pZ -actions on algebraic surfaces

Journal of Algebra 477 (2017) 360–389

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Journal of Algebra www.elsevier.com/locate/jalgebra

Wild Z/pZ-actions on algebraic surfaces Masayoshi Miyanishi 1 Research Center for Mathematical Sciences, Kwansei Gakuin University, 2-1 Gakuen, Sanda 669-1337, Japan

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Article history: Received 25 April 2015 Available online 14 January 2017 Communicated by Steven Dale Cutkosky MSC: primary 14R20 secondary 14R25

a b s t r a c t An Artin–Schreier covering of the affine plane is a hypersurface in A3 defined by an equation z p − z = f (x, y). Modeling on hypersurfaces of this type, we consider in the present article a normal affine domain B with a G := Z/pZ-action which is written in the form B = A[z], z p − sz = a with s, a ∈ A, where A = B G is the G-invariant subring. We also prove a classification result of Z/pZ-actions on the affine plane A2 = Spec k[x, y], which was first announced by S. Kuroda. © 2017 Elsevier Inc. All rights reserved.

Keywords: Z/pZ-action Artin–Schreier covering Affine plane Frobenius sandwich Exotic line

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Liftability of Ga -actions . . . . . . . . . . . . . . . . 2. Free Z/pZ-actions on affine domains . . . . . . . . 3. Wild Z/pZ-actions on the affine plane . . . . . . . 4. Artin–Schreier coverings of the affine plane over

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E-mail address: [email protected]. The author was supported by Grant-in-Aid for Scientific Research (B) 24340006, JSPS.

http://dx.doi.org/10.1016/j.jalgebra.2016.12.025 0021-8693/© 2017 Elsevier Inc. All rights reserved.

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5. Z/pZ-action on a smooth projective surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 6. Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

Introduction Throughout the article, the ground field k is an algebraically closed field of positive characteristic p. In [13], a Z/pZ-covering of an affine variety Spec A is investigated, and it is shown that if A is a noetherian factorial domain satisfying the condition that every projective A-module of rank p, p − 1, p − 2 is free then every Z/pZ-covering of Spec A is given as Spec B, where B = A[z]/(z p − z − a) with a ∈ A. Later in [7], Kambayashi and Srinivas showed that this result holds for an arbitrary affine variety Spec A. In fact, they recognized a Z/pZ-covering Spec B as a Z/pZ-torsor over Spec A in the flat cohomology 1 whose isomorphism class over Spec A corresponds to an element of Hflat (Spec A, Z/pZ) and used the flat cohomology sequence over Spec A for an exact sequence of group schemes F −id

0 −→ Z/pZ −→ Ga −→ Ga −→ 0 , where F is the Frobenius endomorphism x → xp . Thus the isomorphism class of Spec B over Spec A is determined by an element a ∈ A modulo elements of the form cp − c with c ∈ A. We call Spec B an Artin–Schreier covering of Spec A as well. Of real interest in problems under this setting is to look into geometric properties of the covering space. In the present article, we take the first step toward this problem. Here we note that in the projective case an extensive study of Artin–Schreier coverings of projective algebraic surfaces was made by Peskin [18,19] and Takeda [22]. In section one, we begin with some remarks about liftability of Ga -actions as well as αp and μp -actions for Artin–Schreier coverings. In section two, we consider free Z/pZ-actions on affine k-domains and discuss some relationships between Z/pZ-actions and the Makar–Limanov invariants. In section three, we consider Z/pZ-actions on the affine plane. In the talk given on the occasion of the thirteenth meeting of Affine Algebraic Geometry at Umeda, Osaka on March 5, 2015, S. Kuroda announced that such a Z/pZ-action on A2 is given by σ(x, y) = (x + a(y), y) with respect to a suitable system of coordinates, where a(y) ∈ k[y] and σ is a generator of Z/pZ (cf. [10]). We give an independent proof by using the results in [12] together with a geometric approach to related problems. In the case where there exists an A1 -fibration on A2 which is preserved by the Z/pZ-action, the action is completely clarified and the problem is reduced to a problem of determining a polynomial which satisfies a numerical condition. The last case corresponds to the case where the σ-action is an automorphism of de Jonquière type.

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In section four, we also consider a covering z p − z = f (x, y) of the affine plane A2 = Spec k[x, y], where f (x, y) = 0 is the defining equation of a curve isomorphic to the affine line which is called an exotic line. In section five, we discuss the fixed points of Z/pZ-actions on projective surfaces. We also consider a normal affine variety Y = Spec B with a Z/pZ-action such that B = A[z], z p − sz = a with s, a ∈ A = B G . Such an affine variety Y is said to be of quasi-Artin–Schreier type. Throughout the article, the quotient field of an integral domain R is denoted by Q(R). We are indebted to S. Kuroda and the referee for reading the manuscript and pointing out mistakes in the proof of Lemma 2.1 and improving the section 6. 1. Liftability of Ga -actions The liftability of Ga -actions for a finite étale extension is discussed in [11, Theorem 1.1] in the case of characteristic zero. We generalize this result to the case of positive characteristic. Theorem 1.1. Let A be a normal affine domain over the ground field k of positive characteristic p and let B be a finite étale extension of A such that the extension degree [B : A] is not divisible by p. Then any Ga -action on Spec A lifts to Spec B. Proof. By [12, Lemma 1.5, p. 20], there exists an element c ∈ A0 := AGa and an element x ∈ A such that A[c−1 ] = A0 [c−1 ][x], where the coaction ϕ of Ga is given by n

ϕ(x) = x + α0 t + α1 tp + · · · + αn tp

with α0 , α1 , . . . , αn ∈ A0 [c−1 ]. Set C = Spec A0 [c−1 ]. Then Spec A[c−1 ] ∼ = C × A1 . We alg,p denote by π1 the prime to p part of the algebraic fundamental group π1alg . Since alg,p alg,p π1 (Spec A[c−1 ]) = π1 (C) and B[c−1 ] is a finite étale extension of A[c−1 ] of extension degree not divisible by p, it follows that B[c−1 ] = B ⊗A A[c−1 ] = D[x], where D is a finite étale extension of A0 [c−1 ]. The Ga -action on Spec A[c−1 ] extends to Spec B[c−1 ] by defining the coaction ϕ  : B[c−1 ] → B[c−1 ][t] by ϕ(x)  = ϕ(x) and ϕ|  D = id D . We show that ϕ(B)  ⊆ B[t]. For the purpose, it suffices to show that ϕ λ (b) ∈ B for any b ∈ B and λ ∈ k, where ϕ λ = ϕ|  t=λ . Since b satisfies an integral relation bm + a1 bm−1 + · · · + am = 0, a1 , . . . , am ∈ A and ϕλ (ai ) ∈ A, we know that ϕ λ (b) is integral over A. This implies that ϕ λ (b) ∈ B since B is a normal domain. 2 Remark 1.2. A Ga -action does not necessarily lift for an Artin–Schreier extension. In fact, consider an Artin–Schreier extension C of the affine line A1 = Spec k[t] defined by

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z p − z = c0 tn + · · · + cn−1 t , where c0 , c1 , . . . , cn−1 ∈ k, c0 = 0, n > 0 and ci = 0 whenever i ≡ 0 (mod p). Then C has geometric genus (p − 1)(n − 1)/2 (see [13, Lemma 3.1]). The affine line A1 has a natural Ga -action, while the curve C does not if n > 1. We consider the liftability of αp (resp. μp ) action for a finite extension SpecB → SpecA of affine varieties. An αp (resp. μp ) action on Spec A is given by a k-derivation δ on A such that δ p = 0 (resp. δ p = δ), which is called a derivation of additive type (resp. of multiplicative type). Lemma 1.3. Let Spec B → Spec A be an Artin–Schreier extension given by B = A[z]/ (z p − z − a) with a ∈ A. Then any αp (resp. μp ) action on Spec A lifts uniquely to a αp (resp. μp ) action on Spec B. Proof. Let δ be a k-derivation of A corresponding to the αp (resp. μp ) action on Spec A.  If δ is a k-derivation of B which extends δ, then δ(z) = −δ(a). Now define a k-linear   endomorphism of B by δ(z) = −δ(a) and δ|A = δ. Clearly, δ extends to a k-derivation of B and hence δp is a k-derivation of B, too. If δ p = 0, then δp (z) = δp−1 (−δ(a)) = −δ p (z) = 0. Hence δp = 0 as well. If δ p = δ, then δp (z) = δp−1 (−δ(a)) = −δ p (a) =   From these observations follows the assertion. 2 −δ(a) = δ(z). Hence δp = δ. We have also the following result. Lemma 1.4. Let A be a normal affine domain and let B be a finite étale extension of A such that B is a domain and the extension degree [B : A] is prime to p. Then any αp -action on SpecA extends uniquely to an αp -action on SpecB. Similarly, any μp -action on Spec A lifts uniquely to a μp -action on Spec B. Proof. Let δ be a k-derivation of additive type on A which corresponds to the αp -action. Since Derk (B) = Derk (A) ⊗A B, δ extends uniquely to a k-derivation δ on B. We show that δp = 0 holds. Let b be a nonzero element of B. Then it satisfies an integral relation bn + a1 bn−1 + · · · + an = 0, a1 , . . . , an ∈ A

(∗)

where we can take n to be the smallest among all integral relations of b. Since A ⊂ A[b] ⊂ B implies that [B : A] = [B : A[b]] · [A[b] : A], the extension degree [A[b] : A] is prime to p. Since (∗) is also a minimal algebraic equation of b over the quotient field Q(A) because A is normal, n is prime to p. Noting that δp is a k-derivation of B and that δ p = 0, the differentiation of the relation (∗) by δp entails 

 nbn−1 + (n − 1)a1 bn−2 + · · · + an−1 δp (b) = 0 .

