Extension of automorphisms of rational smooth affine curves
aa r X i v : . [ m a t h . AG ] S e p M ath. Res. Lett. (2011), no.
00, 10001–100NN c (cid:13)
International Press 2011
EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTHAFFINE CURVES
J´er´emy Blanc, Jean-Philippe Furter, and Pierre-Marie Poloni
Abstract.
We provide the existence, for every complex rational smooth affine curve Γ,of a linear action of Aut(Γ) on the affine 3-dimensional space A , together with a Aut(Γ)-equivariant closed embedding of Γ into A . It is not possible to decrease the dimensionof the target, the reason for this obstruction is also precisely described.
1. Introduction
Throughout this article, all varieties are algebraic varieties defined over the field C of complex numbers. The affine (resp. projective) n -space is denoted by A n (resp. P n ).It is well known that any smooth affine variety X of dimension n admits a closedembedding into A m , when m ≥ n + 1 [12, Theorem 1]. If moreover m ≥ n + 2,then, by a result of Nori, Srinivas and Kaliman (see [12] and [8]), any two closedembeddings ι, ι ′ : X → A m are equivalent in the sense that there exists f ∈ Aut( A m )such that ι ′ = f ◦ ι .In particular, if ι : X → A m is a closed embedding of a smooth affine variety ofdimension n into some affine space of dimension m ≥ n + 2, then it follows thatevery automorphism ϕ of X extends to an automorphism of the ambient space A m ,since the two embeddings ι ◦ ϕ and ι are equivalent.However, Derksen, Kutzschebauch and Winkelmann showed in [4] that it is notalways possible to extend the group structure of Aut( X ), i.e. to find a closed embed-ding ι : X → A m and an action of Aut( X ) on A m that restricts on X to the actionof Aut( X ) on it. More precisely, they proved that there does not exist, for any inte-ger m , any injective group homomorphism from Aut( C ∗ × C ∗ ) ∼ = GL ( Z ) ⋉ ( C ∗ ) tothe group Diff( R m ) of diffeomorphisms of R m .Recall that, if G is an algebraic group acting on an affine variety X , then X admits a G -equivariant closed embedding into a finite dimensional G -module (see [2,Proposition 1.12, p. 56]). In particular, there exist, for every smooth affine curve Γ,a linear action of Aut(Γ) on an affine space A m and a Aut(Γ)-equivariant closedembedding of Γ into A m . A natural question is then to find the smallest possible m .In this article, we settle the case of rational smooth affine curves. In this setting,the proof of Borel only gives the embedding dimension m = 2 · | Aut(Γ) | , when theautomorphism group Aut(Γ) is finite. However, our main result shows that it isalready possible to obtain m = 3: The authors gratefully acknowledge support by the Swiss National Science Foundation Grant”Birational Geometry” PP00P2 128422 /1 and by the French National Research Agency Grant”BirPol”, ANR-11-JS01-004-01. 100010002 J´er´emy Blanc, Jean-Philippe Furter, and Pierre-Marie Poloni
Theorem 1.
Every rational smooth affine curve Γ admits an Aut(Γ) -equivariantclosed embedding into the affine space A . Furthermore, there exist such embeddingsfor which the action of Aut(Γ) on A is linear. It is easy to construct closed embeddings into the affine plane A for all rationalsmooth affine curves Γ. But it is of course not possible in general to ask for Aut(Γ)-equivariant embeddings into A . Indeed, there exist rational smooth affine curveswhose automorphism groups are isomorphic to the alternating group A , to A , orto the symmetric group S (see Section 6) and it is well known that the group A has no faithful representation of dimension two. Since all finite subgroups of Aut( A )are linearizable, it follows that we cannot embed equivariantly such a curve into theplane, even if we allow non linear actions on A .In fact, we establish stronger impossibility statements showing that it would be alsotoo optimistic in general to look for closed embeddings into A in such a way thatevery single automorphism of the curve extends to an automorphism of the ambientspace (see Corollary 2.6). Theorem 2.
There exist rational smooth affine curves Γ with Aut(Γ) = 1 and suchthat for every closed embedding of Γ into A , the identity on Γ is its only automor-phism that extends to an automorphism of A . Let us also emphasize that Theorem 1 cannot be generalized to all smooth affinecurves. Actually, there even exist, for every natural number n , smooth affine curves Γwhich do not admit any Aut(Γ)-equivariant closed embedding into A n .To see this, recall that every finite group G is equal to the automorphism groupof a smooth projective curve, and thus of an affine one [7], and take a smooth affinecurve Γ n whose automorphism group is isomorphic to ( Z / Z ) n +1 . Then, Γ n does notadmit any Aut(Γ n )-equivariant embedding into A n , because ( Z / Z ) n +1 does not actfaithfully on A n . Indeed, by Smith theory, the action of a finite p -group on A n hasalways a fixed point (see e.g. [3, Th. 7.11, p 145], [9, p. 204], or [4, Proposition 1]) andthe induced tangential (linear) representation at that fixed point should be faithfultoo (see e.g. [4, Lemma 4]).It would however be interesting to know what happens in the case of smooth affinecurves of genus 1. Sathaye proved in [10] that such curves admit closed embeddingsinto A . Nevertheless, we do not know what is the minimal m (if it exists) such thatevery smooth affine curve Γ of genus 1 admits an Aut(Γ)-equivariant closed embeddinginto A m .The article is organized as follows.Section 2 concerns embeddings of rational smooth affine curves into the affineplane. We give examples of automorphisms of such curves that do not extend, andprove Theorem 2 (see Corollary 2.6).Section 3 is devoted to the study of embeddings of smooth rational curves into A whose images are contained in a hyperplane. We prove that they are all equivalentand thus that any two closed embeddings of a rational smooth affine curve into A become equivalent, when seen as embeddings in A (Proposition 3.1). This answersa question of Bhatwadekar and Srinivas in this case.In section 4 we realize every non-empty subset of P that is invariant by a sub-group H of Aut( P ) as the fixed-point set of a H -equivariant endomorphism of P XTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 10003 (Corollary 4.4). This result is used in Section 5 to prove Theorem 1 (see Theorem 5.2).Explicit formulas are given in Section 6.
