We may assume that (S_{0}^{2}) < 0. Then S_{0} (resp. S_{1}) corresponds to the projection from O_{B} ⊕ L to O_{B} (resp. L). Suppose there exists an étale endomorphism φ : X → X. Then π ∘ φ = β ∘ π for an endomorphism β of B. Let now Y be a fiber product X ×_{(B,β)} B and π_{Y} : Y → B be the induced A_{∗}^{1}-fibration. Then Y = P(O_{B} ⊕ β^{∗}(L)) − (S_{0,Y} ∪ S_{1,Y}), where S_{0,Y} = S_{0} ×_{(B,β)} B and S_{1,Y} = S_{1} ×_{(B,β)} B. Let β_{Y} : Y → X be the base change of β. Then φ splits as a composite φ = β_{Y} ∘ ψ, where ψ : X → Y is an étale Bmorphism. Then ψ extends to a B-morphism of P^{1}-bundles \overline{ψ} : P(O_{B}⊕L) → P(O_{B} ⊕β^{∗}(L)) such that \overline{ψ}^{−1}(S_{0,Y}) = S_{0} and \overline{ψ}^{−1}(S_{1,Y}) = S_{1}. Take an affine open covering U = {U_{i}}_{i∈I} of B such that L |_{U_{i}}= O_{U_{i}}e_{i} for every i ∈ I. Write O_{B} = O_{B}e. Then π^{−1}(U_{i}) = Spec A_{i}[z_{i}, z_{i}^{−1}], where z_{i} = e_{i}/e and A_{i} = Γ(U_{i}, O_{B}). Let {a_{ij}} be the transition functions of L with respect to U. Then ψ |_{U_{i}}: π^{−1}(U_{i}) → π_{Y}^{−1}(U_{i}) is given by β^{∗}(z_{i}) = λ_{i}z_{i}^{d} with λ_{i} ∈ A_{i}^{∗} and d := deg ψ. Then it is easy to deduce a relation β^{∗}(a_{ji}) = a_{ji}^{d}λ_{j}λ_{i}^{−1} for i, j ∈ I. This implies that L^{⊗d} ≅ β^{∗}(L). Since deg L^{⊗d} = d deg L and deg β^{∗}(L) = deg β deg L, we have necessarily d = deg β.

Conversely, if we are given β, L, d as above, we can construct an endomorphism φ : X → X as a composite of β_{Y} and ψ, where ψ is a d-th power morphism on the generic fibers. This d-th power morphism is extended to a B-morphism ψ : X → Y because of the condition L^{⊗d} ≅ β^{∗}(L). Since φ = β_{Y} ∘ ψ, deg β_{Y} = deg β and deg ψ = d, we have deg φ = d^{2}. Q.E.D.

Lemma 5.4 For any positive integer n > 1, let n_{B} denote the multiplication by n endomorphism of an elliptic curve B when B is viewed as an abelian variety. Then there exists an invertible sheaf L on B such that deg L > 0 and L^{⊗n^{2}} ≅ n_{B}^{∗}(L). For this invertible sheaf L, let V := P(O_{B} ⊕ L) and let S_{0} (resp. S_{1}) be a cross-section such that O_{S_{0}}(S_{0}) ≅ L^{−1} (resp. O_{S_{1}}(S_{1}) ≅ L). Then the cross-section S_{0} contracts to an elliptic singularity O on a projective normal surface \overline{V}. Let Σ := \overline{V} − S_{1}. Then there exists a G_{m}-action on Σ such that the point O is a unique fixpoint, that every orbit is contained in a fiber of the P^{1}-bundle projection ρ : V → B and that the closure of every orbit contains the point O.

Proof. Take any invertible sheaf L on B with deg L > 0. Then, by [8, (iv),page 75], n_{B}^{∗}(L) ≅ L^{⊗n^{2}} ⊗ M with M ∈ Pic^{0}(B). By [8, (iii), page 75], we have n_{B}^{∗}(N) ≅ N^{⊗n} for any N ∈ Pic^{0}(B). Set L′ := L ⊗ N. Then we compute

n_{B}^{∗}(L′) = n_{B}^{∗}(L ⊗ N) ≅ L^{⊗n^{2}} ⊗ M ⊗ N^{⊗n} .