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This implies that δp (b) = 0 for every b ∈ B because B is a domain. Hence δp = 0 and the αp -action on Spec A lifts uniquely to an αp -action on Spec B. We give a different argument for the μp -action which can be also applied to the case of αp -action. Let δ be a k-derivation on A corresponding to the μp -action. So, we have δ p = δ. By the same reasoning as above, the k-derivation δ lifts uniquely to a k-derivation δ on B. Since δp is a k-derivation on B, δp − δ is a k-derivation on B which restricts to the zero derivation on A. Since Derk (B) = Derk (A) ⊗A B, it follows that δp − δ = 0 on B. This implies that the μp -action on Spec A lifts uniquely to a μp -action on Spec B. 2 If B is not étale over A, an αp -action on Spec A might not extend to an αp -action on Spec B. Example 1.5. Let p = 2, A = k[x] and B = k[x, y]/(y 3 − y − x). Then B is regular, but B is not étale over A. In fact, we have y(y + 1)2 = x. So, the point x = 0 has a ramified point (x, y) = (0, 1) on Spec B. Consider an αp -action on Spec A defined by δ(x) = 1.  If δ extends to a k-derivation δ of B, we must have δ(y) = 1/(y 2 + 1) ∈ B, which is  not the case. So, δ does not extend to B. Let A = k[x, x−1 ] and B  = B ⊗A A =  = (y 2 + 1)−1 = yx−1 ∈ B  . Hence this αp -action lifts k[x, x−1 , y]/(y 3 − y − x). Then δ(y) up to the inverse image of an open set of Spec A. Z-actions on affine domains 2. Free Z /pZ We set G = Z/pZ and let σ denote a generator of G. A finite group action is called wild if the order of the group is divisible by the characteristic of the ground field. We note that if G acts on an affine domain B then the invariant subring A := B G is also an affine domain, and also note that G has no nontrivial multiplicative characters, i.e., there are no nontrivial group homomorphism G → R∗ for any k-domain R. We have the following result due to Samuel [20, Chapter III] (see also [5, Prop. 6.2]). Lemma 2.1. Let B be a factorial affine domain with a G := Z/pZ-action and let A = B G . Suppose that B ∗ = k∗ . Then A is a factorial affine domain. Proof. The result follows from [20, Remark 2, p. 57]. For the convenience of the readers, we give a conceptual proof in the case where G acts freely on Y = Spec B. Consider a standard exact sequence of G − Z-modules 0 → B ∗ → Q(B)∗ → Div(Y ) → C(Y ) → 0, where Div(Y ) (resp. C(Y )) denotes the group of Weil divisors (resp. the divisor class group) of Y . Since B is factorial by assumption, we have C(Y ) = 0 and hence Div(Y ) = P (Y ), where P (Y ) is the group of principal divisors. Hence we have an exact sequence 0 → B ∗ → Q(B)∗ → P (Y ) → 0.

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Now taking the long exact sequence of group cohomologies, we have 0 → (B ∗ )G → (Q(B)∗ )G → P (Y )G → H 1 (G, B ∗ ), where (B ∗ )G = A∗ , (Q(B)∗ )G = Q(A) and H 1 (G, B ∗ ) = H 1 (G, k∗ ) = Homgroup (G, k∗ ) = 0. Hence we have an exact sequence 0 → A∗ → Q(A)∗ → P (Y )G → 0, whence follows P (X) = P (Y )G = Div(Y )G . Let p be any height 1 prime ideal of A. Then pB ∈ Div(Y )G . Hence pB = aB for some a ∈ A. Since B is a faithfully flat extension of A because the G-action on Y is free, we have p = pB ∩ A = aB ∩ A = aA. So, A is factorial. 2 Kuroda [9] has informed the author of the following example. Example 2.2. Let p = 2 and B = k[x, y, z, z −1 ] with a G := Z/2Z-action given by σ(x) = yz, σ(y) = xz and σ(z) = z −1 . Then A := B G is not factorial. We refer to [3] for group cohomologies of G acting freely on an affine domain. Unfortunately, there is a mistake in the proof of [3, Prop. 1.1], where the authors identify the coordinate ring of G with the group algebra k[G] of G over k. This is apparently wrong because the coordinate ring of G viewed as a k-group scheme is k[x]/(xp − x), while k[G] = k[x]/(xp − 1). In fact, the Cartier dual of Z/nZ is isomorphic to Z/nZ if n is prime to p but the Cartier dual of Z/pZ (resp. μp ) is isomorphic to μp (resp. Z/pZ). As the statement is valid, we restate the result and give a different proof. Lemma 2.3. Let B be an affine domain with a free G-action and let A = B G . Then H i (G, B) = 0 for every i > 0. Proof. Let k[G] be the group algebra of G over k. If one can show that B is isomorphic to k[G] ⊗k A as k[G]-modules, then H i (G, B) ∼ = H i ({1}, A) for every i > 0 by Eckmann– Shapiro lemma, where {1} is the trivial group consisting of the identity element. Since 1, σ, σ 2 , . . . , σ p−1 is a A-free basis of k[G] ⊗k A, define a homomorphism of k[G]-modules θ : k[G] ⊗k A → B by θ(1) = z p−1 , θ(σ i ) = σ i (θ(1)) = (z + i)p−1 , 1 ≤ i ≤ p − 1, where B = A[z]/(z p − z − a) and we denote the residue class of z in B by the same letter. In fact, one can show that {z p−1 , (z + 1)p−1 , . . . , (z + p − 1)p−1 } is a free A-basis of B. 2 To a G-action on an affine domain B over k we can associate a k-algebra homomorphism

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ϕ : B → B ⊗k k[ξ], ξ p = ξ which is written as ϕ(b) = b + δ(b)ξ +

1 2 1 δ (b)ξ 2 + · · · + δ p−1 (b)ξ p−1 2! (p − 1)!

where b ∈ B and δ is a k-module endomorphism of B such that δ p = 0 (see [22]). The G-action on B is given as follows when G is identified with Z/pZ = {0, 1, . . . , p − 1}: σ i (b) = ϕ(b)|ξ=i . On the other hand, we define the trace T by T = 1 + σ + σ 2 + · · · + σ p−1 . Then we have the following result. Lemma 2.4. The k-module endomorphisms T and Im (σ − 1) of B are given by the following formulas in terms of δ. (1)

T = δ p−1

(2)

σ−1=δ+

1 2 1 δ + ··· + δ p−1 . 2! (p − 1)!

Proof. (2) is straightforward. To prove (1), we use the following congruence relations  1 i 1 + 2i + · · · + (p − 1)i ≡ 0 (mod p) for 1 ≤ i < p − 1 i!  p−1  1 1 + 2p−1 + · · · + (p − 1)p−1 ≡ 1 (mod p) 2 (p − 1)! Lemma 2.5. Assume that G acts freely on Spec B. Then the following assertions hold. (1) A := B G = Ker (σ − 1) = Ker δ and A = Im δ p−1 . (2) There exists an element z of B such that δ(z) = 1. With this z, we can write B = A[z], where z p − z = a ∈ A. Such an element z is determined up to an element of A. Namely, if δ(z  ) = 1 for z  ∈ B then z  − z = a0 ∈ A. (3) Conversely, if B = A[z] with z p − z = a ∈ A, then δ(z) = i ∈ {1, 2, . . . , p − 1}, and we may assume that δ(z) = 1 by replacing z by z/i if δ(z) = i. Proof. (1) Since σ(b) = b implies σ i (b) = b for 1 ≤ i < p, it is clear that A = Ker (σ − 1). If b ∈ Ker δ, we have ϕ(b) = b. Hence b ∈ A. Suppose that b ∈ Ker (σ − 1). Then we have

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δ(b)ξ +

367

1 2 1 δ (b)ξ 2 + · · · + δ p−1 (b)ξ p−1 = 0 2! (p − 1)!

for ξ = 1, . . . , p − 1. Since the Vandermonde determinant det(ij )1≤i,j≤p−1 is not zero, it follows that δ(b) = 0. So, Ker (σ − 1) = Ker δ. Since δ p = 0 it is clear that Im δ p−1 ⊂ A. Since Ker (σ − 1) = Im T as shown in (2) below, it follows that A ⊂ Im δ p−1 . (2) By [1, Chapter XII, §7] and [3], we have H 0 (G, B) = Ker (σ − 1) and for i > 0 H 2i+1 (G, B) = Ker T /Im (σ − 1), H 2i (G, B) = Ker (σ − 1)/Im T. By Lemma 2.3, we have KerT = Im (σ −1). Since 1 ∈ Kerδ and T = δ p−1 by Lemma 2.4, 1 ∈ Ker T = Im (σ − 1). Hence, by Lemma 2.4, (2), there exists an element z ∈ B such that δ(z) = 1. If δ(z  ) = 1, it is clear that δ(z  − z) = 0. Hence z  = z + a0 with a0 ∈ A. Now σ(z) = z + 1 since σ = ϕ|ξ=1 and δ(1) = 0. Hence σ(z p ) = z p + 1. So, δ(z p ) = 1 and z p − z = a ∈ A. We shall show that B = A[z]. We prove this assertion by induction on the length  defined on B by (b) = max{n| δ n (b) = 0}. Since δ p = 0, we have 0 ≤ (b) < p. If (b) = 0 then b ∈ A. Suppose that (b) = n and write σ(b) = b + δ(b) +

1 2 1 δ (b) + · · · + δ n (b), 2! n!

where δ n (b) = 0 and δ n+1 (b) = 0, whence δ n (b) ∈ A. Meanwhile, since ϕ(z n ) = (z + ξ)n and ϕ(z n ) = z n + δ(z n )ξ +

1 2 n 2 1 δ (z )ξ + · · · + δ n (z n )ξ n 2! n!

= z n + nz n−1 ξ + · · · + ξ n , 1 we have δ n (z n ) = n!. Let b1 = b − δ n (b)z n . Since δ : B → B is an A-module homon! morphism, we have δ n (b1 ) = δ n (b) −

1 n δ (b)δ n (z n ) n!