2. Embeddings of rational smooth affine curves into the plane
Let us recall that every rational smooth affine curve Γ is isomorphic to P \ Λ,where Λ is a finite set of r ≥ A . Indeed, Γ can also be seenas the complement in A of a finite number (possibly zero) of points and we canconsider the closed embedding τ : Γ → A given by x ( x, P ( x ) ), where P ∈ C [ x ] isa polynomial whose roots are exactly the removed points. Note that the image of τ is the curve of A defined by the equation P ( x ) y = 1.Moreover, the automorphism group Aut(Γ) of the curve Γ = P \ Λ is equal to thegroup of automorphisms of P that preserve the set Λ. This gives a group homomor-phism from Aut(Γ) to the symmetric group Sym r . Note that this homomorphism isinjective if and only if r ≥ r is equal to 1 or 2, then Γ is isomorphic to A or A \ { } , and its automorphismgroup is C ∗ ⋉ C or {± } ⋉ C ∗ respectively. If r ≥
3, then Aut(Γ) is a finite group.The Abhyankar-Moh-Suzuki theorem claims that all closed embeddings of A into A are equivalent to the one given by t ( t, A extends to an automorphism of theambiant space. If r ≥ Lemma 2.1.
Let
Γ = A \ ∆ , where ∆ is a non-empty finite set. Then, there existinfinitely many non-equivalent closed embeddings ι : Γ → A such that the identity isthe only automorphism of A that preserves ι (Γ) .Proof. We can assume that ∆ = { , a , . . . , a m } , where a , . . . , a m ∈ C \{ , } , m ≥ k ≥
2, we denote by ι k : Γ → A the embedding given by x (cid:18) x, x − x k Q mi =1 ( x − a i ) (cid:19) . It induces an isomorphism between Γ and the curve ι k (Γ) defined by the equation x = yx k m Y i =1 ( x − a i ) + 1 . We first remark that any automorphism of A that sends ι k (Γ) onto a curve ofdegree at most deg( ι (Γ)) = k + m + 1 is necessarily affine. Indeed, if f : ( x, y ) ( f ( x, y ) , f ( x, y )) is the inverse of such an automorphism, we get:deg( f − f ( f ) k Q mi =1 ( f − a i ) −
1) = ( k + m ) deg f + deg f ≤ k + m + 1 . This implies that deg( f ) = deg( f ) = 1, i.e. that f (and its inverse too) is affine. Inparticular, all above embeddings are non-equivalent. We now show that the identityis the only affine automorphism of A that preserves the curve ι k (Γ). Any such automorphism extends to an automorphism τ of P preserving the lineat infinity given by z = 0 and the curve of equation xz k + m − yx k m Y i =1 ( x − a i z ) − z k + m +1 = 0 . On the line at infinity we get the two points [0 : 1 : 0] and [1 : 0 : 0]. The point [1 : 0 : 0]is smooth with tangent y = 0 and the point [0 : 1 : 0] is singular with tangent conegiven by x k Q mi =1 ( x − a i z ) = 0. Hence, both lines x = 0 and y = 0 are invariant.Therefore, τ is given by a diagonal automorphism of the form [ x : y : z ] [ µx : νy : z ], µ, ν ∈ C ∗ . Replacing in the equation yields µ = ν = 1. (cid:3) The curves A and A \ { } admit closed embeddings into A such that all theirautomorphisms extend to automorphisms of A . Consider for example the curves ofequations y = 0 and xy = 1. However, it is no longer true for the curve A \ { , } . Proposition 2.2.
Let
Γ = A \ { , } . For every closed embedding τ : Γ → A , thereexists an automorphism of Γ that does not extend to A . Before proving this statement, let us recall the following classical result (see e.g. [6,Theorem 2]).
Lemma 2.3.
Every finite subgroup of
Aut( A ) is conjugate to a subgroup of GL(2 , C ) .Proof of Proposition . . Note that the group of automorphisms of Γ is the groupSym of permutations of a set of three elements, corresponding to the three points“at infinity”, i.e. the points of P \ ι (Γ), where ι is any (open) embedding of Γ in P .It is generated by the automorphisms ρ : x / (1 − x ) and σ : x − x and wehave Aut(Γ) = h σ, ρ | σ = ρ = 1 , σρσ − = ρ − i = Sym . Suppose for contradiction that there exists a closed embedding τ : Γ → A for whichevery automorphism of Γ extends. Since the identity is the only automorphism of A that restricts to the identity on a closed curve isomorphic to A \{ , } (see Lemma 2.4below), we would have a subgroup G ⊂ Aut( A ) isomorphic to Sym whose restrictionto τ (Γ) yields Aut(Γ).We now prove that this is impossible. First, recall that G is conjugate to a subgroupof GL(2 , C ) (see Lemma 2.3 above). Then, one easily checks that G is conjugate tothe subgroup G ′ of GL(2 , C ) generated byˆ ρ : ( x, y ) ( y, − x − y ) and ˆ σ : ( x, y ) ( y, x ) . We let f ∈ Aut( A ) be an automorphism such that f Gf − = G ′ and we considerthe embedding ˆ τ = f ◦ τ of Γ in A . The automorphism group of Γ extends thento G ′ for this embedding.Remark that the set { ω | ω − ω + 1 = 0 } ⊂ Γ, which is the set of fixed points of ρ ,is an orbit of size 2 of Aut(Γ). But one checks that G ′ ⊂ GL(2 , C ) does not have anyorbit of size 2 in the affine plane A . This gives a contradiction. (cid:3) Lemma 2.4.
The set of fixed points of a plane polynomial automorphism is either afinite set of points ( possibly empty ) , a finite disjoint union of subvarieties isomorphicto A , or the whole plane. XTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 10005
Proof.
Using the amalgamated structure of Aut( A ), it is observed in [5] that a planepolynomial automorphism is conjugate either to a triangular automorphism ( x, y ) ( ax + p ( y ) , by + c ) with a, b, c ∈ C and p ( y ) ∈ C [ y ], or to some cyclically reducedelement (see [11, I.1.3] or [5, p. 70] for the definition of a cyclically reduced element).In the first case, an obvious computation shows that the set of fixed points is eitherempty, a point, a finite disjoint union of subvarieties isomorphic to A , or the wholeplane. In the second case, by [5, Theorem 3.1], the set of fixed points is a non-emptyfinite set of points. (cid:3) Using tools of birational geometry, we can actually specify the statement of Propo-sition 2.2. Indeed, Theorem 2.5 below shows that there is no closed embedding of thecurve A \ { , } into A such that the automorphism ρ : x / (1 − x ) extends to anautomorphism of the affine plane.Before we state this result, let us recall that any automorphism f of P of finiteorder n > x : y ] [ x : ξy ], where ξ is a primitive n -th root of unity.In particular, it has the following properties:(1) the automorphism f fixes exactly two points of P ;(2) all other orbits under the action of f have size n .Thus, the following holds for every automorphism g ∈ Aut(Γ) of order n > g fixes 0, 1 or 2 points of Γ;(2) all other orbits under the action of g have size n . Theorem 2.5.
