EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 14R20, 14H45

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1 EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES JÉRÉMY BLANC, JEAN-PHILIPPE FURTER, AND PIERRE-MARIE POLONI Abstact. We povide the existence, fo evey complex ational smooth affine cuve Γ, of a linea action of Aut(Γ) on the affine 3-dimensional space A 3, togethe with a Aut(Γ)-equivaiant closed embedding of Γ into A 3. It is not possible to decease the dimension of the taget, the eason fo this obstuction is also pecisely descibed. 4R20, 4H45. Intoduction Thoughout this aticle, all vaieties ae algebaic vaieties defined ove the field C of complex numbes. The affine (esp. pojective) n-space is denoted by A n (esp. P n ). It is well known that any smooth affine vaiety X of dimension n admits a closed embedding into A m, when m 2n + [Si9, Theoem ]. If moeove m 2n + 2, then, by a esult of Noi, Sinivas and Kaliman (see [Si9] and [Ka9]), any two closed embeddings ι, ι : X A m ae equivalent in the sense that thee exists an automophism f Aut(A m ) such that ι = f ι. In paticula, if ι: X A m is a closed embedding of a smooth affine vaiety of dimension n into some affine space of dimension m 2n + 2, then it follows that evey automophism ϕ of X extends to an automophism of the ambient space A m, since the two embeddings ι ϕ and ι ae equivalent. Howeve, Deksen, Kutzschebauch and Winkelmann showed in [DKW99] that it is not always possible to extend the goup stuctue of Aut(X), i.e. to find a closed embedding ι: X A m and an action of Aut(X) on A m that esticts on X to the action of Aut(X) on it. Moe pecisely, they poved that thee does not exist, fo any intege m, any injective goup homomophism fom Aut(C C ) = GL 2 (Z) (C ) 2 to the goup Diff(R m ) of diffeomophisms of R m. In this aticle, we study the case of ational smooth affine cuves. Ou main esult is the following. Theoem. Let Γ be a ational smooth affine cuve. Thee exist a closed embedding ι: Γ A 3 and a linea action of Aut(Γ) on A 3 that peseves ι(γ) and such that ι becomes Aut(Γ)-equivaiant. Moeove, we pove that it is not possible to decease the dimension of the taget (Coollay 2.4). The authos gatefully acknowledge suppot by the Swiss National Science Foundation Gant "Biational Geomety" PP00P2_28422 / and by the Fench National Reseach Agency Gant "BiPol", ANR--JS

2 2 JÉRÉMY BLANC, JEAN-PHILIPPE FURTER, AND PIERRE-MARIE POLONI Theoem 2. Thee exist ational smooth affine cuves Γ with Aut(Γ) and such that fo evey closed embedding of Γ in A 2, the identity on Γ is its only automophism that extends to A 2. Recall that, if G is an algebaic goup acting on an affine vaiety X, then X admits a G- equivaiant closed embedding into a finite dimensional G-module (see [Bo9, Poposition.2, p. 56]). Theefoe, it was aleady known that thee exists, fo evey ational smooth affine cuve Γ, a linea action of Aut(Γ) on an affine space A m and a Aut(Γ)-equivaiant closed embedding of Γ into A m. Howeve, this geneal esult only gave the embedding dimension m = 2 Aut(Γ), when the automophism goup Aut(Γ) is finite. Let us also ecall a esult of Geenbeg ([G63]) which assets that evey finite goup G is equal to the automophism goup of a smooth pojective cuve, and thus of an affine one. Since the goup Bi(A 3 ) of biational tansfomations of the thee-dimensional space does not contain all finite goups as subgoups (see [Po]), this yields the existence of smooth affine cuves Γ that do not admit closed embeddings into A 3 in such a way that the action of Aut(Γ) on Γ extends to an action on A 3. Theefoe, Theoem cannot be genealised fo all smooth affine cuves. The aticle is oganised as follows. Section 2 concens embeddings of ational smooth affine cuves into the affine plane. We give examples of automophisms of such cuves that do not extend, and pove Theoem 2 (see Coollay 2.4). Section 3 is devoted to the study of embeddings of smooth ational cuves into A 3 whose images ae contained in a hypeplane. We pove that they ae all equivalent and thus that any two closed embeddings of a ational smooth affine cuve into A 2 become equivalent, when seen as embeddings in A 3 (Poposition 3.). This answes a question of Bhatwadeka and Sinivas in this case. In section 4 we ealise evey non-empty subset of P that is invaiant by a subgoup H of Aut(P ) as the fixed-point set of a H-equivaiant endomophism of P (Coollay 4.4). This esult is used in Section 5 to pove Theoem (see Poposition 5.2). Explicit fomulas ae given in Section Embeddings of ational smooth affine cuves into the plane Let us ecall that evey ational smooth affine cuve Γ is isomophic to P \ Λ, whee Λ is a finite set of points. In paticula, it admits a closed embedding into A 2. Indeed, Γ can also be seen as the complement in A of a finite numbe (possibly zeo) of points and we can conside the closed embedding τ : Γ A 2 given by x (x, P (x)), whee P C[x] is a polynomial whose oots ae exactly the emoved points. Note that the image of τ is the hypesuface of A 2 defined by the equation P (x)y =. Moeove, the automophism goup Aut(Γ) of the cuve Γ = P \ Λ is equal to the goup of automophisms of P that peseve the set Λ. This gives a goup homomophism fom Aut(Γ) to the symmetic goup Sym. Note that this homomophism is injective if and only if 3. If is equal to o 2, then Γ is isomophic to A o A \ {0}, and its automophism goup is C C o {±} C espectively. If 3, then Aut(Γ) is a finite goup.