= δ n (b) − δ n (b) = 0. Hence (b1 ) < n. So, b1 ∈ A[z] by induction hypothesis and hence b ∈ A[z]. This implies B = A[z].

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(3) Conversely, suppose that z p − z = a ∈ A. Since ξ p = ξ we have p  1 2 1 2 p−1 p−1 δ ϕ(z ) = z + δ(z)ξ + δ (z)ξ + · · · + (z)ξ 2! (p − 1)! 1 1 (δ p−1 (z))p ξ p−1 = z p + δ(z)p ξ + (δ 2 (z))p ξ 2 + · · · + 2! (p − 1)! 1 1 δ p−1 (z)ξ p−1 . ϕ(z + a) = (z + a) + δ(z)ξ + δ 2 (z)ξ 2 + · · · + 2! (p − 1)! p

Hence we have δ(z)p = δ(z), (δ 2 (z))p = δ 2 (z), . . . , (δ p−1 (z))p = δ p−1 (z). In particular, δ(z)p = δ(z) implies δ(z) ∈ {1, 2, . . . , p − 1}. Here the case δ(z) = 0 is excluded because z then belongs to A. So, δ(z) = i = 0. We may assume that δ(z) = 1 by replacing z by z/i. 2 Hereafter in this section, a G-action on Y = Spec B is nontrivial but not necessarily free. Let D = {D0 , D1 , . . .} be a locally finite iterative higher derivation, which is by definition (see [12]) a collection of k-linear endomorphisms Di of B such that  (1) D0 = id and Di (b1 b2 ) = j+=i Dj (b1 )D (b2 ) for i > 0 and b1 , b2 ∈ B. (2) For any b ∈ B, Dm (b) = 0 for m  0.   (3) Di Dj = i+j i Di+j for all i, j ≥ 0. The condition (3) above is equivalent to (4) Di = of i.

(D1 )i0 (Dp )i1 · · · (Dpr )ir , where i = i0 + i1 p + · · · + ir pr is the p-adic expansion (i0 )!(i1 )! · · · (ir )!

Define a k-algebra homomorphism ϕ : B → B[t] by ϕ(b) =



Di (b)ti .

i≥0

Then a ϕ : Ga × Y → Y defines a Ga -action τD . For λ ∈ k, ϕ(b)|t=λ is denoted by exp(λD)(b). The k-algebra exp(λD) : B → B for λ ∈ k defines a k-algebra homomorphism exp(tD) : B → B[t] which is equal to ϕ and called the exponential map associated to D. The invariant subalgebra B τD is equal to Ker D := i≥0 Ker Dpi . Conversely, it is known (see [12, Chapter 1]) that any Ga -action τ is determined as τ = τD for some locally finite iterative higher derivation D. The intersection of the subalgebras B τD for all locally finite iterative higher derivation D is denoted by ML(Y ) and called the Makar– Limanov invariant of Y . It is often denoted by ML(B). The k-subalgebra of B generated

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by B τD for all possible D is called the Derksen invariant of Y (or B) and denoted by Dk(Y ) (or Dk(B)). Lemma 2.6. Let Y = Spec B is a normal affine variety with a G-action. Then the following assertions hold. (1) ML(Y ) and Dk(Y ) are G-stable subalgebra of B. (2) Suppose that the quotient field of ML(Y ) has transcendence degree less than or equal to 2 over k. Then ML(Y ) is a normal affine subalgebra over k. (3) Suppose that dim Y − tr.degk Q(ML(Y )) = 1. Then Ker τ = ML(Y ) for any Ga -action τ on Y . Suppose further that ML(Y ) is finitely generated over k. Let Z = Spec ML(Y ). Then the morphism q : Y → Z induced by the inclusion ML(Y ) → B is the quotient morphism for any Ga -action τ on Y . Proof. (1) Write G = σ as before. Let τ be a Ga -action on Y associated to a locally finite iterative higher derivation D. Then σDσ −1 which consists of σDi σ −1 for i ≥ 0 is also a locally finite iterative higher derivation such that Ker (σDσ −1 ) = σ(Ker D). This implies that ML(Y ) and Dk(Y ) are σ-stable. (2) Let K = Q(B) and K0 = Q(ML(Y )). We show that ML(Y ) = K0 ∩ B. It is clear that ML(Y ) ⊆ K0 ∩ B. Let τ = τD be any Ga -action on Y , where D is the associated higher derivation which is locally finite and iterative. Then K0 ∩ B ⊆ Q(Ker D) ∩ B = KerD, where the last equality follows from [12, Chapter 1, Lemma 1.3.2]. Hence it follows that K0 ∩ B ⊆ ML(Y ). If tr.degk Q(ML(Y )) ≤ 2, then ML(Y ) is an affine subalgebra by a theorem of Zariski. Since B is normal, it follows that ML(Y ) is integrally closed in Q(ML(Y )). (3) Let τ = τD be a Ga -action on Y . Then ML(Y ) ⊆ Ker D by definition and tr.degk Q(Ker D) = dim Y − 1. Hence Ker D is algebraic over ML(Y ). We show that ML(Y ) is integrally closed in B. Let b be an element of B which is integral over ML(Y ). Hence there is an integral relation bn + a1 bn−1 + · · · + an−1 b + an = 0,

a1 , . . . , an ∈ ML(Y ).

(∗)

Let λ b be the translate of b by λ ∈ k according to the action τ . Since λ ai = ai for any i, λ b is a root of the equation (∗). Hence λ b = b, i.e., b ∈ Ker D. Since τ is an arbitrary Ga -action on Y , it follows that b ∈ ML(Y ). For any element b ∈ Ker D, there is an element a = 0 of ML(Y ) such that ab is integral over ML(Y ) because Q(Ker D) is algebraic over Q(ML(Y )). Hence ab ∈ ML(Y ). This implies Q(ML(Y )) = Q(Ker D). On the other hand, Ker D is factorially closed in B, i.e., b = b1 b2 with b ∈ Ker D and b1 , b2 ∈ B implies b1 , b2 ∈ Ker D, and hence so is ML(Y ) in B. This implies that ML(Y ) is factorially closed in Ker D. As shown above, for any element b ∈ Ker D, we have ab = a1 ∈ ML(Y ) for some nonzero element a ∈ ML(Y ). Then the factorial closedness

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of ML(Y ) in Ker D implies b ∈ ML(Y ). So, ML(Y ) = Ker D. Now it is clear that the morphism q : Y → Z is the quotient morphism of the Ga -action τ . 2 The following result is a consequence of Lemma 2.6. Theorem 2.7. Let Y = Spec B be a normal affine variety and let R := ML(Y ) be the Makar–Limanov invariant of Y . Assume that (1) tr.degk Q(ML(Y )) = dim Y − 1. (2) R is finitely generated over k. Let σ be a G = Z/pZ-action on Y , where σ is a generator of G and identified with a k-algebra automorphism of B. Then σ is uniquely determined by the restriction of σ onto R. Namely, if σ, σ  are such k-automorphisms of B, then σ = σ  if σ|R = σ  |R . In particular, assumptions (1) and (2) are satisfied if dim Y = 2 and tr.degk Q(R) > 0. Proof. By Lemma 2.6, (3), R = Ker D for the locally finite iterative higher derivation D associated to a Ga -action τ on Y . Then the quotient morphism q : Y → Z = Spec R defines generically an A1 -bundle by [12, Chapter 1, Lemma 1.5]. Namely, there exists a nonzero element c ∈ R such that B[c−1 ] = R[c−1 ][t], where t ∈ B is algebraically independent over R. Let K0 = Q(R). Then B⊗R K0 = K0 [t]. Since R is G-stable, we may extend uniquely the G-action on B to the G-action on K0 [t]. Then σ(t) = a(σ)t + b(σ), where a(σ) ∈ K0∗ and b(σ) ∈ K0 . In fact, a(σ) (resp. b(σ)) satisfies a 1-cocycle condition with coefficients in K0∗ (resp. K0 ). By Hilbert’s Theorem 90 with coefficients in K0∗ (resp. K0 ), we may replace t by a new element so that we may assume that σ(t) = t. Now any element b ∈ B is written as ab = f (t) for some a ∈ R\{0} and f (t) ∈ R[t]. If we write f (t) = a0 tn + a1 tn−1 + · · · + an−1 t + an , we write f σ (t) = σ(a0 )tn + σ(a1 )tn−1 + · · · + σ(an−1 )t + σ(an ). Then σ(b) = f σ (t)/σ(a). Hence the action of σ on B is uniquely determined by its restriction on R. 2 In the above process of making t a σ-invariant element, we replaced t by an element t = αt + β with α, β ∈ K0 . By replacing further t by t = dt with a σ-invariant element d ∈ R, we may assume that t is σ-invariant as well as t ∈ B. With this remark in mind, we can show the following result. 

Corollary 2.8. Let Y be a normal affine variety with a G-action and satisfying the conditions (1), (2) in Theorem 2.7. Let X be the quotient variety Y //G. Then X has also a nontrivial Ga -action.