Let Γ be a rational smooth affine curve and let g ∈ Aut(Γ) be anautomorphism. (1) If g fixes at most one point of Γ , there is a closed embedding τ : Γ → A suchthat g extends to an automorphism of A . (2) If g is of finite order n > with n odd and if it fixes exactly two pointsof Γ , then there is no closed embedding τ : Γ → A such that g extends to anautomorphism of A .Proof. (1) Let P ∈ C [ x ] be a non-zero polynomial such that Γ is isomorphic to A \ { x ∈ A | P ( x ) = 0 } . Let g ∈ Aut(Γ) be an automorphism that fixes at most onepoint of Γ. Let us denote also by g its extension as an automorphism of P . We canassume that g fixes the point of P at infinity, so that it is of the form x ax + b ,for some a ∈ C ∗ and b ∈ C . Moreover P ( ax + b ) = µP ( x ) for some µ ∈ C ∗ .When we embed Γ into A via the map x ( x, P ( x ) ), the automorphism g extendsto ( x, y ) ( ax + b, µ − y ).(2) Let g ∈ Aut(Γ) be of finite order n > n odd such that it fixes 2 pointsof Γ. Suppose, for contradiction, that there exists a closed embedding τ : Γ → A for which g extends to an automorphism h of A . Since g has finite order n , theautomorphism h n ∈ Aut( A ) fixes pointwise the curve τ (Γ). Because g fixes twopoints of Γ, τ (Γ) is not isomorphic to A , hence h n is trivial by Lemma 2.4.Recall that every automorphism of A of finite order is conjugate to a linear one(Lemma 2.3). Thus, there exists f ∈ Aut( A ) such that ˆ h = f ◦ h ◦ f − is linear. More-over, the automorphism g ∈ Aut(Γ) extends to ˆ h , when we consider the embeddingˆ τ = f ◦ τ : Γ → A . The linear automorphism ˆ h extends to an automorphism of P , and the closureof ˆ τ (Γ) in P is a projective rational curve C , having all its singular points on the lineat infinity L = P \ A .If C is smooth, it is isomorphic to P . Hence, it is a conic or a line, and thusintersects L into 1 or 2 points, which contradicts the fact that g acts on C withorder n > C is singular.Denote by η : X → P the blow-up of the points of P that are singular pointsof C , and write C ⊂ X the strict transform of C in X . If C is singular, we denoteby η : X → X the blow-up of the points of X that are singular points of C , andwrite C the strict transform of C in X . We continue like this until we end with asmooth curve C m ⊂ X m such that the intersection of C m with all curves contractedby η η . . . η m is transversal. Note that C m is isomorphic to P . For i = 1 , . . . , m ,the lift of ˆ h yields an automorphism h i of X i which preserves the curve C i . It alsopreserves the pull-back of A in X i , which is again isomorphic to A .For i = 1 , . . . , m , we denote by B i ⊂ C i the (finite) set of points not lying in A .Each point p ∈ B i has a multiplicity m ( p ) as a point of C i . This multiplicity is apositive integer and it is equal to 1 if and only if C i is smooth at this point p . Denoteby B the set of points of C = C ⊂ X = P not lying in A and let us use the samenotation as above for the multiplicities of the points of B .Writing d the degree of C ⊂ P , the geometric genus of C can be computed withthe following classical formula. (Note that it is equal to 0, since C is rational.)( ⋆ ) 0 = ( d − d − − m X i =0 X p ∈B i m ( p ) · ( m ( p ) − . Let us now prove the following assertion by descending induction on j ≤ m .( ⋄ )Let j ∈ { , . . . , m } and let J ⊂ B j be an orbit under the action of h j .Then m ( p ) = m ( p ′ ) for all p, p ′ ∈ J, and the integer P p ∈ J m ( p ) is a multiple of n.For j = m , the assertion ( ⋄ ) holds for all orbits J ⊂ B m . Indeed, C m is isomorphicto P and the action of h m on B m ⊂ C m is fixed-point-free, so all orbits have size n and all multiplicities are equal to 1.Then, we can prove ( ⋄ ) for j < m , assuming it holds for every integer k with j + 1 ≤ k ≤ m . For this, let J ⊂ B j be an orbit under the action of h j and let usdenote by m J the multiplicity m ( p ) of a point p ∈ J . Note that this multiplicity doesnot depend of the choice of p , since h j acts transitively on J .If m J = 1, all points of J are smooth, and so the pull-back by η j +1 of J consistsof | J | points of multiplicity m j = 1. This implies P p ∈ J m ( p ) ∈ n N , by inductionhypothesis.If m J >
1, then all points of J are singular points of the curve C j and are thusblown-up by η j +1 : X j +1 → X j . The number m J is the multiplicity of the curve C j at the point p ∈ J . Denoting by E p ⊂ X j +1 the curve contracted by η j +1 onto p ,the number m J is the intersection number E p · C j +1 . This latter is equal to the sumof m q ( E p ) · m q ( C j +1 ), where q runs through all points infinitely near to p , and where XTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 10007 m q ( E p ) and m q ( C j +1 ) are the multiplicities of the strict transforms of E p and C j +1 at q , respectively. Note that m q ( E p ) is equal to 0 or 1.Therefore, the sum P p ∈ J m J is equal to a sum of multiplicities of orbits in B k for k ≥ j + 1. By induction hypothesis, it is a multiple of n . This achieves to prove ( ⋄ ).In order to finish the proof, we will show how Equation ( ⋆ ) and Assertion ( ⋄ ) implythat the integers ( d − d − and d are both multiple of n . Since the greatest commondivisor of d and ( d − d − is 1 or 2, this will contradict the assumption n > n divides ( d − d − , we decompose the sum of ( ⋆ ) according to orbits( d − d − m X j =0 X J ⊂B j X p ∈ J m ( p ) · ( m ( p ) − . By Assertion ( ⋄ ), the multiplicities m ( p ) are all equal among the same orbit J , so P p ∈ J m ( p ) · ( m ( p ) −
1) is a multiple of P p ∈ J m ( p ), which is a multiple of n by ( ⋄ ).Since n is odd, P p ∈ J m ( p ) · ( m ( p ) − is also a multiple of n , and so is ( d − d − .It remains to show that d is also a multiple of n . For this, we observe that theintersection number d = L · C is the sum of multiplicities of all points of C that belongto L , as proper or infinitely near points. Since L is invariant under the extension ofthe affine automorphism ˆ h , the union of these points decomposes into orbits of h j andthe sum is then a multiple of n by Assertion ( ⋄ ). (cid:3) Corollary 2.6.