3 EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 3 The cuves A and A \ {0} admit obviously closed embeddings into A 2 such that all thei automophisms extend to automophisms of A 2. (Conside fo example the hypesufaces of A 2 defined by the equation y = 0 and xy =, espectively.) Howeve, it is no longe tue fo the cuve A \ {0, }. Poposition 2.. Let Γ = A \ {0, }. Fo evey closed embedding τ : Γ A 2, thee exists an automophism of Γ that does not extend to A 2. Poof. Note that the goup of automophisms of Γ is the goup Sym 3 of pemutations of a set of thee elements, coesponding to the thee points at infinity, i.e. the points of P \ ι(γ), whee ι is any (open) embedding of Γ in P. It is geneated by the automophisms ρ: x /( x) and σ : x x and we have Aut(Γ) =< σ, ρ σ 2 = ρ 3 =, σρσ = ρ >= Sym 3. Suppose fo contadiction that thee exists a closed embedding τ : Γ A 2 fo which evey automophism of Γ extends. Since the identity is the only automophism of A 2 that esticts to the identity on a closed cuve isomophic to A \ {0, } (see Lemma 2.2 below), we would have a subgoup G Aut(A 2 ) isomophic to Sym 3 whose estiction to τ(γ) yields Aut(Γ). We now pove that this is impossible. Fist, ecall that G is conjugate to a subgoup of GL(2, C). This uses the stuctue of amalgamated poduct of Aut(A 2 ) (see [Kam79]). Then, one easily checks that G is conjugate to the subgoup G of GL(2, C) geneated by ˆρ: (x, y) (y, x y) and ˆσ : (x, y) (y, x). We let f Aut(A 2 ) be an automophism such that fgf = G and we conside the embedding ˆτ = f τ of Γ in A 2. The automophism goup of Γ extends then to G fo this embedding. Remak that the set {ω ω 2 ω + = 0} Γ is an obit of size 2 of Aut(Γ). But one checks that G GL(2, C) does not have any obit of size 2 in the affine plane A 2. This gives a contadiction. Lemma 2.2. The set of fixed points of a plane polynomial automophism is eithe a finite set of points (possibly empty), eithe a finite disjoint union of subvaieties isomophic to A, o eithe the whole plane. Poof. Using the amalgamated stuctue of Aut(A 2 ), it is obseved in [FM89] that a plane polynomial automophism is conjugate eithe to a tiangula automophism (x, y) (ax + p(y), by + c) with a, b, c C and p(y) C[y], o to some cyclically educed element (see [Se80, I..3] o [FM89, p. 70] fo the definition of a cyclically educed element). In the fist case, an obvious computation shows that the set of fixed points is eithe empty, eithe a point, eithe a finite disjoint union of subvaieties isomophic to A o eithe the whole plane. In the second case, by [FM89, Theoem 3.], the set of fixed points is a non-empty finite set of points. Using tools of biational geomety, we can actually specify the statement of Poposition 2.. Indeed, Theoem 2.3 below shows that thee is no closed embedding of the cuve A \ {0, } into A 2 such that the automophism ρ: x /( x) extends to an automophism of the affine plane.

4 4 JÉRÉMY BLANC, JEAN-PHILIPPE FURTER, AND PIERRE-MARIE POLONI Befoe we state this esult, let us ecall that any automophism f of P of finite ode n > is conjugate to [x : y] [x : ξy], whee ξ is a pimitive n-th oot of unity. In paticula, it has the following popeties: () the automophism f fixes exactly two points of P ; (2) all othe obits unde the action of f have size n. Thus, the following holds fo evey automophism g Aut(Γ) of ode n > of a ational smooth cuve Γ. () The automophism g fixes 0, o 2 points of Γ; (2) all othe obits unde the action of g have size n. Theoem 2.3. Let Γ be a ational smooth affine cuve and let g Aut(Γ) be an automophism. () If g fixes at most one point of Γ, thee is a closed embedding τ : Γ A 2 such that g extends to an automophism of A 2. (2) If g is of finite ode n > with n odd and if it fixes exactly two points of Γ, then thee is no closed embedding τ : Γ A 2 such that g extends to an automophism of A 2. Poof. () Let P C[x] be a non-zeo polynomial such that Γ is isomophic to A \ {x A P (x) = 0}. Let g Aut(Γ) be an automophism that fixes at most one point of Γ. Let us denote also by g its extension as an automophism of P. We can assume that g fixes the point of P at infinity, so that it is of the fom x ax + b, fo some a C and b C. Moeove P (ax + b) = µp (x) fo some µ C. When we embed Γ into A 2 via the map x (x, P (x)), the automophism g extends to (x, y) (ax + b, µ y). (2) Let g Aut(Γ) be of finite ode n > with n odd such that it fixes 2 points of Γ. Suppose, fo contadiction, that thee exists a closed embedding τ : Γ A 2 fo which g extends to an automophism h of A 2. Since g has finite ode n, the automophism h n Aut(A 2 ) fixes pointwise the cuve τ(γ). Because g fixes two points of Γ, τ(γ) is not isomophic to A, hence h n is tivial by Lemma 2.2. Recall that evey automophism of A 2 of finite ode is conjugate to a linea one (see [Kam79]). Thus, thee exists an automophism f Aut(A 2 ) such that ĥ = f h f is linea. Moeove, the automophism g Aut(Γ) extends to ĥ, when we conside the embedding ˆτ = f τ : Γ A 2. The linea automophism ĥ extends to an automophism of P2, and the closue of ˆτ(Γ) in P 2 is a pojective ational cuve C, having all its singula points on the line L = P 2 \ A 2. If C is smooth, it is isomophic to P. Hence, it is a conic o a line, and thus intesects L into o 2 points, which contadicts the fact that g acts on C with ode n > 2 and with no fixed point at infinity. This implies that C is singula. Denote by η : X P 2 the blow-up of the points of P 2 that ae singula points of C, and wite C X the stict tansfom of C in X. If C is singula, we denote by η 2 : X 2 X the blow-up of the points of X that ae singula points of C, and wite C 2 the stict tansfom of C in X 2. We continue like this until we end with a smooth cuve C m X m. Note that C m is isomophic to P. Fo i =,..., m, the lift of ĥ yields