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Proof. With the notations in the proof of Theorem 2.7, we have B[c−1 ] = R[c−1 ][t], where we may assume that c is σ-invariant after replacing c by its norm N (c). Let S = RG and A = B G . Then A[c−1 ] = (B[c−1 ])G = S[c−1 ][t]. Write A = k[a1 , . . . , am ] and ai = gi (t)/cn , where gi (t) ∈ S[t]. Now define an exponential map ϕ on S[c−1 ][t] by ϕ|S[c−1 ] = id and ϕ(t) = t + cN ξ, where N > n. Then we have ϕ(ai ) =

gi (t + cN ξ) gi (t) = n + cN −n hi (t, ξ), hi (t, ξ) ∈ S[t][ξ]. n c c

Hence ϕ(ai ) ∈ A[ξ] for each i because S[t] ⊆ A. Then ϕ(A) ⊂ A[ξ]. It is clear that the thus-defined Ga -action on X is non-trivial. 2 We here give an example of a smooth affine surface Y with ML(Y ) = k or tr.degk Q(ML(Y )) = 1, which is a Danielewski surface in positive characteristic. Proposition 2.9. Let Y be an Artin–Schreier covering z p − z = xn y of X := A2 = Spec k[x, y]. Then we have: (1) Y has Ga -actions ϕ defined as follows: ϕ(x) = x, ϕ(z) = z + 1 ϕ(y) = y + n x



m 

i

ai (x)tp ,

i=0

−a0 (x)t +

m 

 (ai−1 (x) − ai (x)) t p

pi

p pm+1

+ am (x) t

i=1

where ai (x) (0 ≤ i ≤ m) are elements of k[x] which are divisible by xn . (2) If n = 1, Y has another Ga -action ψ as above with the roles of x, y exchanged. Hence ML(Y ) = k. (3) If n ≥ 2, then every Ga -action is of the form given in (1) above. In particular, ML(Y ) = k[x]. Proof. (1) Since ϕ(x)ϕ(y) = ϕ(z)p − ϕ(z), we put the data ϕ(x) = x and ϕ(z) = m i z + i=0 ai (x)tp . Then ϕ(y) is accordingly determined by the given formula. The rest is straightforward. (2) is clear. (3) We follow the arguments on weights and homogenization by Crachiola [2, Theorem 2.3]. Let Y = SpecB, where B = k[x, y, z] with z p −z = xn y. Consider the weight on B defined by wt(x) = 0, wt(y) = p and wt(z) = 1. Then the weight defines a Z-filtration on B and the associated graded ring gr(B) is an affine domain k[x, y, z] with xn y = z p . Then the singular locus of Spec (gr(B)) is defined by the ideal (x). Since the singular locus is stable under the associated Ga -action ϕ, we have ϕ(x) = xu, where u is a unit in the domain gr(B)[t] for a variable t. Note that (gr(B))[t]∗ = gr(B)∗ = k∗ . Hence u = 1

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because ϕ(x)|t=0 = x. So, the Ga -invariant subring of gr(B) is (gr(B))ϕ = k[x]. This implies that x is ϕ-invariant because an element of weight < 0 is 0. Since ϕ is a Ga -action on B, B ϕ is a normal affine domain of dimension one containing a polynomial ring k[x]. Since B[x−1 ] = B ϕ [x−1 , z] = k[x, x−1 , z], it follows that Q(k[x]) = Q(B ϕ ). Since B ϕ is an affine normal domain of dimension one and (B ϕ )∗ ⊆ (k[x, x−1 , z])∗ ∩ B ∗ = k ∗ , we conclude that B ϕ = k[x]. Furthermore, since Y has a nontrivial Ga -action by (1), it follows that ML(Y ) = k[x]. 2 Z-actions on the affine plane 3. Wild Z /pZ We consider a nontrivial G-action σ on A2 = Spec k[x, y]. A final answer is given by Theorem 3.5 due to S. Kuroda. Our proof depends on the structure theorem on AutA2 /k and reduces to the case where σ induces an automorphism of A2 of de Jonquière type. We need to prepare several results assuming the following hypothesis (A). Hypothesis A. The coordinates x, y are chosen in such a way that the A1 -fibration ρ on A2 given by (x, y) → x is preserved by the G-action. Namely the translates of the fibers of ρ are again the fibers of ρ. The assumption implies that there is an induced G-action on A1 = Spec (k[x]). If the induced action on A1 is trivial, then σ(x, y) = (x, a(x)y + b(x)) with a(x) ∈ k[x]∗ and b(x) ∈ k[x]. Since a(x)p = 1, we have a(x) = 1 and the action is σ(x, y) = (x, y + b(x)). The invariant subring is then A := k[x, y]G = k[x, y p − b(x)p−1 y]. Lemma 3.1. If the induced action on A1 = Spec k[x] is nontrivial, we can normalize the G-action on A1 so that σ(x) = x + 1. Proof. The condition that the induced action on A1 is nontrivial is equivalent to the condition that there is a fiber F of ρ such that σ(F ) ∩ F = ∅. We may assume that F is defined by x = 0. Let f = 0 be the defining equation of σ(F ). Write f = xh(x, y) + g(y) / k then F ∩ σ(F ) = ∅. So, g(y) = c ∈ k∗ . Since {σ(F ) | with g(y) ∈ k[y]. If g(y) ∈ −1 F = ρ (λ), λ ∈ k} defines an A1 -fibration whose fibers are all irreducible and reduced, f − c = xh(x, y) is irreducible. Hence h(x, y) is a constant. We may write σ(x) = ax + c with a ∈ k∗ and c ∈ k. Then ap = 1, whence a = 1. Since the action is nontrivial, c = 0. Then, by replacing x by c−1 x, we may assume that σ(x) = x + 1. 2 From now on until Theorem 3.5, we assume the hypothesis (A). Lemma 3.2. Suppose that the induced G-action on k[x] is nontrivial. Then the following assertions hold. (1) σ(x, y) = (x + 1, y + a(x)), where a(x) ∈ k[x] such that a(x) + a(x + 1) + · · · + a(x + p − 1) = 0

(∗)

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(2) The polynomial a(x) in the above relation (∗) is an element of a k-vector subspace V of k[x] which is a direct sum V = V1 ⊕ V2 , where (i) V1 is generated freely by monomials xi such that if i = i0 + i1 p + · · · + in pn is the p-adic expansion of i then i0 + i1 + · · · + in < p − 1. We denote by N1 the set of natural numbers i which have the p-adic expansions satisfying the said condition and by N2 the set of natural numbers which do not belong to N1 . (ii) Let V2 be the subspace of k[x] generated by xi with i ∈ N2 . Then V2 is the kernel of the vector space homomorphism ϕ : V2 → k(x) defined by ϕ(xi ) =

p−1  (x + )i =0

=



(−1) ·

0
    i0 in ··· · xi−j j0 jn

with i ∈ N2 , where j ≤p i if and only if j0 ≤ i0 , . . . , jn ≤ in for the p-adic expansion j = j0 + j1 p + · · · + jn p n Proof. (1) Since the G-action preserves the A1 -fibration ρ, we can write σ(x, y) = (x + 1, α(x)y +a(x)), where α(x) ∈ k[x]∗ and a(x) ∈ k[x]. Thus α(x) ∈ k∗ and α(x) = 1 because σ p = 1. It is clear that a(x) satisfies the condition (∗). (2) Suppose that p > 2. Let i ∈ N and let i = i0 + i1 p + · · · + in pn

(0 ≤ ij < p)

be the p-adic expansion of i. By setting 00 = 1, we then compute p−1 

(x + )i =

=0

p−1 

n

(x + )i0 (xp + )i1 · · · (xp + )in

=0

  in    i0 in j0 +j1 +···+jn (i0 −j0 )+(i1 −j1 )p+···+(in −jn )pn ···  x j0 jn jn =0 =0 j0 =0 p−1

    i0 in    i0 in (i0 −j0 )+···+(in −jn )pn j0 +j1 +···+jn = ···  ··· x j0 jn j =0 j =0 =

p−1  i0 

0

···

n

=1

By the following lemma, noting that n

j0 +j1 +···+jn ≡ j0 +j1 p+···+jn p ≡ j

(mod p)

and that j ≡ 0 (mod (p − 1)) if and only if j0 + j1 + · · · + jn ≡ 0 (mod (p − 1)), we have

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p−1  (x + )i =



(−1) ·

0
=0

    i0 in i−j ··· x j0 jn

Thus a(x) is in the kernel of the vector space homomorphism ϕ(x) : k[x] → k[x] defined by ϕ(xi ) =

p−1  (x + )i . =0

In fact, if xi ∈ V1 then ϕ(xi ) = 0. So, V1 ⊆ Ker ϕ. Since k[x] = V1 ⊕ V2 , it follows that Ker ϕ = V1 ⊕ V2 . Suppose that p = 2. Then a(x) is in the kernel of the vector space morphism ϕ : k[x] → k[x] defined by ϕ(xi ) = xi + (x + 1)i . Let i = i0 + i1 2 + · · · + in 2n be the 2-adic expansion. Then it is easily checked that xi + (x + 1)i ≡



xi−j

(mod 2)

0
This is identified with the formula (∗) in the case p = 2. 2 Lemma 3.3. Suppose that p > 2. Let i ∈ N. Then we have p−1 



if i ≡ 0

(mod (p − 1))

−1 if i ≡ 0

(mod (p − 1))

0

i

 =

=1

Proof. Since p−1 ≡ 1 (mod p), it is clear that p−1 

i ≡ p − 1 ≡ −1 (mod p)

if

i ≡ 0 (mod (p − 1)) .

=1

We assume that i ≡ 0 (mod (p − 1)). Let d = (i, p − 1) and write p − 1 = dr, where r > 1. If d = 1 then {i | 1 ≤  < p} = { | 1 ≤  < p}. Hence we have p−1 

i =

=1

1 p(p − 1) ≡ 0 (mod p). 2

Suppose d > 1. Then d is the r-th root of 1 because p−1 = (d )r = 1. Hence, writing i = dj, we have p−1  =1

i =

p−1  (d )j = d =1



(0 )j ,

0 ∈{r-th roots of 1}

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where 0 := d moves d times over all r-th roots of 1 if  moves over { | 0 <  < p}. r Since (j, r) = 1, then j0 moves over all r-th roots of 1. Since j=1 (x − j0 ) = xr − 1, it follows that 

j

(0 ) =

0 ∈{r-th roots of 1}

Hence we conclude that

p−1 =1

r 

j0 = 0.

j=1

i ≡ 0 (mod (p − 1)).