There exist rational smooth affine curves Γ with Aut(Γ) = 1 and suchthat for every closed embedding of Γ in A , the identity on Γ is its only automorphismwhich extends to an automorphism of A .Proof. Let ω = e iπ/ and a = 1. Let a , . . . , a k be complex numbers algebraicallyindependent over Q . We consider the curve Γ = P \ Λ, where Λ is the following setof 3 k points Λ = (cid:8) [ a i ω j : 1] | i = 1 , . . . , k, j = 0 , . . . , (cid:9) . The map h : [ x : y ] [ x : ωy ] is obviously an automorphism of Γ. We will now provethat it generates the whole automorphism group Aut(Γ) if k ≥
3. This will concludethe proof, since h and h do not extend to automorphisms of A by Theorem 2.5.Let g ∈ Aut(Γ) be an automorphism of Γ. It extends to an automorphism of P that preserves the set Λ. Let us denote this latter also by g .Consider the 4-tuple V = (cid:0) [1 : 1] , [ ω : 1] , [ ω : 1] , [ a : 1] (cid:1) . Since a , . . . , a k are al-gebraically independent over Q , the image of V by g is a 4-tuple of points containedin the set S = (cid:8) [1 : 1] , [ ω : 1] , [ ω : 1] , [ a : 1] , [ a ω : 1] , [ a ω : 1] (cid:9) . Indeed, the cross-ratio of g ( V ) must be equal to the cross-ratio of V , i.e. to ω ( ω − a ) / ( a − (cid:0) [1 : 1] , [ ω : 1] , [ ω : 1] , [ a : 1] (cid:1) allows us toconclude that g preserves the set (cid:8) [1 : 1] , [ ω : 1] , [ ω : 1] (cid:9) . Therefore, g is either apower of h , or it is one of the maps ϕ i : [ x : y ] [ y : xω i ] with i = 0 . . . g cannot be one of the ϕ i ’s, since ϕ i sends the point [ a : 1] ontothe point [ 1 a ω i : 1], which does not belong to the set S . (cid:3) Remark . The proof of Corollary 2.6 shows that if k ≥ ⊂ P isgeneral among all sets of distinct 3 k points invariant by the map [ x : y ] [ x : ωy ],then for all closed embeddings of the curve Γ = P \ Λ into A , the identity is theonly automorphism of Γ that extends to an automorphism of A .On the contrary, when k ≤
2, every such curve Γ admits an automorphism of order 2and Proposition 2.8 below implies then that this latter extends to an automorphismof A for a well-chosen closed embedding of Γ into A . Proposition 2.8.
Let Γ be a rational smooth affine curve and let σ ∈ Aut(Γ) be anautomorphism of Γ of order . There exists a closed embedding of Γ in A and anautomorphism ˆ σ ∈ Aut( A ) of order whose restriction to Γ yields σ .Proof. Let Γ = P \ Λ, where Λ is a finite set of points. Let us denote by σ theextension of the automorphism σ ∈ Aut(Γ) as an automorphism of P . If it fixes atmost one point of Λ, the result follows from Theorem 2.5.We can thus assume that the two points of P fixed by (the extension of) σ belongto Γ. Let p be a point of Λ. Its orbit { p, σ ( p ) } is then contained in Λ. Let C be thecurve C = P \ { p, σ ( p ) } . Note that C is isomorphic to A \ { } and that σ restrictsto an automorphism of C . Remark that all automorphisms of A \ { } of order 2 withtwo fixed points are conjugate to the automorphism x x − ∈ Aut(Spec( C [ x, x − ])).Therefore, there is a closed embedding τ : C → A whose image is the curve definedby the equation y − x and such that σ ∈ Aut( C ) extends to the automorphism ˆ σ : ( x, y ) ( − x, y ). More-over, the curve τ (Γ) is then equal to a set of points of τ ( C ) satisfying that Q ni =1 ( y − a i ) = 0, for some n ≥ a , . . . , a n ∈ C \ {± } .Let Y ⊂ A be the closed curve defined by the equation y − x · n Y i =1 ( y − a i ) ! . Consider finally the birational transformation of A defined by( x, y ) (cid:18) x Q ni =1 ( y − a i ) , y (cid:19) , which restricts to an isomorphism between τ (Γ) and Y . Since it commutes with theautomorphism ( x, y ) ( − x, y ), this yields the result. (cid:3)
3. Planar embeddings in the space
The following question of Bhatwadekar and Srinivas is asked at the end of [12]: areany two embeddings of a smooth affine curve in A equivalent, when considered asembeddings in A ?The next result answers positively for the case of rational smooth affine curves. Proposition 3.1.
Let Γ be a rational smooth affine curve. XTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 10009 (1) If τ , τ : Γ → A are two closed embeddings whose images are contained ina hyperplane ( planar embeddings in the space ) , there exists an automorphism α ∈ Aut( A ) such that τ = α ◦ τ , i.e. any two planar embeddings in thespace are equivalent. (2) In particular, fixing a planar embedding Γ → A , every automorphism of Γ extends to A .Proof. Let Γ = A \ { x ∈ A | P ( x ) = 0 } , where P ∈ C [ x ] is a polynomial withsimple roots. Note that the coordinate ring of Γ is C [Γ] = C [ x, P ( x ) ] and recallthat the map x ( x, P ( x ) ) defines a closed embedding of Γ in A . To prove theproposition, it suffices to prove that any planar embedding is equivalent to the onegiven by x ( x, P ( x ) , τ : Γ → A be a planar embedding of Γ. We can compose τ with an affineautomorphism f of A and get an embedding τ = f ◦ τ : Γ → A of the form x (0 , Q ( x ) , R ( x )), where Q, R ∈ C ( x ) are rational functions without poles on Γ.Since τ is a closed embedding of the curve Γ, the equality C [ x, P ( x ) ] = C [ Q ( x ) , R ( x )]holds. In particular, there exists a polynomial A ∈ C [ X, Y ] such that A ( Q ( x ) , R ( x )) = x . Now, we compose τ with the automorphism of A defined by f ( X, Y, Z ) =( X + A ( Y, Z ) , Y, Z ) and obtain the embedding τ : Γ → A given by τ : x ( x, Q ( x ) , R ( x )) . Because of the equality C [ x, P ( x ) ] = C [ Q ( x ) , R ( x )], all zeros of P ( x ) are polesof aQ ( x ) + bR ( x ) for general complex numbers a, b ∈ C . We can thus compose τ with a linear automorphism of the form ( X, Y, Z ) ( X, aY + bZ, Z ) and get anembedding τ : Γ → A of the form τ : x (cid:18) x, Q ( x ) Q ( x ) , R ( x ) R ( x ) (cid:19) , where Q , Q , R , R ∈ C [ x ] are polynomials such that Q and Q (resp. R and R )have no common factor, and such that P ( x ) divides Q ( x ).In particular, there exist two polynomials U, V ∈ C [ x ] such that U Q + V P = 1.It follows 1 P = U Q + V PP = U Q P + V = SU Q Q + V, where S ∈ C [ x ] satisfies P S = Q .This implies C [ x, P ] ⊂ C [ x, Q Q ] and thus C [ x, Q Q , R R ] = C [ x, P ] = C [ x, Q Q ] . Therefore, there exist polynomials