5 EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 5 an automophism h i of X i which peseves the cuve C i. It also peseves the pull-back of A 2 in X i, which is again isomophic to A 2. Fo i =,..., m, we denote by B i C i the (finite) set of points not lying in A 2. Each point p B i has a multiplicity m(p) as a point of C i. This multiplicity is a positive intege and it is equal to if and only if C i is smooth at this point p. Denote by B 0 the set of points of C 0 = C X 0 = P 2 and let us use the same notation as above fo the multiplicities of the points of B 0. Witing d the degee of C P 2, the geometic genus of C can be computed with the following classical fomula. (Note that it is equal to 0, since C is ational.) ( ) 0 = (d )(d 2) 2 m m(p) (m(p) ). 2 p B i i=0 Let us now pove that fo each j {,..., m}, each obit I B j unde the action of h j satisfies ( ) m(p) = m(p ) fo all p, p I, and the intege p I m(p) is a multiple of n. Fo j = m, the assetion ( ) holds fo all obits I B m. Indeed, C m is isomophic to P and the action of h m on B m C m is fixed-point-fee, so all obits have size n and all multiplicities ae equal to. Then, we can pove ( ) fo j < m, assuming it holds fo evey intege k with j + k m. Fo this, let J B j be an obit unde the action of h j and let us denote by m J the multiplicity m(p) of a point p J. Note that this multiplicity does not depend of the choice of p, since h j acts tansitively on J. If m J =, all points of J ae smooth, and so the pull-back by η j+ of J consists of J points of multiplicity m j =. This implies p J m(p) nn, by induction hypothesis. If m J >, then all points of J ae singula points of the cuve C j and ae thus blownup by η j+ : X j+ X j. The numbe m J is the multiplicity of the cuve C j at the point p J. Denoting by E p X j+ the cuve contacted by η j+ onto p, the numbe m J is the intesection numbe E p C j+. This latte is equal to the sum of m q (E p ) m q (C j+ ), whee q uns though all points infinitely nea to p, and whee m q (E p ) and m q (C j+ ) ae the multiplicities of the stict tansfoms of E p and C j+ at q, espectively. Note that m q (E p ) is equal to 0 o. Theefoe, the sum p J m J is equal to a sum of multiplicities of obits in B k fo k j +. By induction hypothesis, it is a multiple of n. This achieves to pove ( ). Since n is odd, Equation ( ) and Assetion ( ) imply that the intege (d )(d 2) 2 is a multiple of n. Computing the intesection numbe d = L C as befoe as a sum of multiplicities, we also get that d is a multiple of n. Since the geatest common diviso of d and (d )(d 2) 2 is o 2, this contadicts the assumption n > 2. This concludes the poof. Coollay 2.4. Thee exist ational smooth affine cuves Γ with Aut(Γ) and such that fo evey closed embedding of Γ in A 2, the identity on Γ is its only automophism which extends to an automophism of A 2.

6 6 JÉRÉMY BLANC, JEAN-PHILIPPE FURTER, AND PIERRE-MARIE POLONI Poof. Let ω = e 2iπ/3 and a =. Let a 2,..., a k be complex numbes algebaically independent ove Q. We conside the cuve Γ = P \ Λ, whee Λ is the following set of 3k points Λ = { [a i ω j : ] i =,..., k, j = 0,..., 2 }. The map h: [x : y] [x : ωy] is obviously an automophism of Γ. We will now pove that it geneates the whole automophism goup Aut(Γ) if k 3. This will conclude the poof, since h and h 2 do not extend to automophisms of A 2 by Theoem 2.3. Let g Aut(Γ) be an automophism of Γ. It extends to an automophism of P that peseves the set Λ. Let us denote this latte also by g. Conside the 4-tuple V = ( [ : ], [ω : ], [ω 2 : ], [a 2 : ] ). Since a 2,..., a k ae algebaically independent ove Q, the image of V by g is a 4-tuple of points contained in the set S = { [ : ], [ω : ], [ω 2 : ], [a 2 : ], [a 2 ω : ], [a 2 ω 2 : ] }. Indeed, the coss-atio of g(v ) must be equal to the coss-atio of V, i.e. to ω(ω a 2 )/(a 2 ). The same agument with the 4-tuple ( [ : ], [ω : ], [ω 2 : ], [a 3 : ] ) allows us to conclude that g peseves the set { [ : ], [ω : ], [ω 2 : ] }. Theefoe, g is eithe a powe of h, o it is one of the maps ϕ i : [x : y] [y : xω i ] with i = Finally, note that g cannot be one of the ϕ i s, since ϕ i sends the point [a 2 : ] onto [ : ], which does not belong to the set S. a 2 ωi Remak 2.5. The poof of Coollay 2.4 shows that if k 3 and if the set Λ P is geneal among all sets of distinct 3k points invaiant by the map [x : y] [x : ωy], then fo all closed embeddings of the cuve Γ = P \ Λ into A 2, the identity is the only automophism of Γ that extends to an automophism of A 2. On the contay, when k 2, evey such cuve Γ admits an automophism of ode 2 and Poposition 2.6 below implies then that this latte extends to an automophism of A 2 fo a well-chosen closed embedding of Γ into A 2. Poposition 2.6. Let Γ be a ational smooth affine cuve and let σ Aut(Γ) be an automophism of Γ of ode 2. Thee exists a closed embedding of Γ in A 2 and an automophism ˆσ Aut(A 2 ) of ode 2 whose estiction to Γ yields σ. Poof. Let Γ = P \ Λ, whee Λ is a finite set of points. Let us denote by σ the extension of the automophism σ Aut(Γ) as an automophism of P. If it fixes a point of Λ, the esult follows fom Theoem 2.3. We can thus assume that the two points of P fixed by (the extension of) σ belong to Γ. Let p be a point of Λ. Its obit {p, σ(p)} is then contained in Λ. Let C be the cuve C = P \ {p, σ(p)}. Note that C is isomophic to A \ {0} and that σ esticts to an automophism of C. Remak that all automophisms of A \{0} of ode 2 with two fixed points ae conjugate to the automophism x x Aut(Spec(C[x, x ])). Theefoe, thee is a closed embedding τ : C A 2 whose image is the hypesuface defined by the equation y 2 = x 2