Hence, if we take N so that M ⊗ N^{⊗n} ≅ N^{⊗n^{2}}, we have n_{B}^{∗}(L′) ≅ L^{′⊗n^{2}}. Meanwhile, since Pic^{0}(B) is a divisible group there exists N ∈ Pic^{0}(B) such that N^{⊗(n^{2}−n)} ≅ M. Then we replace L by L′. The second assertion is well-known. Q.E.D.

Now we shall consider the case where B is a smooth rational curve. Let m_{1}Γ_{1}, …, m_{r}Γ_{r} exhaust the multiple fibers of the A_{∗}^{1}-fibration π : X → B. We denote by {r; m_{1}, m_{2}, …, m_{r}} the number of multiple fibers and the multiplicities with the condition m_{1} ≤ m_{2} ≤ … ≤ m_{r}. Then we have the following:

Lemma 5.5 With the above notations and assumptions, we have \overline{κ}(X) = 1 if r ≥ 3 and except for the following cases :

(i) {4; 2, 2, 2, 2}, {3; 2, 3, 6}, {3; 2, 4, 4}, {3; 3, 3, 3}

(ii) {3; 2, 2, n} (n ≥ 2), {3; 2, 3, 3}, {3; 2, 3, 4}, {3; 2, 3, 5}.

Hence, in the case \overline{κ}(X) = 1, any étale endomorphism φ of X preserves the A_{∗}^{1}-fibration π.

Proof. By Lemma 4.3, there exists a cyclic Galois covering B̃ → B ramifying over the points p_{i} := π(Γ_{i}) with index m_{i}. Let d be the degree of the covering. Then the genus g of B̃ is given by

g = d \{(r − 2) − (\frac{1}{m_{1}} + … + \frac{1}{m_{r}} )\} + 1 .

Hence, if either r ≥ 5 or except for the above-listed cases, g > 1. Let Y be the normalization of the fiber product X ×_{B} B̃. Then, by argueing in the same fashion as in the proof of Lemma 4.5, we know that \overline{κ}(Y) = \overline{κ}(X) = 1. The preservation of the A_{∗}^{1}-fibration follows from [3, Lemma 2.4]. Q.E.D.

Lemma 5.6 In the cases, which we will call the quasi-elliptic case,

{r; m_{1}, …, m_{r}} = {4; 2, 2, 2, 2}, {3; 2, 3, 6}, {3; 2, 4, 4}, {3; 3, 3, 3},

we have \overline{κ}(X) = 0. In the cases, which we will call the Platonic case,

{3; 2, 2, n} (n ≥ 2), {3; 2, 3, 3}, {3; 2, 3, 4}, {3; 2, 3, 5},

we have \overline{κ}(X) = −∞.

Proof. In the first cases, there is a finite Galois covering B̃ → B = P^{1} with ramification indices m_{1}, …, m_{r} such that the normalization X̃ of the fiber product X ×_{B} B̃ is an A_{∗}^{1}-bundle over an elliptic curve B̃. It is easy to show that \overline{κ}(X̃) = 0. Since the canonical morphism X̃ → X is a finite étale covering, we have \overline{κ}(X) = 0. In the second cases, X is a Platonic A_{∗}^{1}-fiber space. Hence \overline{κ}(X) = −∞. Q.E.D.

Lemma 5.7 In the quasi-elliptic case, any étale endomorphism φ of X preserves the A_{∗}^{1}-fibration π : X → B.

Proof. We denote φ : X_{1} → X_{2} to distinguish the upper and lower X. Similarly, π_{i} : X_{i} → B_{i} (1 = 1, 2) denotes the same π : X → B, where B_{1} = B_{2} = B. Let (V_{2}, D_{2}) be a smooth completion of X_{2} with a simple normal crossing divisor D_{2} and let ρ_{2} : V_{2} → B_{2} be the P^{1}-fibration on V_{2} which extends π_{2}. Similarly, we take a smooth completion (V_{1}, D_{1}) of X_{1} and a P^{1}-fibration ρ_{1} : V_{1} → B_{1} extending π_{1}. Furthermore, we may assume that there exists a morphism Φ : V_{1} → V_{2} such that Φ |_{X_{1}}= φ. Then the logarithmic ramification divisor formula implies that there exists an effective divisor R_{Φ} such that

D_{1} + K_{V_{1}} = Φ^{∗}(D_{2} + K_{V_{2}}) + R_{Φ}

and that R_{Φ} contains the ramification locus of the morphism Φ.