B, C ∈ C [ X, Y ] such that B ( x, Q ( x ) Q ( x ) ) = P ( x ) − R ( x ) R ( x ) and C ( x, P ( x ) ) = Q ( x ) Q ( x ) . Finally, we consider the automorphisms of A definedby f ( X, Y, Z ) = (
X, Y, Z + B ( X, Y )) and f ( X, Y, Z ) = (
X, Z, Y − C ( X, Z )). Onechecks that f ◦ f ◦ τ : Γ → A is the desired embedding x ( x, P ( x ) , (cid:3) Note that the proof above is constructive. In particular, a planar embedding of asmooth rational curve Γ in A and an automorphism ϕ of Γ being given, it allows usto construct an explicit automorphism of A which extends ϕ . Example . Let Γ be the curve Γ = A \ { , } and let ρ ∈ Aut(Γ) be the automor-phism of Γ defined by ρ ( x ) = 1 / (1 − x ). We saw in Section 2 that there is no closedembedding of Γ into A such that ρ extends to an automorphism of A . However,it extends to an automorphism of A , when we consider the embedding τ : Γ → A defined by x ( x, /x ( x − , f , f , . . . , f be the automorphismsof A defined by f ( X, Y, Z ) = (
Z, Y, X ), f ( X, Y, Z ) = ( X + Y + 2 − Y Z , Y, Z ), f ( X, Y, Z ) = (
X, aY + bZ, Z ), f ( X, Y, Z ) = (
X, Y, Z − ab [( b + ( a − b ) X )( Y − aX +2 a ) − ( a − b ) ](1 + X )) and f ( X, Y, Z ) = (
X, Z, Y − aX + 2 a + aZ + ( b − a ) XZ ),where a, b ∈ C are general complex numbers.Setting F = f ◦ f ◦ · · · ◦ f , one checks F ◦ τ ◦ ρ = τ . This implies that F − is anextension of the automorphism ρ ∈ Aut(Γ).
Remark . To our knowledge, there is no known example of a smooth affine curveadmitting two non-equivalent embeddings into A . Paradoxically, we do not knowany smooth affine curve such that all its embeddings into A are equivalent!The case of the affine line is of particular interest. On one hand, all closed embed-dings of A into A are equivalent by the famous Abhyankar-Moh-Suzuki theorem.On the other hand, all closed embeddings of A into A n with n ≥
4. Actions of
SL(2 , C ) on End( A ) and of PGL(2 , C ) on P The aim of this section is to construct, for every non-empty subset Λ of P that isinvariant by a subgroup H of Aut( P ), a H -equivariant endomorphism of P whosefixed-point set is equal to the set Λ (Corollary 4.4). We will use this result later onto construct embeddings of every rational smooth affine curve into A in such a waythat the whole automorphism group of the curve extends to a subgroup of Aut( A ).For the rest of the paper we will consider the following actions of the group SL(2 , C )on O ( A ) = C [ x, y ] and End( A ) = C [ x, y ] × C [ x, y ].SL(2 , C ) × O ( A ) → O ( A )( g, P ) g · P := P ◦ g − and SL(2 , C ) × End( A ) → End( A )( g, F ) g · F := g ◦ F ◦ g − . Note that these actions come from the natural action of SL(2 , C ) on A . Indeed,denote by V the space A as a complex vector space of dimension 2 and identify theset of the linear forms on it as the dual space V ∗ . The action of SL( V ) on V yieldsactions on V ∗ , on the symmetric algebra S ( V ∗ ) and on S ( V ∗ ) ⊗ V . The naturalisomorphisms between S ( V ∗ ) and C [ x, y ] = O ( A ) , and between S ( V ∗ ) ⊗ V and C [ x, y ] × C [ x, y ] = End( A ), lead then to the SL(2 , C )-actions that we defined above. Lemma 4.1.
The map ρ : End( A ) → O ( A ) defined by C [ x, y ] × C [ x, y ] → C [ x, y ]( f , f ) f y − f x is SL(2 , C ) -equivariant, when we consider the actions defined above. XTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 10011
Proof.
The result could of course be checked by direct computations, but let us men-tion that it also follows from the fact that ρ corresponds to the morphism S ( V ∗ ) ⊗ V → S ( V ∗ ) given by the composition τ ◦ τ , where τ and τ are the two following homo-morphisms of SL( V )-modules. τ : S ( V ∗ ) ⊗ V → S ( V ∗ ) ⊗ V ⊗ V ∗ ⊗ Vp ⊗ v p ⊗ v ⊗ id , where id denotes the identity element seen as an element of V ∗ ⊗ V = Hom( V, V ),and τ : S ( V ∗ ) ⊗ V ⊗ V ∗ ⊗ V → S ( V ∗ ) p ⊗ v ⊗ v ∗ ⊗ v det( v , v )( pv ∗ ) . (cid:3) Lemma 4.2.
Let G ⊂ SL(2 , C ) be a finite subgroup of SL(2 , C ) and let P ∈ C [ x, y ] .The following conditions are equivalent: (1) The polynomial P satisfies P (0 ,
0) = 0 and is fixed by G . (2) There exists an endomorphism F = ( f , f ) of A that is fixed by G and suchthat ρ ( F ) = f y − f x = P .Proof. Let E P ⊂ End( A ) be the set E P = ρ − ( P ) = { ( f , f ) ∈ C [ x, y ] × C [ x, y ] | f y − f x = P } . This defines an affine subset of the C -vector space End( A ), since the endomorphism( λf + (1 − λ ) f , λf + (1 − λ ) f ) belongs to E P , for any ( f , f ) , ( f , f ) ∈ E P andany λ ∈ C . Moreover, E P is non-empty if and only if P (0 ,
0) = 0.If F ∈ End( A ) is fixed by G and belongs to E P , then g · P = g · ρ ( F ) = ρ ( g · F ) = ρ ( F ) = P hold for any g ∈ G . This shows (2) ⇒ (1).If P is fixed by G , then the set E P is invariant by G , since ρ ( g · F ) = g · ρ ( F ) = g · P = P hold for any F ∈ E P and g ∈ G .Therefore, if F belongs to E P , then | G | P g ∈ G g · F is an element of E P that isfixed by G . This shows (1) ⇒ (2) and concludes the proof. (cid:3) Proposition 4.3.
Let H ⊂ PGL(2 , C ) = Aut( P ) be a finite subgroup and set G = π − ( H ) , where π : SL(2 , C ) → PGL(2 , C ) is the canonical surjective map. Let Λ ⊂ P be a non-empty H -invariant finite subset. (1) There exist homogeneous polynomials f , f ∈ C [ x, y ] of the same degree suchthat ( f , f ) is an endomorphism of A fixed by G and such that Λ = (cid:8) [ x : y ] ∈ P | f ( x, y ) y − f ( x, y ) x = 0 (cid:9) . (2) The morphism δ : P → P defined by [ x : y ] [ f ( x, y ) : f ( x, y )] is H -equivariant, for all pairs ( f , f ) given by the statement (1) above. (3) There exist polynomials f , f satisfying the statement (1) and also the extraproperty Λ = (cid:8) q ∈ P | δ ( q ) = q (cid:9) . This latter holds moreover for all pairs ( f , f ) given by the statement (1) , inthe case where the set Λ consists of exactly one orbit of H .Proof. (1) We let p ∈ C [ x, y ] be the (unique up a nonzero constant) square-freehomogeneous polynomial whose roots correspond to the points of Λ. Because Λ isinvariant by H , there exists a character χ : G → C ∗ such that p ◦ g = χ ( g ) p, for all g ∈ G . Since G is finite, there exists a positive integer d such that thepolynomial P = p d is fixed by G .By Lemma 4.2, there exists an endomorphism ( f , f ) ∈ C [ x, y ] × C [ x, y ] of A that is fixed by G and such that f y − f x = P . Since P is homogeneous and sincethe action of G on End( A ) is linear and preserves the filtration by degrees, we canassume that f and f are homogeneous of the same degree. This proves (1).Statement (2) follows directly from the fact that the endomorphism ( f , f ) is fixedby G .(3) Since δ is H -equivariant, its fixed-point set is invariant by H . Let us denote itby Ω δ and write f = α ˜ f and f = α ˜ f , where α, ˜ f , ˜ f are homogeneous polynomialssuch that ˜ f and ˜ f have no common root in P . Then, δ ([ x : y ]) = [ ˜ f ( x, y ) : ˜ f ( x, y )]holds for all [ x : y ] ∈ P . The set Ω δ = { q ∈ P | δ ( q ) = q } is thus the zero setof ˜ f y − ˜ f x . In particular, it is non-empty. Moreover, the equalities P = f y − f x = α ( ˜ f y − ˜ f x ) imply that Ω δ is contained in Λ.If Λ consists of exactly one orbit of H , then Ω δ = Λ follows from the fact that Ω δ is invariant by H .Let us now consider the general case, where Λ consists of r > H andwrite Λ = S ri =1 Λ i , where Λ , . . . , Λ r are disjoint orbits of H . For each i , there exist,by the previous argument, homogeneous polynomials f i, , f i, of the same degree suchthat the zero set of P i = f i, y − f i, x is equal to Λ i and such that the pair ( f i, , f i, )defines an endomorphism of A which is fixed by G .Set g = 1 r r X i =1 f i, Y j = i P j and g = 1 r r X i =1 f i, Y j = i P j . Note that g and g are homogeneous of the same degree and satisfy the equality g y − g x = Q ri =1 P i . Moreover, the endomorphism ( g , g ) ∈ End( A ) is fixed by G .In other words, it satisfies the statement (1) of the lemma.We will now show that the set Ω ˜ δ of fixed points of the morphism ˜ δ : P → P defined by ˜ δ ([ x : y ]) = [ g ( x, y ) : g ( x, y )] is equal to Λ, which will conclude theproof. Note that it is contained in Λ and invariant under the action of H , since ˜ δ is H -equivariant.Let us write g = β ˜ g and g = β ˜ g , where β, ˜ g , ˜ g are homogeneous and ˜ g , ˜ g have no common root in P . Note that the set Ω ˜ δ is equal to the zero set of thehomogeneous polynomial ˜ g y − ˜ g x . XTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 10013
We claim that none of the P i divides β . Indeed, otherwise such a P i would divideboth g and g and thus also f i, Q j = i P j and f i, Q j = i P j . Since P i has no commonroot with any of the P j , this would imply that P i divides f i, and f i, . This isimpossible, since P i = f i, y − f i, x , hence P i has degree bigger than f i, and f i, .Therefore, it follows from the equalities r Y i =1 P i = g y − g x = β ( ˜ g y − ˜ g x )that, for every index i , at least one point of Λ i is contained in Ω ˜ δ . This latter setbeing invariant by H and Λ i being an orbit under the action of H , we get that thewhole set Λ i is contained in Ω ˜ δ , for each i = 1 . . . r . This achieves the proof. (cid:3) Corollary 4.4.
Let H ⊂ PGL(2 , C ) = Aut( P ) be a finite subgroup and let Λ ⊂ P be a finite subset. The following conditions are equivalent: (1) The set Λ is non-empty and invariant by H . (2) There exists a H -equivariant morphism δ : P → P such that Λ = { q ∈ P | δ ( q ) = q } . Proof.
The implication (1) ⇒ (2) follows directly from Proposition 4.3. Let us provethe other one.Let δ : P → P be a H -equivariant morphism whose fixed-point set is equal to Λ.The set Λ is then invariant under the action of H , since δ ( h ( q )) = h ( δ ( q )) = h ( q )hold for all h ∈ H and all q ∈ Λ.Furthermore, let f , f ∈ C [ x, y ] be two homogeneous polynomials of the samedegree and without common root in P such that δ ([ x : y ]) = [ f ( x, y ) : f ( x, y )] forall points [ x : y ] ∈ P . Since Λ is the zero set of f y − f x , it is clearly non-empty. (cid:3)
5. Equivariant embeddings into the affine three-space
Let us recall that the following morphism P × P ֒ → P ([ y : y ] , [ z : z ]) [ y z : y z : y z : y z ]is a classical closed embedding of P × P into P and that it induces an isomorphismbetween P × P and the quadric in P defined by the equation x x = x x . Moreover,since this embedding is canonical (it is given by the linear system |− K P × P | ), everyautomorphism of P × P extends to a unique automorphism of P .Identifying A with the complement in P of the hyperplane defined by the equation x = x , we obtain a closed embedding ( P × P ) \ ∆ ֒ → A , where ∆ denotes thediagonal curve ∆ = { ( q, q ) | q ∈ P } ⊂ P × P .Consider the diagonal action of PGL(2 , C ) = Aut( P ) on P × P . Note that eachautomorphism of P × P coming from this action extends to an automorphism of P which preserves the plane of equation x = x . This yields an action of PGL(2 , C )on A for which the closed embedding ( P × P ) \ ∆ ֒ → A , that we defined above,becomes PGL(2 , C )-equivariant.After a change of coordinates in A , we obtain a PGL(2 , C )-equivariant embeddingof ( P × P ) \ ∆ into A , where the action of PGL(2 , C ) on A is linear. Lemma 5.1.