7 EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 7 and such that the automophism σ Aut(C) extends to the automophism (x, y) ( x, y). Moeove, the cuve τ(γ) is then equal to a set of points of τ(c) satisfying that n (y a i) 0, fo some n 0 and distinct a,..., a n C \ {±}. Let Y A 2 be the closed cuve defined by the equation ( n 2 y 2 = x 2 (y a i )). Conside finally the biational tansfomation of A 2 defined by ( ) x (x, y) n (y a i), y, which esticts to an isomophism between τ(γ) and Y. automophism (x, y) ( x, y), this yields the esult. Since it commutes with the 3. Plana embeddings in the space The following question of Bhatwadeka and Sinivas is asked at the end of [Si9]: ae any two embeddings of a smooth affine cuve in A 2 equivalent, when consideed as embeddings in A 3? Fo a ational smooth affine cuve, the following esult gives a positive answe. Poposition 3.. Let Γ be a ational smooth affine cuve. () If τ, τ 2 : Γ A 3 ae two closed embeddings whose images ae contained in a hypeplane (plana embeddings in the space), thee exists an automophism α Aut(A 3 ) such that τ 2 = α τ, i.e. any two plana embeddings in the space ae equivalent. (2) In paticula, fixing a plana embedding Γ A 3, evey automophism of Γ extends to A 3. Poof. Let Γ = A \ {x A P (x) = 0}, whee P C[x] is a polynomial with simple oots. Note that the coodinate ing of Γ is C[Γ] = C[x, P (x)] and ecall that the map x (x, P (x) ) defines a closed embedding of Γ in A2. To pove the poposition, it suffices to pove that any plana embedding is equivalent to the one given by x (x, P (x), 0). Let τ : Γ A 3 be a plana embedding of Γ. We can compose τ with an affine automophism f of A 3 and get an embedding τ 2 = f τ : Γ A 3 of the fom x (0, Q(x), R(x)), whee Q, R C(x) ae ational functions without poles on Γ. Since τ 2 is a closed embedding of the cuve Γ, the equality C[x, P (x)] = C[Q(x), R(x)] holds. In paticula, thee exists a polynomial A C[X, Y ] such that A(Q(x), R(x)) = x. Now, we compose τ 2 with the automophism of A 3 defined by f 2 (X, Y, Z) = (X + A(Y, Z), Y, Z) and obtain the embedding τ 3 : Γ A 3 given by τ 3 : x (x, Q(x), R(x)). Because of the equality C[x, P (x) ] = C[Q(x), R(x)], all zeos of P (x) ae poles of aq(x) + br(x) fo geneal complex numbes a, b C. We can thus compose τ 3 with

8 8 JÉRÉMY BLANC, JEAN-PHILIPPE FURTER, AND PIERRE-MARIE POLONI a linea automophism of the fom (X, Y, Z) (X, ay + bz, Z) and get an embedding τ 4 : Γ A 3 of the fom ( τ 4 : x x, Q (x) Q 2 (x), R ) (x), R 2 (x) whee Q, Q 2, R, R 2 C[x] ae polynomials such that Q and Q 2 (esp. R and R 2 ) have no common facto, and such that P (x) divides Q 2 (x). In paticula, thee exist two polynomials U, V C[x] such that UQ + V P =. It follows P = UQ + V P P whee S C[x] satisfies P S = Q 2. This implies C[x, P ] C[x, Q Q 2 ] and thus = U Q P + V = SU Q Q 2 + V, C[x, Q Q 2, R R 2 ] = C[x, P ] = C[x, Q Q 2 ]. Theefoe, thee exist polynomials B, C C[X, Y ] such that B(x, Q (x) Q 2 (x) ) = P (x) R (x) R 2 (x) and C(x, P (x) ) = Q (x) Q 2 (x). Finally, we conside the automophisms of A3 defined by f 4 (X, Y, Z) = (X, Y, Z + B(X, Y )) and f 5 (X, Y, Z) = (X, Z, Y C(X, Z)). One checks that f 5 f 4 τ 4 : Γ A 3 is the desied embedding x (x, P (x), 0). Note that the poof above is constuctive. In paticula, a plana embedding of a smooth ational cuve Γ in A 3 and an automophism ϕ of Γ being given, it allows us to constuct an explicit automophism of A 3 which extends ϕ. Example 3.2. Let Γ be the cuve Γ = A \{0, } and let ρ Aut(Γ) be the automophism of Γ defined by ρ(x) = /( x). We saw in Section 2 that thee is no closed embedding of Γ into A 2 such that ρ extends to an automophism of A 2. Howeve, it extends to an automophism of A 3, when we conside the embedding τ : Γ A 3 defined by x (x, /x(x ), 0). Following the poof of Poposition 3., we let f, f 2,..., f 5 be the automophisms of A 3 defined by f (X, Y, Z) = (Z, Y, X), f 2 (X, Y, Z) = (X +Y +2 Y Z 2, Y, Z), f 3 (X, Y, Z) = (X, ay +bz, Z), f 4 (X, Y, Z) = (X, Y, Z ab [(b+(a b)x)(y ax +2a) (a b)2 ](+X)) and f 5 (X, Y, Z) = (X, Z, Y ax + 2a + az + (b a)xz), whee a, b C ae geneal complex numbes. Setting F = f 5 f 4 f, one checks F τ ρ = τ. This implies that F is an extension of the automophism ρ Aut(Γ). Remak 3.3. To ou knowledge, thee is no known example of a smooth affine cuve admitting two non-equivalent embeddings into A 3. Paadoxically, we do not know any smooth affine cuve such that all its embeddings into A 3 ae equivalent! The case of the affine line is of paticula inteest. On one hand, all closed embeddings of A into A 2 ae equivalent by the famous Abhyanka-Moh-Suzuki theoem. On the othe hand, all closed embeddings of A into A n with n 4 ae also equivalent (see [Si9] o [Ka9]).