2

The following result determines a polynomial a(x) satisfying the condition (∗) in more explicit terms. Theorem 3.4. With the same assumption as in Lemma 3.2, let a(x) ∈ k[x] be a polynomial satisfying the condition (∗). Then there exists a polynomial b(x) ∈ k[x] such that a(x) = b(x) − b(x + 1). Hence the following sequence of k-vector spaces ψ

ϕ

k[x] −→ k[x] −→ k[x] is exact, where

ϕ(f (x)) =

p−1 

f (x + )

and

ψ(g(x)) = g(x) − g(x + 1).

=0

Proof. A proof given below is more geometric. A more straightforward proof is to apply Lemma 2.3 to the free G-action on k[x] because ϕ = 1 + σ + · · · + σ p−1 and ψ = 1 − σ. Let Y = Spec k[x, y], X = Y //G and q : Y → X the quotient morphism, where X = Spec A with A = k[x, y]G . Since G acts freely on Y , the quotient morphism q is a finite, étale morphism of degree p. Hence X is smooth. Since the action of G on Y preserves the A1 -fibration ρ : Y → A1 = Spec k[x], the images of the fibers of ρ by q form a pencil L = {q(F ) | F = q(ρ−1 (λ)), λ ∈ k} which is parametrized by A1 = Spec k[u], where u = xp − x. Let F η be the generic fiber of the pencil L. Then F η ⊗k(u) k(x) is the generic fiber of ρ. Since the extension k(x)/k(u) is separable algebraic and the generic fiber of ρ is isomorphic to A1k(x) , F η is isomorphic to A1k(u) . Hence the pencil L defines an A1 -fibration ρ : X → A1 = Spec k[u]. Since every closed fiber of ρ is the image of a disjoint union of p fibers of ρ which are isomorphic to A1 , it follows that every closed fiber of ρ is reduced and isomorphic to A1 . This entails that X is isomorphic to A2 = Spec k[u, y1 ]. Since k[x, y] is a finite extension of A of degree p, it follows that k[u, y1 ] ⊗k[u] k[x] ∼ = k[x, y]. Namely we have k[x, y] = k[x, y1 ], and both y and y1 are the fiber coordinates of the fibration ρ. So, y1 = α(x)y + b(x) with α(x), b(x) ∈ k[x]. Since (x, y) → (x, y1 ) is an automorphism of k[x, y], it follows that α(x) = c ∈ k[x]∗ = k∗ . Replace y1 by c−1 y1 . We may assume that α(x) = 1. Since σ(y1 ) = y1 , we have

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y + b(x) = y1 = σ(y1 ) = σ(y + b(x)) = y + a(x) + b(x + 1), which gives a relation a(x) = b(x) − b(x + 1). This implies that Ker ϕ ⊆ Im ψ. Since Im ψ ⊆ Ker ϕ is clear, we have Ker ϕ = Im ψ. 2 We next give a proof of the following theorem of S. Kuroda by making use of results in [12]. Note that the hypothesis (A) is used in the proof only in a partial case where we use Theorem 3.4. Theorem 3.5. A G-action on the affine plane A2 is given by σ(x, y) = (x + a(y), y) with respect to a suitable system of coordinates {x, y} and a(y) ∈ k[y]. Proof. Write G = σ. By [12, Lemma 3.5.3] and with the notations therein, σ ∈ A2 or σ ∈ J2 . We show that the case σ ∈ A2 is reduced to the case σ ∈ J2 . Write ⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎪ ⎪ x⎪ a b⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪x⎪ ⎪+⎪ ⎪α⎪ ⎪, σ⎪ ⎩y⎪ ⎭=⎪ ⎩c d⎪ ⎭⎪ ⎩y⎪ ⎭ ⎪ ⎩β⎪ ⎭

⎧ ⎫ ⎪ a b⎪ ⎪ ⎪ ∈ GL(2). ⎪ ⎩c d⎪ ⎭

Then we have ⎧ ⎫ ⎧ ⎫i ⎧ ⎫ ⎧ ⎫i−1 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ x⎪ x⎪ α⎪ a b⎪ a b⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ ⎪ ⎩y⎪ ⎩y⎪ ⎩β⎪ ⎭=⎪ ⎭+⎪ ⎭ ⎩c d⎪ ⎩c d⎪ ⎭ ⎪ ⎭ ⎧ ⎧ ⎫ ⎫i−2 ⎧ ⎫ ⎪ ⎪ ⎪ α⎪ a b⎪ ⎪ ⎪ ⎪ ⎪α⎪ ⎪ ⎪. ⎪ +⎪ ⎩β⎪ ⎭ + ··· + ⎪ ⎩c d⎪ ⎩β⎪ ⎭ ⎭ i

Since σ p = 1, we have the relation (1): ⎧ ⎫p ⎧ ⎫ ⎪ ⎪ a b⎪ 1 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩c d⎪ ⎩0 1⎪ ⎭ =⎪ ⎭ and the relation (2):

⎧ ⎧ ⎫p−1 ⎧ ⎫p−2 ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a b a b 1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪α⎪ ⎪=⎪ ⎪0⎪ ⎪. ⎪ ⎪ ⎪ +⎪ + ··· + ⎪ ⎩c d⎪ ⎩c d⎪ ⎩0 1⎪ ⎭ ⎭ ⎭ ⎪ ⎩β⎪ ⎭ ⎪ ⎩0⎪ ⎭ By the relation (1) and by the Jordan canonical form, we may assume that ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ a b⎪ 1 b⎪ ⎪ ⎪ ⎪ ⎪. ⎪ ⎩c d⎪ ⎭=⎪ ⎩0 1⎪ ⎭ Then the relation (2) is written as

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⎧ ⎧ ⎫p−1 ⎧ ⎫p−2 ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ 1 b⎪ 1 b⎪ 1 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪α⎪ ⎪ ⎪ +⎪ + ··· + ⎪ ⎩0 1⎪ ⎩0 1⎪ ⎩0 1⎪ ⎭ ⎭ ⎭ ⎪ ⎩β⎪ ⎭

⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 (p − 1)b 1 (p − 2)b 1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪α⎪ ⎪ ⎪ ⎪ ⎪ = ⎪ ⎩0 ⎭+⎪ ⎩0 ⎭ + ··· + ⎪ ⎩0 1⎪ ⎭ ⎪ ⎩β⎪ ⎭ 1 1 ⎫⎧ ⎫ ⎧ ⎫ ⎧ p(p−1) ⎪α⎪ ⎪0⎪ ⎪ b⎪ 0 ⎪ ⎪. ⎪ 2 ⎪⎪ ⎪ ⎪ ⎪=⎪ ⎪ =⎪ ⎭ ⎩ ⎭ ⎩0⎪ ⎭ ⎩ β 0 0 Hence we have b = 0 if p = 2 and β = 0. If αβ = 0 then replace x by x = βx − αy. Since σ(x) = x + by + α, σ(y) = y + β, we have σ(x ) = x + βby and σ(y) = y + β. Hence we may assume that α = 0. If β = 0, replace y by y  = y/β and assume that σ(y) = y + 1. If p = 2 and β = 0, then σ(x, y) = (x, y + 1). If p = 2, then one can find a quadric polynomial c(y) such that by = c(y) − c(y + 1). Set x = x + c(y). Then σ(x ) = x . If α = β = 0 then σ(x) = x + by and σ(y) = y. This is a required form of σ. If σ ∈ J2 then σ(x) = αx + a(y) and σ(y) = βy + γ, where α, β ∈ k∗ , γ ∈ k and a(y) ∈ k[y]. Since σ p = 1, it follows that αp = β p = 1, whence α = β = 1. If γ = 0, then σ(x) = x + a(y) and σ(y) = y. This is the required form of σ. Assume that γ = 0. Replacing y by yγ −1 , we may assume that σ(y) = y + 1. Then σ(x) = x + a(y) and σ(y) = y + 1. This implies that σ preserves the A1 -fibration (x, y) → y. By Theorem 3.4, there exists b(y) ∈ k[y] such that a(y) = b(y) − b(y + 1). Set x = x + b(y). Then σ(x ) = x . So, by taking coordinates (y, x ) instead of (x, y), we have σ(x) = x + 1 and σ(y) = y. 2 Corollary 3.6. With the notations in Theorem 3.5, the G-invariant subring of B := k[x, y] is a polynomial ring A := k[xp − a(y)p−1 x, y]. Proof. Note that t := xp − a(y)p−1 x is the norm of x under the G-action. It is clear that Q(B)G = Q(B G ) = k(t, y) and B G = B ∩ Q(B G ), where Q(R) denotes the quotient field of a domain R. Since B is a finite A-module, B G = B ∩ Q(B G ) is a finite A-module as well. Since Q(A) = Q(B G ) and since A and B G are integrally closed, it follows that A = BG. 2 4. Artin–Schreier coverings of the affine plane over exotic lines In the case of positive characteristic, a curve C on A2 isomorphic to A1 needs not be a coordinate line of A2 contrary to the characteristic zero case. Such a curve C, if not a coordinate line, defines a fibration ρ : A2 → A1 = Spec k[f ] with C defined by f = 0 such that the generic fiber of ρ has a unique place whose residue field is purely inseparable over k(f ). According to Ganong [4], we call such a line C an exotic line of A2 . An Artin–Schreier covering Y of A2 defined by z p − z − f = 0 is said to be defined over the curve C.