The morphism ι : ( P × P ) \ ∆ ֒ → A ([ y : y ] , [ z : z ]) (cid:18) y z + y z y z − y z , y z y z − y z , y z y z − y z (cid:19) is a closed embedding whose image is the hypersurface of A defined by the equation yz = x − .Moreover, ι is PGL(2 , C ) -equivariant, when we consider the actions of PGL(2 , C ) on ( P × P ) \ ∆ and A defined by PGL(2 , C ) × ( P × P ) \ ∆ → ( P × P ) \ ∆ (cid:0)(cid:0) a bc d (cid:1) , ([ y : y ] , [ z : z ]) (cid:1) ([ ay + by : cy + dy ] , [ az + bz : cz + dz ]) and PGL(2 , C ) × A → A (cid:18) a bc d (cid:19) , xyz ad − bc ad + bc ac bd ab a b cd c d · xyz . Proof.
Let Q denotes the quadric hypersurface of A defined by the equation yz = x −
1. One checks that the morphism ι induces an isomorphism between ( P × P ) \ ∆and Q whose inverse morphism is given by Q → ( P × P ) \ ∆( x, y, z ) (cid:26) ([ x + 1 : z ] , [ y : x + 1]) if x = − , ([ y : x − , [ x − z ]) if x = 1 . It is also straightforward to check that ι is PGL(2 , C )-equivariant for the given actions. (cid:3) Combining the latter lemma with the results of the previous section, we finally getAut(Γ)-equivariant embeddings of every smooth affine rational curve Γ into A . Theorem 5.2.
For every rational smooth affine curve Γ , there exist a linear actionof Aut(Γ) on A and a closed embedding τ : Γ ֒ → A which is Aut(Γ) -equivariant forthis action.Proof.
If Γ = A , it suffices to consider the embedding τ : A → A defined by τ ( t ) = ( t, , { x ax + b | a ∈ C ∗ , b ∈ C } act on A via themaps ( x, y, z ) ( ax + b ( y + 1) , y, z ).If Γ = C ∗ , we consider the embedding τ : Γ → A defined by τ ( t ) = ( t, /t, A defined by the equations z = 0 and xy = 1. Recall thatAut(Γ) = { ϕ λ : x λx | λ ∈ C ∗ } ∪ (cid:8) ψ λ : x λx − | λ ∈ C ∗ (cid:9) . The embedding τ becomes Aut(Γ)-equivariant, when we let Aut(Γ) act on A via themaps Φ λ : ( x, y, z ) ( λx, λ − y, z ) and Ψ λ : ( x, y, z ) ( λy, λ − x, z ).If Γ is equal to P \ Λ, where Λ is a finite set of at least 3 points, then its au-tomorphism group H = Aut(Γ) is the finite subgroup of PGL(2 , C ) = Aut( P ) thatpreserves the set Λ. Applying Corollary 4.4, let δ : P → P be a H -equivariant mor-phism such that Λ = (cid:8) q ∈ P | δ ( q ) = q (cid:9) . This allows us to define a closed embeddingˆ τ : Γ → ( P × P ) \ ∆ by letting ˆ τ ( q ) = ( q, δ ( q )) for all q ∈ Γ = P \ Λ. The morphism ˆ τ is moreover H -equivariant, when H acts diagonally on ( P × P ) \ ∆. XTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 10015
Composing ˆ τ with the PGL(2 , C )-equivariant closed embedding ι : ( P × P ) \ ∆ ֒ → A that we defined in Lemma 5.1, we obtain a closed embedding τ : Γ → A which is H -equivariant, as desired. (cid:3)
6. Explicit formulas for the equivariant embeddings into A The proof of Theorem 5.2 is constructive and already contains explicit Aut(Γ)-equivariant embeddings into A for the curves Γ = A and Γ = A \ { } . Let usnow describe the construction for the other cases, i.e., when the automorphism groupAut(Γ) is finite.We consider the curves Γ = P \ Λ, where Λ is a set of at least 3 points of P . Letus denote by H the subgroup of Aut( P ) = PGL(2 , C ) that restricts to Aut(Γ), anddenote as before by G its pull-back in SL(2 , C ), which is a finite group of order 2 | H | .The set Λ decomposes into r orbits Λ = S ri =1 Λ i of H . An orbit Λ i of H is given bythe zero set of a homogeneous polynomial p i ∈ C [ x, y ]. Some power P i = p d i i of p i is invariant by the action of G on P defined in Section 4. For each i , Lemma 4.2yields the existence of a G -invariant pair ( f i, , f i, ) ∈ End( A ) which satisfy f i, y − f i, x = P i . The H -equivariant morphism δ : P → P given by Proposition 4.3 (orCorollary 4.4) is thus δ : [ x : y ] [ f ( x, y ) : f ( x, y )], where f = 1 r r Y i =1 P i ! r X i =1 f i, P i and f = 1 r r Y i =1 P i ! r X i =1 f i, P i . Moreover, ( f , f ) is invariant by G and satisfies f y − f x = Q ri =1 P i .Following the proof of Theorem 5.2, we define a closed embedding Γ = P \ Λ → ( P × P ) \ ∆ by [ x : y ] ([ x : y ] , [ f : f ]). We compose then this latter with theembedding ι : ( P × P ) \ ∆ → A defined by Lemma 5.1, and obtain the followingAut(Γ) = H -equivariant closed embedding of Γ into A .Γ = P \ Λ → A [ x : y ] r r X i =1 xf i, + yf i, xf i, − yf i, , r r X i =1 xf i, xf i, − yf i, , r r X i =1 yf i, xf i, − yf i, ! . So it suffices to determine the polynomials f i, and f i, , which depend on H and Λ,to get explicit embeddings.Recall that any finite subgroup of Aut( P ) = PGL(2 , C ) is isomorphic to Z /n Z (thecyclic group of order n ), D n (the dihedral group of order 2 n ), A (the tetrahedralgroup), S (the octahedral or cubic group) or A (the icosahedral or dodecahedralgroup) and that there is only one conjugacy class for each of these groups (see e.g. [1]).1) In the cyclic case, we can assume that H ⊂ PGL(2 , C ) is generated by [ x : y ] [ ξ n x : y ], where ξ n is a primitive n -th root of unity. Its pullback G ⊂ SL(2 , C )is then generated by (cid:18) ζ ζ − (cid:19) , where ζ is a primitive 2 n -th root of unity. Anorbit Λ i of H is given by the zero set of a polynomial p i = a i x n + b i y n for some( a i , b i ) ∈ C \ { (0 , } (the cases where a i = 0 or b i = 0 provide a fixed point withmultiplicity n ). We thus get P i = ( p i ) ∈ O ( A ) G and ( f i, , f i, ) = (cid:0) b i y n − ( a i x n + b i y n ) , − a i x n − ( a i x n + b i y n ) (cid:1) ∈ End( A ) G which satisfy f i, y − f i, x = P i (note that the f i, and f i, are here not unique,and could also be chosen without common factor). The corresponding embeddingΓ = P \ Λ → A is given by[ x : y ] r r X i =1 a i x n − b i y n a i x n + b i y n r r X i =1 − b i xy n − a i x n + b i y n r r X i =1 a i x n − ya i x n + b i y n .