9 EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 9 4. Actions of SL(2, C) on End(A 2 ) and of PGL(2, C) on P The aim of this section is to constuct, fo evey non-empty subset Λ of P that is invaiant by a subgoup H of Aut(P ), a H-equivaiant endomophism of P whose fixedpoint set is equal to the set Λ (Coollay 4.4). We will use this esult late on to constuct embeddings of evey ational smooth affine cuve into A 3 in such a way that the whole automophism goup of the cuve extends to a subgoup of Aut(A 3 ). Fo the est of the pape we will conside the following actions of the goup SL(2, C) on O(A 2 ) = C[x, y] and End(A 2 ) = C[x, y] C[x, y]. and SL(2, C) O(A 2 ) O(A 2 ) (g, P ) g P := P g SL(2, C) End(A 2 ) End(A 2 ) (g, F ) g F := g F g. Note that these actions come fom the natual action of SL(2, C) on A 2. Indeed, denote by V the space A 2 as a complex vecto space of dimension 2 and identify the set of the linea foms on it as the dual space V. The action of SL(V ) on V yields actions on V, on the symmetic algeba S(V ) and on S(V ) V. The natual isomophisms between S(V ) and C[x, y] = O(A 2 ), and between S(V ) V and C[x, y] C[x, y] = End(A 2 ), lead then to the SL(2, C)-actions that we defined above. Lemma 4.. The map ρ: End(A 2 ) O(A 2 ) defined by C[x, y] C[x, y] C[x, y] (f, f 2 ) f y f 2 x is SL(2, C)-equivaiant, when we conside the actions defined above. Poof. The esult could of couse be checked by diect computations, but let us mention that its also follows fom the fact that ρ coesponds to the mophism S(V ) V S(V ) given by the composition τ 2 τ, whee τ and τ 2 ae the two following homomophisms of SL(V )-modules. τ : S(V ) V S(V ) V V V p v p v id, whee id denotes the identity element seen as an element of V V = Hom(V, V ), and, v 3 )(pv2 ). τ 2 : S(V ) V V V S(V ) p v v2 v 3 det(v Poposition 4.2. Let G SL(2, C) be a finite subgoup of SL(2, C) and let P C[x, y]. The following conditions ae equivalent: () The polynomial P satisfies P (0, 0) = 0 and is fixed by G. (2) Thee exists an endomophism F = (f, f 2 ) of A 2 that is fixed by G and such that ρ(f ) = f y f 2 x = P.

10 0 JÉRÉMY BLANC, JEAN-PHILIPPE FURTER, AND PIERRE-MARIE POLONI Poof. Let E P End(A 2 ) be the set E P = ρ (P ) = {(f, f 2 ) C[x, y] C[x, y] f y f 2 x = P }. This defines an affine subset of the C-vecto space End(A 2 ), since the endomophism (λf + ( λ)f 3, λf 2 + ( λ)f 4 ) belongs to E P, fo any (f, f 2 ), (f 3, f 4 ) End(A 2 ) and any λ C. Moeove, E P is non-empty if and only if P (0, 0) = 0. If F End(A 2 ) is fixed by G and belongs to E P, then g P = g ρ(f ) = ρ(g F ) = ρ(f ) = P hold fo any g G. This shows (2) (). If P is fixed by G, then the set E P is invaiant by G, since ρ(g F ) = g ρ(f ) = g P = P hold fo any F E P and g G. Theefoe, if F belongs to E P, then G g G g F is an element of E P that is fixed by G. This shows () (2) and concludes the poof. Lemma 4.3. Let H PGL(2, C) = Aut(P ) be a finite subgoup and set G = π (H), whee π : SL(2, C) PGL(2, C) is the canonical sujective map. Let Λ P be a nonempty H-invaiant finite subset. () Thee exist homogeneous polynomials f, f 2 C[x, y] of the same degee such that (f, f 2 ) is an endomophism of A 2 fixed by G and such that Λ = { [x : y] P f (x, y)y f 2 (x, y)x = 0 }. (2) The mophism δ : P P defined by [x : y] [f (x, y) : f 2 (x, y)] is H-equivaiant, fo all pais (f, f 2 ) given by the statement () above. (3) Thee exist polynomials f, f 2 satisfying the statement () and also the exta popety Λ = { q P δ(q) = q }. This latte holds moeove fo all pais (f, f 2 ) given by the statement (), in the case whee the set Λ consists of exactly one obit of H. Poof. () We let p C[x, y] be the (unique up a nonzeo constant) squae-fee homogeneous polynomial whose oots coespond to the points of Λ. Because Λ is invaiant by H, thee exists a chaacte χ: G C such that p g = χ(g)p, fo all g G. Since G is finite, thee exists a positive intege d such that the polynomial P = p d is fixed by G. By Poposition 4.2, thee exists an endomophism (f, f 2 ) C[x, y] C[x, y] of A 2 that is fixed by G and such that f y f 2 x = P. Since P is homogeneous and since the action of G on End(A 2 ) is linea and peseves the filtation by degees, we can assume that f and f 2 ae homogeneous of the same degee. This poves (). Statement (2) follows diectly fom the fact that the endomophism (f, f 2 ) is fixed by G.