Now let ℓ be a general fiber of the P^{1}-fibration ρ_{1}. Then (D_{1}+K_{V_{1}} ·ℓ) = 0. Since |ℓ| is a linear pencil and since \overline{κ}(X) = 0 implies |n(D_{2} + K_{V_{2}})| ≠ ∅ for some n > 0, we have (Φ^{∗}(D_{2} + K_{V_{2}}) · ℓ) ≥ 0 and (R_{Φ} · ℓ) ≥ 0. Hence we must have

(Φ^{∗}(D_{2} + K_{V_{2}}) · ℓ) = (R_{Φ} · ℓ) = 0 .

This, in particular, implies that ℓ does not meet the ramification locus of the morphism Φ. It then follows that Φ(ℓ) is a nonsingular rational curve with (Φ(ℓ)^{2}) = 0. Since (D_{2} + K_{V_{2}} · Φ(ℓ)) = 0 we have (D_{2} · Φ(ℓ)) = 2. Hence Φ(ℓ) ∩ X_{2} = φ(F) ≅ A_{∗}^{1}, where F = ℓ ∩ X_{1} is a general fiber of π_{1}.

Now consider X̃ and the induced A_{∗}^{1}-fibration π̃ : X̃ → B̃. Let σ : X̃ → X be the canonical morphism. Then σ is a finite étale covering. Then each connected component of σ^{−1}(φ(F)) is a finite étale covering of A_{∗}^{1}. Hence each connected component must be isomorphic to A_{∗}^{1}. Then it is mapped to a point by π̃ because B̃ is an elliptic curve. Namely, each connected component of σ^{−1}(φ(F)) is a fiber of π̃. Hence φ(F) is a fiber of π_{2} : X_{2} → B_{2}. So, φ preserves the A_{∗}^{1}-fibration. Q.E.D.

We shall give an example of a nontrivial étale endomorphism φ : X → X in the quasi-elliptic case {4; 2, 2, 2, 2}.

Lemma 5.8 Let n be a positive integer > 1 such that d := (n^{2} − n)/2 is odd. Let C be an elliptic curve which we view as an abelian variety by fixing one point as a point of origin. Let L be an invertible sheaf on C such that deg L > 0 and n_{C}^{∗}(L) ≅ L^{⊗n^{2}} (cf. Lemma 5.4). Let M := L^{⊗(n^{2}−n)/2}. Let (−1)_{C} : C → C be the involution p ↦ −p. Then the following assertions hold:

(1) n_{C}^{∗}(M) ≅ M^{⊗n^{2}} and (−1)_{C}^{∗}(M) ≅ M.

(2) Let Y = P(O_{C} ⊕M)−(S_{0}∪S_{1}), where S_{0} (resp. S_{1}) is the cross-section corresponding to the projection O_{C} ⊕M → O_{C} (resp. O_{C} ⊕M → M).

Let q : Y → C be the canonical projection of A_{∗}^{1}-bundle. Then there exists an étale endomorphism ψ : Y → Y such that n_{C} ∘ q = q ∘ ψ (cf. Lemmas 5.3 and 5.4).

(3) Let {U_{i}}_{i∈I} be an open covering of C such that M |_{U_{i}}= O_{U_{i}}e_{i} for every i ∈ I. Then q^{−1}(U_{i}) = Spec O_{U_{i}}[e_{i}, e_{i}^{−1}]. Define an involution ι : Y → Y locally by ι^{∗}(e_{i}) = −e_{i} for i ∈ I. Then the involution ι is well-defined and satisfies the conditions (−1)_{C} ∘ q = q ∘ ι and ψ ∘ ι = ι ∘ ψ.

(4) Let B := C/⟨(−1)_{C}⟩ and let X := Y/⟨ι⟩. Then the projection q : Y → C induces an A_{∗}^{1}-fibration π : X → B, where B ≅ P^{1} and π has four multiple fibers of multiplicity two. Furthermore, the multiplication by n endomorphism n_{C} of C induces an endomorphism β : B → B.

(5) The étale endomorphism ψ : Y → Y induces an étale endomorphism φ : X → X such that π ∘ φ = β ∘ π. The endomorphism φ is not an automorphism.

Proof. (1) By [8, Corollary 3, page 59], we have for any invertible sheaf L

n_{C}^{∗}(L) ≅ L^{(\frac{n^{2}+n}{2})} ⊗ (−1)_{C}^{∗}L^{(\frac{n^{2}−n}{2})} .