2) In the dihedral case, we can assume that H is generated by the maps [ x : y ] [ ξ n x : y ] and [ x : y ] [ y : x ]. So G is generated by (cid:18) ζ ζ − (cid:19) and (cid:18) ii (cid:19) ,where i denotes the imaginary unit √− i of H is given by the zero set of p i = a i ( x n + y n ) + 2 b i x n y n for some( a i , b i ) ∈ C \ { (0 , } and we thus get P i = ( p i ) ∈ O ( A ) G and ( f i, , f i, ) = (cid:0) y n − ( b i x n + a i y n ) p i , − x n − ( a i x n + b i y n ) p i (cid:1) ∈ End( A ) G which satisfy f i, y − f i, x = P i (note that P i = p i is also possible if n is even, and thatas before the polynomials f i, , f i, are not unique, and could also be chosen withoutcommon factor). This leads to the embedding Γ = P \ Λ → A defined by[ x : y ] r r X i =1 a i ( x n − y n ) a i ( x n + y n ) + 2 b i x n y n r r X i =1 − xy n − ( b i x n + a i y n ) a i ( x n + y n ) + 2 b i x n y n r r X i =1 x n − y ( a i x n + b i y n ) a i ( x n + y n ) + 2 b i x n y n .
3) In the case of the tetrahedral group, we can assume that H ∼ = A is generatedby the maps [ x : y ] [ i ( x + y ) : x − y ] and [ x : y ] [ x : − y ]. This implies that G is generated by (cid:18) i − i − i + 1 − i − (cid:19) and (cid:18) − i i (cid:19) . An orbit Λ i of H is given bythe zero set of p i = 6 a i ( x y − xy ) + b i ( x + y )( x + y − x y ) , XTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 10017 for some ( a i , b i ) ∈ C \ { (0 , } . We thus get P i = p i ∈ O ( A ) G f i, = a i ( x y − x y + 5 x y ) + b i ( − x y − x y + y ) f i, = − a i (5 x y − x y + xy ) − b i ( x − x y − x y )which satisfy ( f i, , f i, ) ∈ End( A ) G and f i, y − f i, x = P i as before. This gives theembedding Γ = P \ Λ → A defined by[ x : y ] r r X i =1 a i x y ( x + y )( x − y ) + b i ( x − x y + 11 x y − y )6 a i ( x y − xy ) + b i ( x + y )( x + y − x y )1 r r X i =1 − x ( a i ( x y − x y + 5 x y ) + b i ( − x y − x y + y ))6 a i ( x y − xy ) + b i ( x + y )( x + y − x y )1 r r X i =1 y ( a i (5 x y − x y + xy ) + b i ( x − x y − x y ))6 a i ( x y − xy ) + b i ( x + y )( x + y − x y ) . It is also possible to describe similarly the other cases ( S and A ), but the formulasare even more intricate. References [1] A. Beauville,
Finite subgroups of
PGL ( K ), in Vector bundles and complex geometry, Vol. 522of Contemp. Math. , 23–29, Amer. Math. Soc., Providence, RI (2010).[2] A. Borel, Linear algebraic groups, Vol. 126 of
Graduate Texts in Mathematics , Springer-Verlag,New York, second edition (1991), ISBN 0-387-97370-2.[3] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London (1972). Pure and Applied Mathematics, Vol. 46.[4] H. Derksen, F. Kutzschebauch, and J. Winkelmann,
Subvarieties of C n with non-extendableautomorphisms , J. Reine Angew. Math. (1999) 213–235.[5] S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms , ErgodicTheory Dynam. Systems (1989), no. 1, 67–99.[6] M. Furushima, Finite groups of polynomial automorphisms in C n , Tohoku Math. J. (2) (1983), no. 3, 415–424.[7] L. Greenberg, Maximal Fuchsian groups , Bull. Amer. Math. Soc. (1963) 569–573.[8] S. Kaliman, Extensions of isomorphisms between affine algebraic subvarieties of k n to auto-morphisms of k n , Proc. Amer. Math. Soc. (1991), no. 2, 325–334.[9] T. Petrie and J. D. Randall, Finite-order algebraic automorphisms of affine varieties , Comment.Math. Helv. (1986), no. 2, 203–221.[10] A. Sathaye, On planar curves , Amer. J. Math. (1977), no. 5, 1105–1135.[11] J.-P. Serre, Trees, Springer-Verlag, Berlin-New York (1980), ISBN 3-540-10103-9. Translatedfrom the French by John Stillwell.[12] V. Srinivas, On the embedding dimension of an affine variety , Math. Ann. (1991), no. 1,125–132.0018 J´er´emy Blanc, Jean-Philippe Furter, and Pierre-Marie Poloni
J. Blanc, Universit¨at Basel, Mathematisches Institut, Rheinsprung , CH- Basel,Switzerland.
E-mail address : [email protected] URL : http://jones.math.unibas.ch/ blanc/ J.-P. Furter, Dpt. of Math., Univ. of La Rochelle, av. Cr´epeau, 17000 La Rochelle,France
E-mail address : [email protected] URL : http://perso.univ-lr.fr/jpfurter/ P.-M. Poloni, Universit¨at Basel, Mathematisches Institut, Rheinsprung , CH- Basel, Switzerland.
E-mail address : [email protected] URL ::