11 EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES (3) Since δ is H-equivaiant, its fixed-point set is invaiant by H. Let us denote it by Ω δ and wite f = α f and f 2 = α f 2, whee α, f, f 2 ae homogeneous polynomials such that f and f 2 have no common oot in P. Then, δ([x : y]) = [ f (x, y) : f 2 (x, y)] holds fo all [x : y] P. The set Ω δ = {q P δ(q) = q} is thus the zeo set of f y f 2 x. In paticula, it is non-empty. Moeove, the equalities P = f y f 2 x = α( f y f 2 x) imply that Ω δ is contained in Λ. If Λ consists of exactly one obit of H, then Ω δ = Λ follows fom the fact that Ω δ is invaiant by H. Let us now conside the geneal case, whee Λ consists of > obits of H and wite Λ = Λ i, whee Λ,..., Λ ae disjoint obits of H. Fo each i, thee exist, by the pevious agument, homogeneous polynomials f i,, f i,2 of the same degee such that the zeo set of P i = f i, y f i,2 x is equal to Λ i and such that the pai (f i,, f i,2 ) defines an endomophism of A 2 which is fixed by G. Set g = f i, P j and g 2 = f i,2 P j. j i Note that g and g 2 ae homogeneous of the same degee and satisfy the equality g y g 2 x = P i. Moeove, the endomophism (g, g 2 ) End(A 2 ) is fixed by G. In othe wods, it satisfies the statement () of the lemma. We will now show that the set Ω δ of fixed points of the mophism δ : P P defined by δ([x : y]) = [g (x, y) : g 2 (x, y)] is equal to Λ, which will conclude the poof. Note that it is contained in Λ and invaiant unde the action of H, since δ is H-equivaiant. Let us wite g = β g and g 2 = β g 2, whee β, g, g 2 ae homogeneous and g, g 2 have no common oot in P. Note that the set Ω δ is equal to the zeo set of the homogeneous polynomial g y g 2 x. We claim that none of the P i divides β. Indeed, othewise such a P i would divide both g and g 2 and thus also f i, j i P j and f i,2 j i P j. Since P i has no common oot with any of the P j, this would imply that P i divides f i, and f i,2. This is impossible, since P i = f i, y f i,2 x, hence P i has degee bigge than f i, and f i,2. Theefoe, it follows fom the equalities P i = g y g 2 x = β( g y g 2 x) that, fo evey index i, at least one point of Λ i is contained in Ω δ. This latte set being invaiant by H and Λ i being an obit unde the action of H, we get that the whole set Λ i is contained in Ω δ, fo each i =.... This achieves the poof. Coollay 4.4. Let H PGL(2, C) = Aut(P ) be a finite subgoup and let Λ P be a finite subset. The following conditions ae equivalent: () The set Λ is non-empty and invaiant by H. (2) Thee exists a H-equivaiant mophism δ : P P such that Λ = {q P δ(q) = q}. Poof. The implication () (2) follows diectly fom Lemma 4.3. Let us pove the othe one. j i

12 2 JÉRÉMY BLANC, JEAN-PHILIPPE FURTER, AND PIERRE-MARIE POLONI Let δ : P P be a H-equivaiant mophism whose fixed-point set is equal to Λ. The set Λ is then invaiant unde the action of H, since δ(h(q)) = h(δ(q)) = h(q) hold fo all h H and all q Λ. Futhemoe, let f, f 2 C[x, y] be two homogeneous polynomials of the same degee and without common oot in P such that δ([x : y]) = [f (x, y) : f 2 (x, y)] fo all points [x : y] P. Since Λ is the zeo set of f y f 2 x, it is clealy non-empty. 5. Equivaiant embeddings into the affine thee-space Let us ecall that the following mophism P P P 3 ([y 0 : y ], [z 0 : z ]) [y 0 z 0 : y 0 z : y z 0 : y z ] is a classical closed embedding of P P into P 3 and that it induces an isomophism between P P and the quadic in P 3 defined by the equation x 0 x 3 = x x 2. Moeove, since this embedding is canonical (it is given by the linea system 2 K P P ), evey automophism of P P extends to a unique automophism of P 3. Identifying A 3 with the complement in P 3 of the hypeplane defined by the equation x = x 2, we obtain a closed embedding (P P )\ A 3, whee denotes the diagonal cuve = {(q, q) q P } P P. Conside the diagonal action of PGL(2, C) = Aut(P ) on P P. Note that each automophism of P P coming fom this action extends to an automophism of P 3 which peseves the plane of equation x = x 2. This yields an action of PGL(2, C) on A 3 fo which the closed embedding (P P ) \ A 3, that we defined above, becomes PGL(2, C)-equivaiant. Afte a change of coodinates in A 3, we obtain a PGL(2, C)-equivaiant embedding of (P P ) \ into A 3, whee the action of PGL(2, C) on A 3 is linea. Lemma 5.. The mophism ι: (P P ) \ A( 3 ) y0 z + y z 0 2y 0 z 0 2y z ([y 0 : y ], [z 0 : z ]),, y 0 z y z 0 y 0 z y z 0 y 0 z y z 0 is a closed embedding whose image is the hypesuface of A 3 defined by the equation yz = x 2. Moeove, ι is PGL(2, C)-equivaiant, when we conside the actions of PGL(2, C) on (P P ) \ and A 3 defined by (( PGL(2, ) C) (P P ) \ ) (P P ) \ a b, ([y c d 0 : y ], [z 0 : z ]) ([ay 0 + by : cy 0 + dy ], [az 0 + bz : cz 0 + dz ]) and ( a b c d PGL(2, C) A 3 A 3 ), x y z ad bc ad + bc ac bd 2ab a 2 b 2 2cd c 2 d 2 x y z.