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Let K be a field of positive characteristic p. An algebraic curve X defined over K is a K-form of the affine line if there exists a finite algebraic extension K  of K such that X ⊗K K  is K  -isomorphic to A1 . It is known that K  can be taken as a purely inseparable extension of K. The height of X is the least non-negative integer n such that −n X ⊗K K p ∼ = A1,K p−n (see [6] for the relevant definitions and results). We prove the following result. Theorem 4.1. Let C be an affine plane curve defined by a parametrization r

x = t + θ(tp )

s

y = tp

and

where r, s are positive integers with s > r and θ(t) ∈ k[t] with n := deg θ(t) not divisible by p and greater than 1. Then the following assertions hold. (i) The curve C is defined by f = 0, where s

r

s

f = y + θ(p ) (y p ) − xp s

s

s

with θ(p ) (tp ) = θ(t)p . (ii) The curve Cλ defined by f = λ is isomorphic to A1 for all λ ∈ k. (iii) The generic fiber Cη = Spec k(f )[x, y] is a nontrivial form of A1 with height less than or equal to s − r. Define an Artin–Schreier covering Y = Spec B of X = Spec k[x, y] defined by B = k[x, y, z], where s

r

s

z p − z = f = y + θ(p ) (y p ) − xp . Introduce inductively elements z1 , z2 , . . . , zr by z + y = z1p , z1 + y = z2p , . . . , zr−1 + y = zrp . Let u = zr . Then the following assertions hold. (1) z1 , z2 , . . . , zr are elements of B such that B = k[x, y, z1 ] = k[x, y, z2 ] = · · · = k[x, y, zr ] = k[x, y, u], where u satisfies a relation s−r

up − u = −xp

+ y + θ(p

s−r

)

(y) .

(∗)

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s−r−1

(2) Let v = u + xp

379

. Then v satisfies a relation s−r−1

xp

= v − v p + y + θ(p

s−r

)

(y).

 s−r−1  Hence we have k xp , y, u = k[v, y] and B is a purely inseparable extension of k[v, y] of degree ps−r−1 . The G-action on B is given by σ(z) = z + 1, σ(zi ) = zi + 1 (1 ≤ i ≤ r) and σ(v) = v + 1 and the G-action descends to k[v, y] so that B G = k[x, y] is a purely inseparable extension of k[v, y]G of degree ps−r−1 . (3) If s = r + 1, then B is a polynomial ring k[v, y]. Proof. The assertions (i), (ii) and (iii) are proved in [17, Theorem 4.4]. We prove the assertions (1), (2) and (3). (1) By the relation (∗) we have s

r

s

z + y = z p − θ(p ) (y p ) + xp   s−1 r−1 s−1 p = z − θ(p ) (y p ) + xp . Let z1 = z − θ(p

s−1

)

r−1

(y p

s−1

) + xp

. Then z + y = z1p and

z1p − z1 = y + θ(p

s−1

)

r−1

(y p

s−1

) − xp

.

It is clear that k[x, y, z] = k[x, y, z1 ]. Similarly, by the relation (∗∗), we have s−1

r−1

s−1

z1 + y = z1p − θ(p ) (y p ) + xp   s−2 r−2 s−2 p = z1 − θ(p ) (y p ) + xp . Set z2 = z1 − θ(p

s−2

)

r−2

(y p

s−2

) + xp

. Then z1 + y = z2p ,

z2p − z2 = y + θ(p

s−2

)

r−2

(y p

s−2

) − xp

and k[x, y, z1 ] = k[x, y, z2 ]. Arguing inductively, we have a relation   s−r s−r p zr−1 + y = zr−1 − θ(p ) (y) + xp . Set zr = zr−1 − θ(p

s−r

)

s−r

(y) + xp

. Then zr−1 + y = zrp ,

zrp − zr = y + θ(p

s−r

)

s−r

(y) − xp

and k[x, y, zr−1 ] = k[x, y, zr ]. So, if we set u = zr , we obtain a relation

(∗∗)

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380

s−r

up − u = −xp s−r−1

Finally put v = u + xp

+ y + θ(p

s−r

)

(y).

. Then we have

v p = u + y + θ(p

s−r

s−r−1

= v − xp

)

(y)

+ y + θ(p

s−r

)

(y).

So, we have s−r−1

xp

= v − v p + y + θ(p

s−r

)

(y).

s−r−1

Here we have k[xp , y, u] = k[v, y]. Since the G-action on B is given by σ(z) = z + 1, it is straightforward to show that σ(zi ) = zi + 1 (1 ≤ i ≤ r) and σ(v) = v + 1. Hence the G-action descends to k[v, y] and s−r−1

k[v, y]G = k[v p − v, y] = k[−xp

+ y + θ(p

s−r

)

s−r−1

(y), y] = k[xp

, y].

So, B G = k[x, y] is a purely inseparable extension of k[v, y]G of degree ps−r−1 . If s = r+1, then B = k[x, y, u] = k[v, y] which is a polynomial ring. 2 As a special case, we consider the example by M. Nagata. Example 4.2. With the notations in Theorem 4.1, let θ(t) = tn , r = 1 and s = 2. Then f = y − (xp − y n )p . We have the following assertions. (1) Let B = k[x, y, z]. Then Y := Spec B is a plane sandwich. (2) Write B = k[v, y] as in Theorem 4.1. Then the G-action on B is given by σ(v) = v +1 and σ(y) = y. The invariant subring B G is k[v p − v, y] = k[x, y]. Proof. (1) Let ξ = xp −y n and η = y −(xp −y n )p . Then y = ξ p +η and xp = ξ +(ξ p +η)n . p Hence k[ξ, η] = k[xp , y]. Let z  = z 1/p and ξ  = ξ 1/p . Then η = z p − z = (z  − z  )p , p whence η 1/p = z  − z  . We have y = ξ p + η = ξ p + (z  − z  )p = (ξ  + z  − z  )p p

p

p

x = ξ  + (ξ  + z  − z  )n p

p

because xp = ξ + y n = ξ  + (ξ  + z  − z  )pn . Hence we have p

p

p

B = k[x, y, z] = k[ξ  + (ξ  + z  − z  )n , (ξ  + z  − z  )p , z  ] ⊂ k[ξ  , z  ]. p

p

p

p

p

Since k[x, y] ⊃ k[xp , y] = k[ξ, η] is a purely inseparable extension of degree p and since B = k[x, y, z] = k[ξ, η, z] ⊗k[ξ,η] k[x, y] and k[ξ, η, z] = k[ξ, z] as z p − z = η, it follows

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that B ⊃ k[ξ, z] is a purely inseparable extension of degree p. This implies also that k[ξ  , z  ] ⊃ B is a purely inseparable extension of degree p. Hence Y = Spec B is a plane sandwich. (2) We set u = xp − y n + z and v = x + u. Then x = v − v p + y + y n . Since the G-action on B is given by σ(z) = z + 1, σ(x) = x and σ(y) = y, it follows that σ(u) = u + 1 and σ(v) = v + 1. Hence B G = k[v p − v, y]. 2 For a curve C(f ) on A2 = Spec k[x, y] defined by f = 0 with f ∈ k[x, y], we denote by B(f ) the Artin–Schreier extension of k[x, y] given by B(f ) = k[x, y, z] with z p −z = f . For curves C(f ) and C(g) on A2 , we say that C(f ) and C(g) (or f and g) are AS-equivalent if B(f ) is isomorphic to B(g), i.e., Spec B(f ) is isomorphic to Spec B(g) without regard to Z/pZ-torsors over Spec A. In the above example of Nagata, the curve C defined by the equation f = y − (xp − y n )p is AS-equivalent to the affine line, while the curve C is not transformed to the line by an automorphism of A2 . It is an interesting problem to clarify the AS-equivalence class of a given curve on A2 . Let c ∈ k. Then αp − α = c for some α ∈ k. By replacing z by z + α, the curve C(f ) is AS-equivalent to C(f + c). Similarly, C(f ) is AS-equivalent to C(f + h − hp ) for any h ∈ k[x, y]. Meanwhile, if λ ∈ k∗ , we do not know if C(f ) is AS-equivalent to C(λf ). In special cases, e.g., f is a weighted homogeneous polynomial in x, y with certain weights, it is the case. Z-action on a smooth projective surface 5. Z /pZ We shall show the following result. See [16] as a reference. Theorem 5.1. Let V be a smooth rational projective surface with a G := Z/pZ-action. Then V has a G-fixed point. Proof. We show that the assumption that V has no G-fixed points leads to a contradiction. So, we assume that V has no G-fixed points. Let W be the quotient surface V //G and let q : V → W be the quotient morphism. Then q is a finite étale morphism, whence W is also a smooth surface. This implies that KV = q ∗ KW . If E is a (−1)-curve on W , we show that q ∗ (E) consists of disjoint p curves q ∗ (E) = C1 + · · · + Cp , where G acts transitively on the set {C1 , . . . , Cp }. In fact, otherwise, either q ∗ (E) = C or q ∗ (E) = pC for an irreducible curve C. If q ∗ (E) = pC then q|C : C → E is an isomorphism and C is contained in the G-fixed point locus. This contradicts the assumption. Hence C = q ∗ (E) and C is a smooth curve. Then C · KV = q ∗ (E) · q ∗ (KW ) = p(E · KW ) = −p and C 2 = (q ∗ E)2 = p(E 2 ) = −p. The adjunction formula implies C 2 + C · KV = −2p = 2g − 2, where g = g(C). This is impossible. Write q ∗ (E) = C1 +· · ·+Cp , where G acts transitively on the set {C1 , . . . , Cp }. We show that C1 , . . . , Cp are disjoint from each other. Let

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(C1 ·C2 +· · ·+Cp ) = N . Then the G-transitivity shows that every Ci meets the remaining p − 1 components with multiplicity N . Then we have p(E 2 ) = q ∗ (E)2 = (C1 + · · · + Cp )2 =

p  

 (Ci )2 + (Ci · C1 + · · · + Ci−1 + Ci+1 + · · · + Cp )

i=1

= p(n + N ), where n = (C12 ). Then we have −p = p(n + N ), i.e., −1 = n + N . Since N ≥ 0, it follows that n < 0. On the other hand, since p(C1 · KV ) = (q ∗ (E) · KV ) = p(E · KW ) = −p, we have (C1 · KV ) = −1. Hence C1 as well as the other Ci is a (−1)-curve on V . This implies that N = 0 because n = −1. Hence the components C1 , . . . , Cp are mutually disjoint. Then we can contract C1 , . . . , Cp simultaneously to obtain a surface V1 which inherits naturally the G-action on V . It is clear that the induced G-action on V1 has no fixed points. Hence the quotient surface W1 := V1 //G is smooth and the quotient morphism q1 : V1 → W1 is finite and étale. Repeating the argument, we may assume 2 2 that W is relatively minimal. Hence KW = 8 or 9. Then KV2 = p(KW ) ≤ 9 since V is rational by the hypothesis. This is a contradiction. Hence there is a G-fixed point on V . 2 Remark 5.2. (1) Let W = P1 × P1 with the G-action defined by σ((X0 , X1 ), (Y0 , Y1 )) = ((X0 , X1 + X0 ), (Y0 , Y0 + Y1 )), where (X0 , X1 ) (resp. (Y0 , Y1 )) is the homogeneous coordinates on P1 . Let  (resp. M ) be the line defined by X0 = 0 (resp. Y0 = 0). Set x = X1 /X0 and y = Y1 /Y0 . Then {x, y} is a system of coordinates on the open set U0 = W \ ( ∪ M ) which is isomorphic to A2 . The induced G-action on U0 is given by σ(x, y) = (x + 1, y + 1). On the lines  and M the point Q∞ := ((0, 1), (0, 1)) is the unique G-fixed point. So, the G-fixed point locus on W consists of a single point Q∞ . Let V := W//G which is a normal projective surface and let q : W → V be the quotient morphism. Then the point P∞ := q(Q∞ ) is the isolated singular point of V . In fact, if V were smooth at P∞ , the purity of branch loci implies that q is a finite étale morphism and hence there are p distinct points over P∞ . But this is not the case. So, P∞ is a unique singular point of V . (2) Let A be an abelian surface with positive p-rank. Let G be a subgroup of p-torsion points of A. Then the translation of A by the elements of G defines a G-action on A without G-fixed points.

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(3) The fixed point locus is not necessarily connected. In fact, we construct a smooth algebraic curve with G-action such that the fixed point locus has two or more points. Let C be an Artin–Schreier covering of A1 which is defined by an equation y p − y = c0 tn + · · · + cn−1 t, where c0 , . . . , cn−1 ∈ k, c0 = 0, n > 0, and ci = 0 whenever i ≡ 0 (mod p). Namely, C is the normalization of P1 in the function field k(t, y), where t is an inhomogeneous coordinate of P1 . Then C is totally ramified over the point P∞ defined by 1/t = 0 and C has geometric genus (p − 1)(n − 1)/2 (see [13, Lemma 3.1]). Let  be a prime number not equal to p and let D be a cyclic covering of P1 of degree  totally ramified over two points P0 , P1 other than P∞ . Then the composite of k(C) and k(D) over k(P1 )  is the function field of an irreducible, reduced curve C ×P1 D, which we denote by C.  Then C is a smooth projective curve with G-action which is the natural lifting of the  → D is the quotient morphism by G-action on C. Hence the second projection p2 : C  the G-action. Similarly, C has a Z/Z-action which is the lifting of that on the curve D. (1) ()  has  distinct points P∞ Then C , . . . , P∞ over P∞ which are G-fixed points. This is a desired example. Let A be a normal affine domain over k and let B = A[z] with z satisfying a relation z p − sz = a, where s, a ∈ A. Let X = Spec A, Y = Spec B and q : Y → X defined by the inclusion A → B. Set K0 = Q(A). Lemma 5.3. With the above notations, the following assertions hold. (1) Suppose s = 0. Then B is reduced, and B is reducible if and only if there exist a separable algebraic extension K1 of K0 and an element a1 of K1 such that the minimal equation of K1 over K0 divides the equation T p−1 = s and a = ap1 − sa1 . (2) B has a G-action such that σ(z) = z + d with d ∈ A \ (0) and σ|A = id if and only if s = dp−1 . (3) Suppose s = 0. Then there exists a flat finite extension A → A1 such that B1 := B ⊗A A1 has a G-action over A1 . If s ∈ A∗ then A1 is a finite étale extension of A. Proof. (1) Since the defining equation of an A-algebra B is z p − sz = a, B is a free A-module of rank p. Then the canonical ring homomorphism B → B ⊗A K0 , b → b ⊗ 1, is an inclusion. Hence if B ⊗A K0 is reduced, so is B. Let s1 be an element of an algebraic closure of K0 such that sp−1 = s and let K1 = K0 (s1 ). Since the equation 1 p−1 T − s = 0 is separable, the extension K1 /K0 is separable algebraic. The minimal polynomial Φ(T ) of the extension K1 /K0 divides T p−1 − s. Since A is normal and all roots of T p−1 − s = 0 are integral over A, Φ(T ) is a monic polynomial with coefficients in A. Then B1 := B ⊗A K1 = K1 [z] with z p − sp−1 z = a. Then there exists a G-action on 1 B1 defined by σ(z) = z+s1 and σ|K1 = id. Compare the difference with Lemma 2.5. Since

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B1 is a finite K1 -algebra of rank p with a free G-action, B1 is either a field extension of Artin–Schreier type or a direct sum of p-copies of K1 as a K1 -algebra. In fact, if Spec B1 has two or more points, then G acts transitively on Spec B1 and Spec B1 consists of p points rational over K1 . Since B1 is reduced in either case, so is B. Suppose that B1 is reducible, then each irreducible component of B1 is a connected component isomorphic to K1 . Then z = a1 ∈ K1 on one component. Then z = a1 + is1 for other connected components of B1 , where i ∈ {1, 2, . . . , p − 1}. Then we have ap1 − sa1 = a. It is clear that the converse holds. (2) If B has a G-action σ(z) = z + d with d ∈ A \ (0), then we have dp = sd, whence s = dp−1 . The converse also holds. (3) With the notations in the proof of the assertion (1), let A1 = A[s1 ], where Φ(s1 ) = 0 and Φ(T ) is a monic polynomial with coefficients in A. Then A1 is a free A-module and hence A1 is a finite flat extension of A. Let B1 = B ⊗A A1 = A1 [z] with z satisfying the equation z p − sp−1 z = a. It is clear that one can define a G-action on B1 1 by σ(z) = z + s1 . If s ∈ A∗ , then A1 is étale over A. 2 Let B be an affine k-domain with a free G-action. Then, by [7], B = A[z], where z p − z = a ∈ A and A = B G . When we drop the assumption that the G-action is free, we may ask if B is written as B = A[z] with elements s, a ∈ A, where z p − sz = a and A = BG. Lemma 5.4. Let B be a normal affine k-domain with a nontrivial G-action and let A := B G . Suppose that B is written as A[z] with z satisfying a relation z p − sz = a, where s, a ∈ A. Then the following assertions hold. (1) s = 0. (2) The group G acts as σ(z) = z + d, where d ∈ A, d = 0 and dp−1 = s. (3) The fixed point locus of the G-action on Y = SpecB is the closed set V (s) = {s = 0}. Hence either G acts freely on Y or the fixed point locus has codimension one in Y . Proof. (1) If s = 0 then σ(z) = z. So, the G-action is trivial, and this is against the assumption. (2) We have σ(z)p − sσ(z) = a, whence (σ(z) − z)p = s(σ(z) − z). Since the G-action on B is nontrivial, σ(z) − z = 0. So, (σ(z) − z)p−1 = s. Let z1 = σ(z) − z. If σ(z1 ) = z1 , then {σ i (z1 ) | 0 ≤ i ≤ p − 1} is a set of p elements. In fact, suppose that σ i (z1 ) = σ j (z1 ) with i < j. Then σ j−i (z1 ) = z1 , where 0 < j − i < p, and the stabilizer group of z1 is not trivial, and hence is G itself. This is a contradiction. Since (σ i (z1 ))p−1 = s for 0 ≤ i ≤ p − 1. This is absurd because an algebraic equation of degree p − 1 has distinct p roots. So, σ(z1 ) = z1 and z1 ∈ A. This shows that σ(z) = z + d with d ∈ A. By Lemma 5.3, (3), we have s = dp−1 . (3) The fixed point locus on Y is the set {P ∈ Y | d(P ) = 0}. Namely, it is the set {P ∈ Y | s(P ) = 0}. 2

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Let Y = Spec B be a normal affine variety over k endowed with a G-action. If B is written as B = A[z], where A = B G and z p − sz = a with s, a ∈ A, we say that Y is of quasi-Artin–Schreier type. The next result shows that Y is not necessarily of quasi-Artin–Schreier type. Proposition 5.5. Let W be a normal projective surface with a nontrivial G-action. Assume that the G-fixedpoint locus consists of a single isolated point Q. Let H be an effective p−1 very ample divisor on W such that Q ∈ / H and let D = i=0 σ i (H). Let Y = W \ D. Then Q ∈ Y and Y is not of quasi-Artin–Schreier type. / H and Q is a fixed point, Q ∈ / σ i (H) for every 0 ≤ i < p. Hence Proof. Since Q ∈ Q∈ / D. Since D is a G-stable, very ample divisor, Y is a normal affine surface with a nontrivial G-action and with an isolated fixed point. If Y is of quasi-Artin–Schreier type, the G-fixedpoint locus is either the emptyset or a curve by Lemma 5.4. This is not the case. Hence Y is not of quasi-Artin–Schreier type. 2 The surface W = P1 × P1 with a G-action constructed in Remark 5.2 satisfies the conditions for W in Proposition 5.5. Notwithstanding this example, we have the following positive result. Theorem 5.6. Let Y = Spec B be a factorial affine variety such that Y has a nontrivial G-action and B ∗ = k∗ . Let A := B G , let X = Spec A and let q : Y → X be the quotient morphism. Assume that the fixed point locus Γ has pure codimension one. Then Y is the normalization of an affine variety Y0 of quasi-Artin–Schreier type. Proof. Let Γ = Γ1 + · · · + Γn be the irreducible decomposition and write Γi = V (si ), where si is a prime element of B. Let t = s1 · · · sn . Since Γi is G-stable, σ(si ) = ui (σ)si , where ui (σ) ∈ B ∗ . Since B ∗ = k∗ by the assumption, ui (σ) ∈ k and hence ui (σ)p = 1. This implies that si ∈ A. Let Y  = Y \ Γ and X  = X \ q(Γ). Then Y  = Spec B[t−1 ] and X  = Spec A[t−1 ]. Since the G-action on Y  is free and B[t−1 ]G = A[t−1 ], it is possible to write B[t−1 ] = A[t−1 ][z  ], (z  )p − z  = a , where a ∈ A[t−1 ]. Write z  = zs−1 , where mn 1 s = sm with mi ≥ 0 and z ∈ B such that z and si are coprime for every i. 1 · · · sn In fact, we need to prove that mi ≥ 0. We consider the case i = 1 and assume that mi < 0. Let Y1 = Y \ (Γ2 ∪ · · · ∪ Γn ). Since (z  )p − z  = a and σ(a ) = a , we have (σ(z  ) − z  )p = (σ(z  ) − z  ). Hence σ(z  ) = z  + i with 0 < i < p. Replacing z  by z  /i, 1  we may assume that σ(z  ) = z  + 1. Write z  = s−m z with z  ∈ B1 := Γ(Y1 , OY1 ). 1     1 1 Since Y1 is G-stable, σ(z ) − z ∈ B1 , while σ(z ) = z + sm and sm / B1 as m1 < 0. 1 1 ∈   This is a contradiction. If m1 = 0, then σ(z ) = z + 1 and the G-action is nontrivial on the component Γ1 . This is also a contradiction. Hence mi > 0 for every 1 ≤ i ≤ n. Plugging z  = zs−1 in the equation (z  )p − z  = a , we have z p − sp−1 z = sp a , where sp a ∈ B and sp a is G-invariant. Hence a := sp a ∈ A. We have thus obtained an equation z p − sp−1 z = a. Let B0 = A[z]. Then G acts on Y0 := Spec B0 by σ(z) = z + s.

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Now the inclusion B0 → B defines a morphism π : Y → Y0 such that π is G-equivariant and q0 · π = q, where q0 : Y0 → X is the G-quotient morphism. By the construction, π is birational and finite. Let Y0 be the normalization of Y0 . Then Y = Y0 by Zariski’s main theorem. 2 6. Open problems We try to prove Theorem 3.5 from a different geometric point of view. This will provide interesting problems in affine algebraic geometry. In Theorem 3.4, we considered the case where the coordinates x, y are chosen in such a way that the A1 -fibration ρ on A2 given by (x, y) → x is preserved by the G-action. In general, the existence of an A1 -fibration on Y = Spec k[x, y] preserved by the G-action is not guaranteed. Let q : Y → X be the quotient surface with X := Y //G. We assume that the G-action is free. A non-free G-action on A2 exists in general. For example, if we consider a G-action σ(x) = x + a(y) and σ(y) = y then the set of points {(0, α)|α ∈ k, a(α) = 0} is the fixed point locus for the G-action. Since the G-action is free by assumption, q is a finite étale morphism, whence X is smooth. We can view the ring B as a flat finite A-module of rank p, where B = k[x, y] and A = B G . Conjecture 6.1. Let Y be a smooth affine surface with a free G-action, whence X := Y //G is smooth and the quotient morphism is a finite étale morphism. If κ(Y ) = −∞ then κ(X) = −∞. The condition that the logarithmic Kodaira dimension κ(Y ) being −∞ for a smooth affine surface is equivalent to the condition that Y has an A1 -fibration (see [15, Lemma 1.3.1 and Theorem 1.3.2, Chapter 3]). Suppose that Y = A2 . Then Lemma 2.1 implies that A is factorial and A∗ = k∗ . If the above conjecture is true, κ(X) = −∞. Hence X ∼ = A2 by an algebraic characterization of A2 (see [15, Theorem 2.2.1]) which states that a smooth affine surface X = Spec (A) defined over k is isomorphic to A2 if and only if the following three conditions are satisfied: (1) A is a factorial domain. (2) A∗ = k∗ . (3) κ(X) = −∞. In the case of characteristic zero, a flat cofinite k-subalgebra of a polynomial ring k[x, y] is a polynomial ring by [14]. However, this result does not hold in the case of positive characteristic. A counterexample is given in [17, Theorem 3.7] as a smooth hypersurface which is a plane sandwich. In fact, the hypersurface has an A1 -fibration over A1 with several multiple fibers. Hereafter in this section, we assume that A = k[u, v] is a polynomial ring. Hence the arguments given below are nothing more than conjectural. By Lemma 2.5, we can write

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B = k[x, y] = A[z]/(z p − z − a) where a := f (u, v) ∈ A and σ(z) = z + 1. Let Cλ be a curve on X defined by f (u, v) = λ and let Λ = {Cλ | λ ∈ k} be the pencil defined by a on X. Then q −1 (Cλ ) is a disjoint (1) (p) union of curves Cλ , . . . , Cλ which are isomorphic to Cλ . In fact, z p − z = λ has p (i) distinct roots λ(1) , . . . , λ(p) and Cλ is defined by z = λ(i) , 1 ≤ i ≤ p. Since q −1 (Cλ ) =  (i) (i) 1≤i≤p Cλ , each Cλ is isomorphic to Cλ . Here we use the following conjecture which is to be called the -adic Suzuki–Zaidenberg formula. In the case of characteristic zero, the corresponding result follows from the Suzuki–Zaidenberg theorem (see [21,23]). For the -adic cohomology, see [8] for example. Conjecture 6.2. Let  be a prime number distinct from the characteristic p. We consider -adic Euler–Poincare characteristics of algebraic varieties. Let ρ : X → A1 be a surjective fibration from a smooth affine surface to A1 such that: (1) There exists a smooth projective completion X of X such that ρ is extended to a fibration2 ρ : X → P1 and the generic fiber F η of ρ is smooth and has no places outside X which are inseparable over k(η). (2) Except for a finite set of points {P1 , . . . , Ps } ⊂ A1 , any fiber ρ−1 (Q) is smooth if Q∈ / {P1 , . . . , Ps }. Then the -adic Suzuki–Zaidenberg formula holds: χ(X) = (1 − s)χ(F ) +

s 

χ(Fi ),

Fi = ρ−1 (Pi )

i=1

where χ(Fi ) ≥ χ(F ) for a general fiber F of ρ and the equality holds if and only if Fi is smooth. We assume the validity of Conjectures 6.1 and 6.2 and apply these conjectures to the quotient morphism q : Y → X, where Y ∼ = A2 and X is assumed to be isomorphic to A2 . p Since B = A[z]/(z − z − a), B is identified with the tensor product k[z] ⊗k[a] A relative to a finite étale extension k[a] ⊂ k[z]. Let ρ : X → A1 = Spec k[a] be the fibration defined by the pencil Λ. We assume that the fibration ρ satisfies the condition (1) in Conjecture 6.2. The composite ρ · q : Y → A1 is factored as ρ

Y −→ A1 = Spec k[z] → Spec k[a]. Let F be a general fiber of ρ, let {F1 , . . . , Fs } exhaust all singular fibers of ρ, let g be the geometric genus of F and let r be the number of places at infinity of F . Note that 2 By fibration, we mean that the field extension k(X)/k(P1 ) is a regular extension. This in turn implies that the generic fiber F η is geometrically connected.

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q −1 (Fi ) is a disjoint union of p copies of Fi and hence the number of singular fibers of the fibration ρ is ps. By Conjecture 6.2, we have 1 = χ(X) = (2 − 2g − r)(1 − s) +

s 

χ(Fi ),

i=1

1 = χ(Y ) = (2 − 2g − r)(1 − ps) +

s 

pχ(Fi ).

i=1

Therefore we have (p − 1)

s 

χ(Fi ) = (2 − 2g − r)(p − 1)s .

i=1

s So, we have i=1 χ(Fi ) = (2 −2g −r)s and χ(Fi ) ≥ χ(F ) = 2 −2g −r for every 1 ≤ i ≤ s. Hence χ(Fi ) = χ(F ) for every i and every Fi is smooth. Thus we may assume that s = 0 from the beginning. Then formula gives 1 = (2 − 2g − r), i.e., 2g + r = 1, where r ≥ 1 because F is an affine curve. Hence g = 0 and r = 1. Namely, F ∼ = A1 . This implies that a = f (u, v) = 0 defines a coordinate of the polynomial ring A = k[u, v]. In fact, Theorem of Abhyankar–Moh–Suzuki does not hold any more in the case of positive characteristic. But the assumption (1) in Conjecture 6.2 on the places of F η lying at infinity implies that the place of F η at infinity is in fact rational over k(η) and hence the fibration ρ : X → A1 is an A1 -fibration. Then the fibration ρ : Y → A1 = Spec k[z] is an A1 -fibration and preserved by the G-action. Thus we have the following restatement of Kuroda’s theorem whose proof is based on the validity of two conjectures. Theorem 6.3. Suppose that a given free G-action on Y = Spec k[x, y] satisfies the following conditions. (1) The quotient X = Y //G is isomorphic to A2 = Spec k[u, v]. (2) Let B = k[x, y] and A = k[u, v] and write B = A[z]/(z p − z − a), where a := f (u, v) defines a fibration ρ : X → A1 = Spec k[a] satisfying the condition (1) of Conjecture 6.2. Then a = f (u, v) is a coordinate of k[u, v] and the lifting of the fibration ρ defines an A1 -fibration ρ : Y → A1 = Spec k[z] which is preserved by the G-action. Hence, after a suitable change of coordinates x, y on Y , the G-action is normalized as σ(x, y) = (x + 1, y). References [1] H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, NJ, 1956, xv+390 pp.

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