Since n_{C}^{∗}(L) ≅ L^{⊗n^{2}} we have

(−1)_{C}^{∗}L^{(\frac{n^{2}−n}{2})} ≅ L^{(\frac{n^{2}−n}{2})} .

Hence we have n_{C}^{∗}(M) ≅ M^{⊗n^{2}} and (−1)_{C}^{∗}M ≅ M.

(2) The assertion follows from Lemmas 5.3 and 5.4.

(3) Clear by the definition.

(4) The fixed points in C by (−1)_{C} are the 2-torsion points, and there are four of them. The quotient morphism C → B ramifies at these four points, and B is therefore isomorphic to P^{1}. Since (−1)_{C} ∘ n_{C} = n_{C} ∘ (−1)_{C}, n_{C} induces an endomorphism β : B → B.

(5) The involution ι : Y → Y has no fixed points by the definition. Hence the quotient morphism Y → X is a finite étale morphism. This implies that the fibers of q : Y → C lying over the four 2-torsion points of C produce the four multiple fibers of π : X → B with multiplicity 2. The rest of the assertion is now readily ascertained. Q.E.D.

The case (ii) of Lemma 5.5 is the case of Platonic A_{∗}^{1}-fiber spaces. In this case, we pose the following

Conjecture Let π : X → B be a Platonic A_{∗}^{1}-fiber space and let φ : X → X be an étale endomorphism. Then there exists an endomorphism β : B → B such that β ∘ π = π ∘ φ.

Let π : X → B be a Platonic fiber space. Then there exists a normal completion V of X as given at the beginning of the section 2. With the notations there we may further assume that the fibers G_{i} of the P^{1}-fibration p : V → B containing the multiple fibers m_{i}Γ_{i} have the linear chains as the dual graphes. The end components of the fibers G_{i} meet the cross-sections S_{0} and S_{1} of p. The components of the G_{i} other than m_{i}Γ_{i} contract to the cyclic quotient singular points lying on the sections S_{0} and S_{1}. Let V̂ be a normal projective surface obtained by contracting all the components of G_{1}, G_{2}, G_{3} except for m_{1}Γ_{1}, m_{2}Γ_{2}, m_{3}Γ_{3}. Then V̂ inherits the P^{1}-fibration p̂ : V̂ → B. The images Ŝ_{0}, Ŝ_{1} of S_{0}, S_{1} are the cross-sections of p̂. Let φ : X → X be an étale endomorphism. By Lemma 2.2, φ extends to an étale endomorphism φ̃ on A^{2} which fixes the point of origin O and commutes with the group G := π_{1}(X). In particular, φ̃ induces a local analytic isomorphism between A^{2}/G and its copy near the point O. This implies that φ extends to a regular endomorphism φ̂ : V̂ − Ŝ_{1} → V̂ − Ŝ_{1}.

Theorem 5.9 Suppose that φ̂ is also regular on the cross-section Ŝ_{1}. Then φ preserves the A_{∗}^{1}-fibration ρ : X → B.

Proof. Under the hypothesis, we know by modifying the logarithmic ramification formula that the following equality among the Q-Caritier divisors holds:

D̂_{1} + K_{V_{1}} = φ̂^{∗}(D̂_{2} + K_{V_{2}}) + R_{φ} ,

where V̂_{1} and V̂_{2} are copies of V̂, which are numbered to clarify the source and the target of the endomorphism φ̂, and similarly D̂_{1} = D̂_{2} = Ŝ_{0} + Ŝ_{1}, and where the divisor R_{φ} is an effective divisor. On the other hand, we can show that N(D̂ + K_{V}) is a Cartier divisor for some integer N > 0 and |−N(D̂+K_{V})| = |ℓ|, where ℓ is a general fiber of the P^{1}-fibration p̂: V̂ → B (for the computation, see [4, p. 81]). Namely we have the equality

−N(D̂_{1} + K_{V_{1}}) + R_{φ} = φ̂^{∗}(−N(D̂_{1} + K_{V_{1}})) .

Then it follows that

(−N(D̂_{1} + K_{V_{1}}) · φ̂^{∗}(ℓ)) = (R_{φ} · φ̂^{∗}(ℓ)) = 0 .

This implies that φ̂^{∗}(ℓ) consists of several fibers of the P^{1}-fibration p̂. So, φ preserves the A_{∗}^{1}-fibration φ. Q.E.D.

6 A_{∗}^{1}-fibrations given by G_{m}-actions

In this section we consider when a given A_{∗}^{1}-fibration comes from a G_{m}-action. Theorem 6.1 below asserts that in most cases the untwisted A_{∗}^{1}-fibrations are given by G_{m}-actions provided one of the two connected components in the boundary is contractible.

Let Σ be an affine normal algebraic surface with a unique singular point O and let X := Σ − {O}. Suppose that there exists a surjective morphism π : X → B from X to a nonsingular projective curve B of genus g such that every fiber of π is irreducible and is isomorphic to A_{∗}^{1} if taken with reduced structure. Let m_{1}Γ_{1}, …, m_{r}Γ_{r} be the multiple fibers of π. Then our result is stated as follows.

Theorem 6.1 Except for the cases (i) g = 0, r = 1 and (ii) g = 0, r = 2, m_{1} ≠ m_{2}, there exists a G_{m}-action on the surface Σ such that :

(1) Σ has a unique fixpoint O, and the point O is in the closure of every orbit.

(2) B ≅ X/G_{m} and the quotient morphism coincides with the given A_{∗}^{1}- fibration π.

Proof. Our proof consists of several steps.

(I) Let p_{i} = π(Γ_{i}) for 1 ≤ i ≤ r. By Lemma 4.3, except for the cases (i) and (ii) above, there exists a Galois covering τ : C → B with group G which ramifies over p_{1}, …, p_{r} with respective ramification indices m_{1}, …, m_{r}. Let Y be the normalization of the fiber product X ×_{B} C and let ρ : Y → C be the composition of the normalization morphism Y → X ×_{B} C and the canonical projection X ×_{B} C → C. By Lemma 4.7 it then follows that ρ : Y → C is an A_{∗}^{1}-fiber bundle, i.e., there are no singular fibers.

The Galois group G acts on C. Hence G acts on the fiber product X ×_{B} C via the factor C. Then the G-action extends to a G-action on the normalization Y and its quotient Y/G is isomorphic to X. Let p : Y → X be the quotient morphism.

(II) We can write Y as Y = P(O_{C} ⊕ L) − (S_{0} ∪ S_{1}), where L is an invertible sheaf over the curve C and S_{0}, S_{1} are two cross-sections of the P^{1}-fibration P(O_{C} ⊕ L) → C. We assume that (S_{0}^{2}) < 0. Now note that X is an open set of the normal affine surface Σ. Let Σ̃ be the normalization of Σ in the function field of X ×_{B} C (or equivalently in the function field of Y). Then Σ̃ is a normal affine surface such that the group G acts on Σ̃ and Y is an open set of Σ̃. In fact, Σ̃ − Y consists of a single point Õ. To see this, let Z = P(O ⊕ L) − S_{1}. Then the morphism p : Y → X extends to a morphism p̂: Z → Σ. Then the morphism p̂ factors as

p̂: Z ⟶^{σ} Σ̃ ⟶^{ν} Σ ,

where σ : Z → Σ̃ is the contraction of the section S_{0} and ν : Σ̃ → Σ is the normalization morphism. Actually, Z is a minimal resolution of the singular point Õ. Then the G-action on Σ̃ lifts to a G-action on Z. In particular, the conormal bundle I/I^{2} of the section S_{0} is G-stable, where I is the defining sheaf of ideals of S_{0} in Z. We know that the conormal bundle I/I^{2} is isomorphic to the bundle L. Thus we have shown that L is a G-bundle in the sense that G acts on L and the canonical projection L → C is a G-equivariant morphism, where we identify intentionally an invertible sheaf and the associated bundle.

(III) Since L is a G-line bundle, O_{C} ⊕ L is a G-vector bundle over C. Hence the P^{1}-bundle V = P(O_{C} ⊕ L) has a G-action which induces the G-actions on Y and Z. Now choose an affine open covering U = {U_{i}}_{i∈I} such that L |_{U_{i}}= O_{U_{i}}e_{i} for every i ∈ I. Write O_{C} = O_{C}e, where one can take e = 1. Then π^{−1}(U_{i}) ≅ Spec A_{i}[z_{i}, z_{i}^{−1}], where U_{i} = Spec A_{i} and z_{i} = e_{i}/e. In U_{i} × P^{1}, the cross-section S_{0} is defined by z_{i} = 0. Now consider a G_{m}-action locally on π^{−1}(U_{i}) by t · z_{i} = tz_{i} for t ∈ G_{m}. Then the local G_{m}-actions on the π^{−1}(U_{i}) glue together to give a G_{m}-action on Y which commutes with the G-action. Hence the G_{m}-action on Y descends on the quotient X := Y/G. Since Σ is a normal surface and X = Σ − {O}, the G_{m}-action on X extends to Σ. In fact, let σ_{t} be the automorphism of X induced by an action of t ∈ G_{m}. Then σ_{t} : Σ → Σ is a birational mapping which is regular except for the point O. Then σ_{t} is regular everywhere on Σ because Σ is a normal surface. By the above construction, one can readily show that the G_{m}-action on Σ satisfies two conditions (1) and (2) in the statement of the theorem.

Q.E.D.

Theorem 6.2 Let Σ be an affine normal algebraic surface as in Theorem 6.1, let X := Σ − {O} and let π : X → B be the given A_{∗}^{1}-fibration. Suppose that one of the following cases occurs:

(1) g > 1.

(2) g = 1 and there is at least one multiple fiber.

(3) g = 0, r ≥ 3 and {r; m_{1}, …, m_{r}} is none of the cases listed as the exceptional cases in Lemma 5.5.

Then Aut (Σ) = Aut (X) satisfies an exact sequence

0 → G_{m} → Aut (X) → Aut_{∗}(B),

where Aut_{∗}(B) is a subgroup of Aut(B) consisting of elements which permute the points of B over which lie the multiple fibers of π. In any one of the above cases, Aut_{∗}(B) is a finite group. Hence the connected component Aut^{0}(X) of Aut (X) is isomorphic to the one-dimensional algebraic torus G_{m}.

Proof. We employ the notations in the proof of Theorem 6.1. Since O is the unique singular point of Σ, every automorphism φ fixes the point O. Conversely, any automorphism ψ of X is regular at O as a birational mapping from Σ into itself. Hence Aut (Σ) = Aut (X) holds.

Note that \overline{κ}(X) = 1 in each of the above three cases. Hence every automorphism φ preserves the A_{∗}^{1}-fibration π. Namely, π ∘ φ = β ∘ φ. Hence we have a canonical group automorphism Aut (X) → Aut (B), whose kernel we denote by Aut_{π}(X). In fact, the image is contained in Aut_{∗}(B) by the observations in the previous sections. In what follows, we shall show that Aut_{π}(X) = G_{m}.

Let φ be an element of Aut _{π}(X). As in the proof of Theorem 6.1, let τ : C → B be a finite Galois covering which ramifies over the points p_{i} with ramification index m_{i} if there are multiple fibers of π and let Y be the normalization of the fiber product X ×_{B} C. If there are no multiple fibers we take B as C and X as Y. Then the induced A_{∗}^{1}-fibration on Y over C has no multiple fibers. The automorphism φ extends naturally to a C-automorphism of Y, which we denote by the same symbol φ. Then Y = P(O_{C} ⊕ L) − (S_{0} ∪ S_{1}) as in the step (III) of the proof of Theorem 6.1. With the notations there, φ induces an A_{i}-algebra automorphism α = φ^{∗} of A_{i}[z_{i}, z_{i}^{−1}].Then α(z_{i}) = a_{i}z_{i}^{±} with a_{i} ∈ A_{i}^{∗} for every i ∈ I. Hence we know that φ extends to a C-automorphism of V = P(O_{C} ⊕ L) which fixes the cross-sections S_{0} and S_{1} or exchanges S_{0} and S_{1}. This implies that if α(z_{i}) = a_{i}z_{i} (resp. α(z_{i}) = a_{i}z_{i}^{−1}) for some i ∈ I then α(z_{j}) = a_{j}z_{j} (resp. α(z_{j}) = a_{j}z_{j}^{−1}) for every j ∈ I. Suppose that α(z_{i}) = a_{i}z_{i}^{−1} for every i ∈ I. Then we have

a_{j}a_{ji}^{−1}z_{i}^{−1} = a_{j}z_{j}^{−1} = α(z_{j}) = α(a_{ji}z_{i}) = a_{ji}α(z_{i}) = a_{ji}a_{i}z_{i}^{−1}

over U_{ij} = U_{i} ∩ U_{j}, where {a_{ji}} is a family of transition functions of L with respect to the affine open covering U = {U_{i}}_{i∈I}. Then a_{ji}^{2} = a_{j}a_{i}^{−1}. This implies that L^{2} ≅ O_{C}. But this is impossible because deg L = −(S_{0}^{2}) > 0. So, α(z_{i}) = a_{i}z_{i} for every i ∈ I. Then a_{i} = a_{j} for all i, j ∈ I by a calculation similar to the above. Hence a_{i} = a ∈ Γ(C, O_{C})^{∗} = C^{∗}. Then it is clear that φ comes from the G_{m}-action on X whose existence is shown in Theorem 6.1. Q.E.D.

Next we consider the case of Platonic A_{∗}^{1}-fiber spaces.

Theorem 6.3 Let π : X → B be a Platonic A_{∗}^{1}-fiber space and let p_{1}, p_{2}, p_{3} be the points of B such that π^{∗}(p_{i}) is a multiple fiber of π for i = 1, 2, 3. Then there exists an exact sequence

0 → G_{m} → Aut (X) → Aut_{∗}(B),

where Aut_{∗}(B) consists of all automorphisms of B permuting the set {p_{1}, p_{2}, p_{3}}.

Proof. Write X = A^{2}/G − {0} as in section 2, where G is a non-cyclic small finite subgroup of GL(2, C). Let φ be an automorphism of X. Then φ lifts to an automorphism ψ of A^{2} such that ψ commutes with the linear G-action and ψ(O) = O (cf. section 2). Then ψ induces an automorphism on Σ := A^{2}/G whose restriction on X is the given automorphism φ. We denote by the same symbol φ the induced automorphism of Σ. We have φ(O) = O. Let h : Σ̂ → Σ be the minimal resolution of the singularity O. Then the exceptional locus h^{−1}(O) is an admissible rational fork F = B_{0} +T_{1} +T_{2} +T_{3} (see [6] for the definitions), where B_{0} is the central component and T_{1}, T_{2}, T_{3} are three maximal rational twigs meeting B_{0}. We can identify B_{0} with the base curve B so that B_{0} ∩ T_{i} = {p_{i}} for i = 1, 2, 3. Then φ lifts to an automorphism φ̂ of Σ̂ such that h∘φ̂ = φ∘h, φ̂(B_{0}) = B_{0} and φ |_{B_{0}} induces a permutation on the set {p_{1}, p_{2}, p_{3}}. Thus there exists a group homomorphism

Aut (X) → Aut_{∗}(B),

whose kernel we denote by Aut_{0}(X).

Let φ be anew an element of Aut_{0}(X). Let C be an irreducible component of the twig T_{1} such that B_{0} ∩ C_{1} = {p_{1}}. Then φ̂(C_{1}) = C_{1}. If φ̂ |_{C_{1}} is the identity then φ̂ (and hence φ) is the identity. In general, there are two points of C_{1} which are fixed by φ̂ |_{C_{1}}. Namely they are the point B_{0} ∩ C_{1} and the point where C_{1} meets the adjacent component of the fiber of π containing C_{1}. By the restriction φ ↦ φ̂ ↦ φ̂ |_{C_{1}}, we have an injective group homomorphism

Aut_{0}(X) → G_{m}.

In particular, Aut_{0}(X) is an abelian group.

On the other hand, there exists a G_{m}-action by Theorem 6.1 which gives rise to the A_{∗}^{1}-fibration π. We can find an element t ∈ G_{m} such that φ̂ |_{C_{1}} is the multiplication by t. Then t^{−1} ∘ φ̂ |_{C_{1}} is the identity. Hence φ̂ = t. It then follows that Aut_{0}(X) = G_{m}. Q.E.D.

If we state Theorem 6.3 in terms of the automorphisms on A^{2} we obtain the following result.

Corollary 6.4 Let G be a non-cyclic small finite subgroup of GL(2, C). Let Aut _{G}(A^{2}) be the subgroup of Aut (A^{2}) consisting of automorphisms which commutes with the linear G-action. Then Aut _{G}(A^{2}) is an algebraic group such that its connected component containing the identity element is isomorphic to G_{m}.

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Department of Mathematics Himeji Institute of Technology 2167 Shosha, Himeji 671-2201, Japan e-mail: kayo@sci.himeji-tech.ac.jp and Department of Mathematics Graduate School of Science Osaka University 1-1 Machikaneyama-cho, Toyonaka 560-0043, Japan e-mail: miyanisi@math.sci.osaka-u.ac.jp