13 EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 3 Poof. Let Q denotes the quadic hypesuface of A 3 defined by the equation yz = x 2. One checks that the mophism ι induces an isomophism between (P P ) \ and Q whose invese mophism is given by Q { (P P ) \ ([x + : z], [y : x + ]) if x, (x, y, z) ([y : x ], [x : z]) if x. It is also staightfowad to check that ι is PGL(2, C)-equivaiant fo the given actions. Combining the latte lemma with the esults of the pevious section, we finally get Aut(Γ)-equivaiant embeddings of evey smooth affine ational cuve Γ into A 3. Poposition 5.2. Fo evey ational smooth affine cuve Γ, thee exist a linea action of Aut(Γ) on A 3 and a closed embedding τ : Γ A 3 which is Aut(Γ)-equivaiant fo this action. Poof. If Γ = A, it suffices to conside the embedding τ : A A 3 defined by τ(t) = (t, 0, 0), and to let Aut(Γ) = {x ax + b a C, b C} act on A 3 via the maps (x, y, z) (ax + b(y + ), y, z). If Γ = C, we conside the embedding τ : Γ A 3 defined by τ(t) = (t, /t, 0). Its image is the cuve in A 3 defined by the equations z = 0 and xy =. Recall that the automophism goup of Γ is Aut(Γ) = {ϕ λ : x λx λ C } { ψ λ : x λx λ C }. The embedding τ becomes Aut(Γ)-equivaiant, when we let Aut(Γ) act on A 3 via the maps Φ λ : (x, y, z) (λx, λ y, z) and Ψ λ : (x, y, z) (λ y, λx, z). If Γ is equal to P \Λ, whee Λ is a finite set of at least 3 points, then its automophism goup H = Aut(Γ) is the finite subgoup of PGL(2, C) = Aut(P ) that peseves the set Λ. Applying Coollay 4.4, let δ : P P be a H-equivaiant mophism such that Λ = { q P δ(q) = q }. This allows us to define a closed embedding ˆτ : Γ (P P )\ by letting ˆτ(q) = (q, δ(q)) fo all q Γ = P \. The mophism ˆτ is moeove H- equivaiant, when H acts diagonally on (P P ) \. Composing ˆτ with the PGL(2, C)-equivaiant closed embedding ι: (P P ) \ A 3 that we defined in Lemma 5., we obtain a closed embedding τ : Γ A 3 which is H- equivaiant, as desied. 6. Explicit fomulas fo the equivaiant embeddings into A 3 The poof of Poposition 5.2 is constuctive and aleady contains explicit Aut(Γ)- equivaiant embeddings into A 3 fo the cuves Γ = A and Γ = A \ {0}. Let us now descibe the constuction fo the othe cases, i.e., when the automophism goup Aut(Γ) is finite. We conside the cuves Γ = P \ Λ, whee Λ is a set of at least 3 points of P. Let us denote by H the subgoup of Aut(P ) = PGL(2, C) that esticts to Aut(Γ), and denote as befoe by G its pull-back in SL(2, C), which is a finite goup of ode 2 H. The set Λ decomposes into obits Λ = Λ i of H. An obit Λ i of H is given by the zeo set of a homogeneous polynomial p i C[x, y]. Some powe P i = p d i i of p i is invaiant by the action of G on P defined in Section 4. Fo each i, Poposition 4.2 yields the existence of a G-invaiant pai (f i,, f i,2 ) End(A 2 ) which satisfy f i, y f i,2 x = P i.

14 4 JÉRÉMY BLANC, JEAN-PHILIPPE FURTER, AND PIERRE-MARIE POLONI The H-equivaiant mophism δ : P P given by Lemma 4.3 (o Coollay 4.4) is thus δ : [x : y] [f (x, y) : f 2 (x, y)], whee ( ( f = f i, P i) and f 2 = f i,2 P i). P i P i Moeove, (f, f 2 ) is invaiant by G and satisfies f y f 2 x = P i. Following the poof of Poposition 5.2, we define a closed embedding Γ = P \ Λ (P P ) \ by [x : y] ([x : y], [f : f 2 ]). We compose then this latte with the embedding ι: (P P ) \ A 3 defined by Lemma 5., and obtain the following Aut(Γ) = H-equivaiant closed embedding of Γ into A 3. Γ = P \ Λ A( 3 [x : y] xf i,2 + yf i, xf i,2 yf i,, 2xf i, xf i,2 yf i,, 2yf i,2 xf i,2 yf i, So it suffices to detemine the polynomials f i, and f i,2, which depend on H and Λ, to get explicit embeddings. Recall that any finite subgoup of Aut(P ) = PGL(2, C) is isomophic to Z/nZ (the cyclic goup of ode n), D 2n (the dihedal goup of ode 2n), A 4 (the tetahedal goup), S 4 (the octahedal o cubic goup) o A 5 (the icosahedal o dodecahedal goup) and that thee is only one conjugacy class fo each of these goups (see e.g. [Beau0]). ) In the cyclic case, we can assume that H PGL(2, C) is geneated by [x : y] [ξ n x : y], whee ( ξ n is a pimitive ) n-th oot of unity. Its pullback G SL(2, C) is then ζ 0 geneated by 0 ζ, whee ζ is a pimitive 2n-th oot of unity. An obit Λ i of H is given by the zeo set of a polynomial p i = a i x n + b i y n fo some (a i, b i ) C 2 \ {(0, 0)} (the cases whee a i = 0 o b i = 0 povide a fixed point with multiplicity n). We thus get P i = (p i ) 2 O(A 2 ) G and (f i,, f i,2 ) = ( b i y n (a i x n + b i y n ), a i x n (a i x n + b i y n ) ) End(A 2 ) G which satisfy f i, y f i,2 x = P i (note that the f i, and f i,2 ae hee not unique, and could also be chosen without common facto). The coesponding embedding is Γ = P \ Λ A( 3 [x : y] a i x n b i y n a i x n + b i y n, 2b i xy n a i x n + b i y n, ) ) 2a i x n y a i x n + b i y n. 2) In the dihedal case, we can assume that H ( is geneated ) by the ( maps )[x : y] ζ 0 0 i [ξ n x : y] and [x : y] [y : x]. So G is geneated by 0 ζ and. i 0 An obit Λ i of H is given by the zeo set of p i = a i (x 2n + y 2n ) + 2b i x n y n fo some (a i, b i ) C 2 \ {(0, 0)} and we thus get and P i = (p i ) 2 O(A 2 ) G (f i,, f i,2 ) = ( y n (b i x n + a i y n )p i, x n (a i x n + b i y n )p i ) End(A 2 ) G.

15 EXTENSION OF AUTOMORPHISMS OF RATIONAL SMOOTH AFFINE CURVES 5 which satisfy f i, y f i,2 x = P i (note that P i = p i is also possible if n is even, and that as befoe the polynomials f i,, f i,2 ae not unique, and could also be chosen without common facto). This leads to the embedding Γ = P \ Λ A 3 which sends a point [x : y] Γ onto ( a i (x 2n y 2n ) a i (x 2n + y 2n ) + 2b i x n y n, 2xy n (b i x n + a i y n ) a i (x 2n + y 2n ) + 2b i x n y n, ) 2x n y(a i x n + b i y n ) a i (x 2n + y 2n ) + 2b i x n y n. 3) In the case of the tetahedal goup, we can assume that H = A 4 is geneated by the maps [x : y] ( [i(x + y) : x) y] and ( [x : y] ) [x : y]. This implies that G is geneated by i i i 0 2 and. An obit Λ i + i 0 i i of H is given by the zeo set of p i = 6a i (x 5 y xy 5 ) 2 + b i (x 4 + y 4 )(x 8 + y 8 34x 4 y 4 ), fo some (a i, b i ) C 2 \ {(0, 0)}. We thus get P i = p i O(A 2 ) G f i, = a i (x 0 y 6x 6 y 5 + 5x 2 y 9 ) + b i ( x 8 y 3 22x 4 y 7 + y ) f i,2 = a i (5x 9 y 2 6x 5 y 6 + xy 0 ) b i (x 22x 7 y 4 x 3 y 8 ) which satisfy (f i,, f i,2 ) End(A 2 ) G and f i, y f i,2 x = P i as befoe. This gives the embedding Γ = P \ Λ A 3 defined by 4a i x 2 y 2 (x 4 +y 4 )(x 4 y 4 )+b i (x 2 x 8 y 4 +x 4 y 8 y 2 ) 6a i (x 5 y xy 5 ) 2 +b i (x 4 +y 4 )(x 8 +y 8 34x 4 y 4 ) 2x(a i (x [x : y] 0 y 6x 6 y 5 +5x 2 y 9 )+b i ( x 8 y 3 22x 4 y 7 +y )) 6a i (x 5 y xy 5 ) 2 +b i (x 4 +y 4 )(x 8 +y 8 34x 4 y 4 ) A 3. 2y(a i (5x 9 y 2 6x 5 y 6 +xy 0 )+b i (x 22x 7 y 4 x 3 y 8 )) 6a i (x 5 y xy 5 ) 2 +b i (x 4 +y 4 )(x 8 +y 8 34x 4 y 4 ) It is also possible to descibe similaly the othe cases (S 4 and A 5 ), but the fomulas ae even moe inticate. Refeences [Beau0] A. Beauville, Finite subgoups of PGL 2(K), Vecto bundles and complex geomety, 23 29, Contemp. Math. 522, Ame. Math. Soc., Povidence, RI, 200. [Bo9] A. Boel, Linea algebaic goups, Second edition, Gaduate Texts in Mathematics 26, Spinge-Velag, New Yok, 99, 288 pp., ISBN [DKW99] H. Deksen, F. Kutzschebauch, J. Winkelmann, Subvaieties of C n with non-extendable automophisms, J. Reine Angew. Math. 508 (999), [FM89] S. Fiedland, J. Milno, Dynamical popeties of plane polynomial automophisms, Egod. Th & Dyn. Syst. 9 (989), [G63] L. Geenbeg, Maximal Fuchsian goups, Bull. Ame. Math. Soc. 69 (963), [Ka9] S. Kaliman, Extensions of isomophisms between affine algebaic subvaieties of k n to automophisms of k n, Poc. Ame. Math. Soc. 3, no. 2 (99), [Kam79] T. Kambayashi, Automophism goup of a polynomial ing and algebaic goup action on an [Po] affine space, J. Algeba 60 (979) Yu. Pokhoov, p-elementay subgoups of the Cemona goup of ank 3, Classification of algebaic vaieties, , EMS Se. Cong. Rep., Eu. Math. Soc., Züich, 20. [Se80] J.-P. See, Tees, Spinge Velag, Belin, Heidelbeg, New Yok, 980. [Si9] V. Sinivas, On the embedding dimension of an affine vaiety. Math. Ann. 289 (99)

16 6 JÉRÉMY BLANC, JEAN-PHILIPPE FURTER, AND PIERRE-MARIE POLONI J. Blanc, Univesität Basel, Mathematisches Institut, Rheinspung 2, CH-405 Basel, Switzeland. addess: J.-P. Fute, Dpt. of Math., Univ. of La Rochelle, av. Cépeau, 7000 La Rochelle, Fance addess: P.-M. Poloni, Univesität Basel, Mathematisches Institut, Rheinspung 2, CH-405 Basel, Switzeland. addess: