The decomposition groups of plane conics and plane rational cubics
aa r X i v : . [ m a t h . AG ] J un THE DECOMPOSITION GROUPS OF PLANECONICS AND PLANE RATIONAL CUBICS
TOM DUCAT, ISAC HED´EN, AND SUSANNA ZIMMERMANN
Abstract.
The decomposition group of an irreducible plane curve X ⊂ P is thesubgroup Dec( X ) ⊂ Bir( P ) of birational maps which restrict to a birational map of X . We show that Dec( X ) is generated by its elements of degree ≤ X is eithera conic or rational cubic curve. Introduction
Preliminaries.
We work over an algebraically closed field k of any characteristic.By elementary quadratic transformation we will mean a birational map ϕ ∈ Bir( P ) ofdegree 2 with only proper base points. Definition 1.1.
For an irreducible curve X ⊂ P , the decomposition group Dec( X ) of X is the subgroup of Bir( P ) of all birational maps ϕ ∈ Bir( P ) which restrict to abirational map ϕ | X : X X .Similarly, the inertia group Ine( X ) of X is the subgroup of Bir( P ) of all birationalmaps ϕ ∈ Bir( P ) which restrict to the identity map ϕ | X = id X .Elements of Dec( X ) are said to preserve the curve X , whilst elements of Ine( X ) aresaid to fix X . We will write Aut( P , X ) = Dec( X ) ∩ PGL for the subgroup of linearmaps Aut( P ) = PGL which preserve X .The focus of this paper is on the group Dec( X ) in the case that X ⊂ P is a planerational curve of degree ≤
3. In this case X is either a line, a smooth conic, a nodalcubic or a cuspidal cubic. Remark 1.2.
A line X ⊂ P (resp. conic, nodal cubic, cuspidal cubic) is projectivelyequivalent to any other line X ′ ⊂ P (resp. conic, nodal cubic, cuspidal cubic), i.e. thereis an automorphism λ ∈ PGL with λ ( X ) = X ′ . For rational curves of degree ≥ Motivation.
The decomposition and inertia groups of plane curves have appearedin a number of places.1.2.1.
Decomposition and inertia groups of plane curves of genus ≥ . The inertiagroups of plane curves of geometric genus ≥ Mathematics Subject Classification. for curves of genus ≤
1. The inertia groups of smooth cubic curves have been studiedby Blanc [2].Decomposition groups were introduced by Gizatullin [9], who used them as a tool togive sufficient conditions for Bir( P ) to be a simple group. This group is not simple, asshown later by Cantat–Lamy [5] for algebraically closed fields, and by Lonjou [11] forarbitrary fields. The decomposition groups of plane curve of genus ≥ X ⊂ P of Kodaira dimension κ ( P , X ) = 0 or 1.For curves X ⊂ P with κ ( P , X ) = −∞ , the pair ( P , X ) is birationally equivalentto ( P , L ) where L ⊂ P is a line, and a description of Dec( L ) is given by Theorem 1below. As X ⊂ P is the image of L under a birational transformation ϕ of P , wehave an isomorphism Dec( X ) ≃ Dec( L ), given by ψ ϕ − ψϕ . Although it is notdegree-preserving, this isomorphism shows that Dec( X ) is not finite.1.2.2. The decomposition group of a line.
The classical Noether–Castelnuovo Theo-rem [7] states that the Cremona group Bir( P ) has a presentation given by:Bir( P ) = (cid:10) PGL , σ (cid:11) where σ is any choice of elementary quadratic transformation. The second two au-thors [10] have shown that an analogous statement holds for the decomposition groupof a line: Theorem 1 ([10]) . Let L ⊂ P be a line. Then Dec( L ) = (cid:10) Aut( P , L ) , σ (cid:11) for any choice of elementary quadratic transformation σ ∈ Dec( L ) . In particular anymap τ ∈ Dec( L ) can be factored into elementary quadratic transformations inside Dec( L ) . In this article, we present a similar theorem for conic and rational cubic curves. Ue-hara [13, Proposition 2.11] proves that for the cuspidal cubic X ⊂ P , the elements ofthe subset { f ∈ Dec( X ) | f is an automorphism near the cusp } ( Dec( X )can be decomposed into quadratic transformations preserving X . Theorem 3 generaliseshis result to all of Dec( X ).1.2.3. Relationship to dynamics of birational maps.
Birational maps of P preservinga curve of degree ≤ P ) with positive first dynamical degree necessarily has degree ≤ §
1] explore thedynamical behaviour of the family of birational transformations f a,b : ( x, y ) (cid:0) y, y + ax + b (cid:1) ,for a, b ∈ C . In particular, they focus on maps of this kind preserving a curve, and showthat this curve is necessarily cubic. HE DECOMPOSITION GROUPS OF CONICS AND RATIONAL CUBICS 3
Main results.
We will use Theorem 1 to deduce:
Theorem 2.
Let C ⊂ P be a conic. Then any map τ ∈ Dec( C ) can be factored intoelementary quadratic transformations inside Dec( C ) . Moreover, from Theorem 2 we will deduce:
Theorem 3.
Let X ⊂ P be a rational cubic and suppose that the characteristic of k isnot 2. Then any map τ ∈ Dec( X ) can be factored into elementary quadratic transfor-mations inside Dec( X ) . The basic strategy used to prove both Theorems 2 & 3 is the same in each case andis explained in §
2. Given a curve Z ⊂ P , the idea is to conjugate τ ∈ Dec( Z ) to τ ′ ∈ Dec( Y ), for a curve Y ⊂ P of lower degree, and then use the result for Y . Remark 1.3.
The proof of each theorem is elementary and only requires choosingquadratic transformations with base points that lie outside of a collection of finitelymany points and lines. In the cubic case we need to choose base points which avoid allof the tangent lines to a conic which pass through a given point. We must restrict to afield k of characteristic = 2 in this case, since over fields of characteristic 2 every linethrough a given point may be tangent to a conic (see [12, Appendix to § Remark 1.4.
As shown in Proposition 3.5, for a conic C it is still possible to writeDec( C ) = (cid:10) Aut( P , C ) , σ (cid:11) using just one suitably general elementary quadratic trans-formation σ (where ‘suitably general’ means that σ does not contract a tangent line to C ). However, if the base field k is uncountable then we need an uncountable numberof elementary quadratic transformations to generate both Ine( C ) (see Remark 3.6) andDec( X ) for X a nodal cubic (see § Acknowledgements.
We would like to thank Eric Bedford and Jeffrey Diller forhelpful comments. 2.
The main Proposition
Let
Y, Z ⊂ P be two arbitrary irreducible plane curves. Definition 2.1.
Let Φ
Y,Z ⊂ Bir( P ) be the set of all elementary quadratic transforma-tions ϕ which map Y birationally onto Z .Note that Φ Y,Z is a (possibly empty) subset of Bir( P ) and not a subgroup. For any ϕ, ψ ∈ Φ Y,Z we clearly have ϕψ − ∈ Dec( Z ). More generally for any τ ∈ Dec( Y ) wehave ϕτ ψ − ∈ Dec( Z ). Proposition 2.2.
Suppose that Φ Y,Z = ∅ and the following three statements hold:(A) Any τ ∈ Dec( Y ) can be factored into elementary quadratic transformations in-side Dec( Y ) .(B) For any ϕ, ψ ∈ Φ Y,Z the composition ϕψ − ∈ Dec( Z ) can be factored into ele-mentary quadratic transformations inside Dec( Z ) .(C) For any elementary quadratic transformation τ ∈ Dec( Y ) there exist ϕ, ψ ∈ Φ Y,Z such that ϕτ ψ − ∈ Dec( Z ) can be factored into elementary quadratic transfor-mations inside Dec( Z ) .Then any τ ∈ Dec( Z ) can be factored into elementary quadratic transformations inside Dec( Z ) . TOM DUCAT, ISAC HED´EN, AND SUSANNA ZIMMERMANN
Proof.
Suppose that τ ∈ Dec( Z ) and choose any two maps ϕ, ψ ∈ Φ Y,Z = ∅ . Then by(A) we can factor τ ′ := ψ − τ ϕ ∈ Dec( Y ) into elementary quadratic transformations τ ′ = τ n τ n − · · · τ τ with τ i ∈ Dec( Y ) for all i = 1 , . . . , n .By (C) we can find ϕ i , ψ i ∈ Φ Y,Z such that f i := ϕ i τ i ψ − i ∈ Dec( Z ) can be factored intoelementary quadratic transformations inside Dec( Z ) for all i = 1 , . . . , n .Now let ϕ := ϕ and ψ n +1 := ψ . Then by (B) we can factor g i := ψ i +1 ϕ − i ∈ Dec( Z )into elementary quadratic transformations inside Dec( Z ) for all i = 0 , . . . , n .We can write τ = g n f n g n − · · · g f g , according to the diagram: Z Z Z Z Z Z Z ZY Y Y Y ϕ ϕ ϕ n − ϕ n ψ ψ ψ n ψ n +1 τ τ τ n − τ n g f g g n − f n g n · · ·· · · and hence we can factor τ into elementary quadratic transformations inside Dec( Z ). (cid:3) Theorem 2 and Theorem 3 follow from Proposition 2.2, where the three statements (A),(B), (C) appearing in the proposition are proved in each case according to:(A) (B) (C)Theorem 2 Theorem 1 Lemma 3.2 Lemma 3.3Theorem 3 Theorem 2 Lemma 4.2 Lemma 4.33.
The decomposition group of a conic
Throughout this section we let L ⊂ P denote a fixed line and C ⊂ P a conic. Remark 3.1. If ϕ ∈ Bir( P ) is an elementary quadratic transformation belonging toΦ L,C then all three base points of ϕ must lie outside of L . Conversely, given any threenon-collinear points in P \ L we can always find an elementary quadratic transformation ϕ ∈ Φ L,C with these as base points.3.1.
Proof of Theorem 2.
We prove statements (B) & (C) in Proposition 2.2 in thespecial case that Y = L a line and Z = C a conic.3.1.1. Proof of statement (B) for conics.
Lemma 3.2.
Suppose that ϕ , ϕ ∈ Φ L,C . Then the composition ϕ ϕ − ∈ Dec( C ) canbe factored into elementary quadratic transformations inside Dec( C ) .Proof. For i = 1 ,
2, we let P i , Q i , R i be the base points of ϕ i , none of which lie on L . Wemay assume that these six points are in general position, i.e. that no points coincideand that no three points are collinear, as in Figure 1(i). If this is not the case, choosea third map ϕ ∈ Φ L,C whose base points are in general position with respect to both ϕ and ϕ . Then we can write ϕ ϕ − = ( ϕ ϕ − )( ϕ ϕ − ) and decompose each of ϕ ϕ − and ϕ ϕ − into elementary quadratic transformations inside Dec( C ). HE DECOMPOSITION GROUPS OF CONICS AND RATIONAL CUBICS 5
We let ϕ =: ψ , ψ , ψ , ψ := ϕ ∈ Φ L,C be a sequence of elementary quadratic trans-formations with base points:( P , Q , R ) , ( P , Q , R ) , ( P , Q , R ) , ( P , Q , R )and we write ϕ ϕ − = ( ψ ψ − )( ψ ψ − )( ψ ψ − ).By our assumption, ψ and ψ exist since no three points are collinear and we can take ψ , ψ ∈ Φ L,C since none of these points lie on L . Moreover ψ i +1 ψ − i ∈ Dec( C ) is anelementary quadratic transformation for i = 0 , , ψ i and ψ i +1 share exactly twocommon base points and no three base points are collinear. (cid:3) (i) • P • Q • R • P • Q • R (ii) • P • Q • R • S Figure 1.
Configuration of base points in proof of (i) Lemma 3.2 and (ii) Lemma 3.3.3.1.2.
Proof of statement (C) for conics.
In fact we prove a stronger statement thanstatement (C) (since id P is a decomposition into zero elementary quadratic transfor-mations in Dec( C )). Lemma 3.3.
Let τ ∈ Dec( L ) be an elementary quadratic transformation. Then we canfind ϕ, ψ ∈ Φ L,C such that ϕτ ψ − = id P .Proof. Let
P, Q, R be the base points of τ , where P, Q / ∈ L and R ∈ L . Choose a point S / ∈ L as in Figure 1(ii), such that no three of P, Q, R, S are collinear.Since
P, Q, S are non-collinear we let ψ ∈ Φ L,C be an elementary quadratic transfor-mation with these base points. Then ϕ := ψτ − ∈ Φ L,C is also an elementary qua-dratic transformation since ψ and τ share two base points and no three base points arecollinear. Thus ϕτ ψ − = id P . (cid:3) A generating set for
Dec( C ) . It was shown in [10] that, for L ⊂ P a line,Dec( L ) can be generated by Aut( P , L ) and any one elementary quadratic transforma-tion σ ∈ Dec( L ). This is because Aut( P , L ) is still large enough to act transitively onthe set: B = (cid:8) ( P, Q, R ) ∈ ( P ) | P ∈ L and Q, R / ∈ L non-collinear (cid:9) of all possible base points for σ . For the conic C ⊂ P , even though the analogousaction of Aut( P , C ) is no longer transitive, it is still true that Dec( C ) can be gener-ated by Aut( P , C ) and a suitably general elementary quadratic transformation σ ∈ Dec( C ).We fix a model C = V (cid:0) xz − y (cid:1) ⊂ P in order to describe Aut( P , C ). Lemma 3.4.
Aut( P , C ) is given by: Aut( P , C ) = a ab b ac ad + bc bdc cd d ∈ PGL (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ad − bc = 0 ≃ PGL . In particular any α ∈ PGL = Bir( C ) extends uniquely to a linear map in Aut( P , C ) . TOM DUCAT, ISAC HED´EN, AND SUSANNA ZIMMERMANN
It follows from Lemma 3.4 that Ine( C ) ∩ PGL = h id P i . Moreover the sequence1 → Ine( C ) → Dec( C ) → PGL → C ) = Ine( C ) ⋊ PGL is a semidirect product, where PGL acts onIne( C ) by conjugation. Proposition 3.5.
Dec( C ) = h Aut( P , C ) , σ i for any elementary quadratic transfor-mation σ which does not contract a tangent line to C .Proof. Let τ ∈ Dec( C ) be an elementary quadratic transformation and consider theaction of PGL ≃ Aut( P , C ) on the set: B = { ( P, Q, R ) ∈ ( P ) | P, Q ∈ C and R / ∈ C non-collinear } of all possible base points for τ . If P, Q ∈ C and R / ∈ C are the (ordered) base pointsof τ then, by an element of PGL , we can send P (1 : 0 : 0), Q (0 : 0 : 1) and R to a point in the conic Γ d = V ( xz − dy ) for a uniquely determined 1 = d ∈ k. Write B = S d ∈ k \ B d , a decomposition into PGL -invariant sets according to this pencil ofconics Γ d . The sets B d with d = 0 are all PGL -orbits. For the degenerate conic Γ theset B splits into three PGL -orbits B = B , ∪ B , ∪ B , according to the cases: R ∈ Γ , := { ( t : 1 : 0) | t = 0 } , R ∈ Γ , := { (0 : 1 : t ) | t = 0 } , R = (0 : 1 : 0) . As shown in Figure 2, these three orbits correspond to the cases where one or two ofthe lines contracted by τ are tangent to C . • • • (i) • • • (ii) • • • (iii) Figure 2.
The base points of τ belonging to the orbit (i) B d with d = 0,(ii) B , or B , , (iii) B , .Let σ a,b ∈ Dec( C ) be an elementary quadratic transformation with base points (1 : 0 : 0),(0 : 0 : 1) and ( a : 1 : b ) belonging to an orbit B ab with ab = 0. By composing with asuitable linear map we can assume the map is actually in Ine( C ), in which case σ a,b isuniquely determined and given by: σ a,b = (cid:16)(cid:0) − ab (cid:1) xy + a (cid:0) xz − y (cid:1) : xz − aby : (cid:0) − ab (cid:1) yz + b (cid:0) xz − y (cid:1)(cid:17) . Any elementary quadratic transformation σ ∈ Dec( C ) which does not contract a tangentline to C has base points belonging to the same PGL -orbit as σ a,b for some a, b ∈ kwith ab = 0 ,
1. Therefore, to prove the proposition, it is enough to show that given any a, b ∈ k with ab = 0 ,
1, we can use σ a,b to generate at least one elementary quadratictransformation with base points belonging to any other PGL -orbit.Consider the linear map: λ a,b = ( x + 2 ay + a z : bx + (1 + ab ) y + az : b x + 2 by + z )and, for c = 0 , , ∞ , the diagonal map µ c = ( c x : cy : z ). Since ab = 0 we get theformula: σ a ′ ,b ′ = λ − a,b µ − c σ a,b µ c σ − a,b λ a,b HE DECOMPOSITION GROUPS OF CONICS AND RATIONAL CUBICS 7 where a ′ = − abcb ( c − and b ′ = ab − ca ( c − .As c varies the base points of σ a ′ ,b ′ are (1 : 0 : 0), (0 : 0 : 1) and the point R ′ =( a (1 − abc ) : ab ( c −
1) : b ( ab − c )) lying on the line: L a,b = V (cid:0) bx + (1 + ab ) y + az (cid:1) . The point R ′ can be any point on L a,b , except for ( a : 0 : − b ), corresponding to c = 1,and L a,b ∩ C = { ( − b : 1 : − b ) , ( − a : 1 : − a ) } , corresponding to c = 0 , ∞ . Outside ofthese points L a,b intersects every conic Γ d at least once.For all d = 0 this construction gives an elementary quadratic transformation with basepoints in B d .If d = 0 and ab = − L a,b meets Γ , and Γ , giving elementary quadratic trans-formations with base points in B , and B , . If ab = − L a,b ∩ Γ = (0 : 1 : 0)giving an elementary quadratic transformation with base points in B , .It remains to produce an elementary quadratic transformation with base points in B , if ab = − B , and B , if ab = −
1. We can use the construction once to produce σ a ′ ,b ′ with a ′ b ′ = − ab = − a ′ b ′ = − ab = −
1) and then proceed asabove. (cid:3)
Remark 3.6.
If the ground field k is uncountable then the corresponding statement forIne( C ) is not true, i.e. Ine( C ) cannot be generated by linear maps and any countablecollection of elementary quadratic maps. Although Ine( C ) ∩ PGL is trivial, Ine( C )contains a lot of elementary quadratic transformations. Indeed the maps { σ a,b ∈ Ine( C ) | a, b ∈ k , ab = 1 } appearing in the proof of Proposition 3.5 give an uncountable family.4. The decomposition group of a rational cubic
Throughout this section we let C ⊂ P denote a fixed conic and X ⊂ P a rationalcubic. We will distinguish between the nodal and cuspidal cases when necessary. Asexplained in Remark 1.3, we will also assume that the characteristic of k is not 2. Remark 4.1.
Any map ϕ ∈ Φ C,X must have exactly one base point P ∈ C and twobase points Q, R / ∈ C . In this case X is a cuspidal cubic if the line QR is tangent to C and a nodal cubic otherwise, as shown in Figure 3. Moreover, given any three non-collinear points in such a position we can always find a map ϕ ∈ Φ C,X with these threepoints as base points. • P • R • Q • P • R • Q (i) (ii) Figure 3.
Base point configurations for ϕ ∈ Φ C,X when X is (i) a nodalcubic and (ii) a cuspidal cubic. TOM DUCAT, ISAC HED´EN, AND SUSANNA ZIMMERMANN
Proof of Theorem 3.
We now prove statements (B) & (C) in Proposition 2.2 for Y = C a conic and Z = X a rational cubic.4.1.1. Proof of statement (B) for cubics.
Lemma 4.2.
Suppose that ϕ , ϕ ∈ Φ C,X . Then the composition ϕ ϕ − ∈ Dec( X ) canbe factored into elementary quadratic transformations inside Dec( X ) .Proof. For i = 1 ,
2, we let P i , Q i , R i be the base points of ϕ i , where P i ∈ C and Q i , R i / ∈ C . As in the proof of Lemma 3.2, we may intertwine with a third map ϕ ∈ Φ C,X to assume that no base points coincide, no three are collinear and no two lie on a tangentline to C (unless X is a cuspidal cubic, in which case we can assume that only Q , R and Q , R lie on a tangent line to C ). The nodal case: If X is a nodal cubic we let ϕ =: ψ , ψ , ψ , ψ := ϕ ∈ Φ C,X be asequence of elementary quadratic transformations with base points:( P , Q , R ) , ( P , Q , R ) , ( P , Q , R ) , ( P , Q , R )and we write ϕ ϕ − = ( ψ ψ − )( ψ ψ − )( ψ ψ − ).By our assumption ψ and ψ exist since each of these triples is non-collinear and ψ , ψ ∈ Φ C,X since they both have precisely one base point on C and do not contractany tangent line to C . Lastly each composition ψ i +1 ψ − i ∈ Dec( X ) is an elementaryquadratic transformation since ψ i and ψ i +1 share exactly two common base points andno three base points are collinear. • P • Q • R • P • Q • R (i) • P • Q • R • P • Q • R • SL L (ii) Figure 4.
Configuration of base points in (i) the nodal case and (ii) thecuspidal case.
The cuspidal case: If X is a cuspidal cubic then we must be a little bit more careful toensure that each of our intermediate maps ψ i contracts a tangent line to C .For i = 1 , L i be the tangent line to C passing through Q i which does not contain R i . By our assumption on the position of the base points, the point S = L ∩ L iswell-defined, S / ∈ C and S is not equal to any P i , Q i , R i . Moreover, no three of the sevenpoints P , P , Q , Q , R , R , S are collinear.Now we let ϕ =: ψ , ψ , ψ , ψ , ψ := ϕ ∈ Φ C,X be a sequence of elementary quadratictransformations with base points:( P , Q , R ) , ( P , Q , S ) , ( P , Q , S ) , ( P , Q , S ) , ( P , Q , R )and we write ϕ ϕ − = ( ψ ψ − )( ψ ψ − )( ψ ψ − )( ψ ψ − ). HE DECOMPOSITION GROUPS OF CONICS AND RATIONAL CUBICS 9
As before, ψ , ψ , ψ exist since each triple of base points is non-collinear and ψ , ψ , ψ ∈ Φ C,X since they all have precisely one base point on C and contract a tangent line to C .Lastly each composition ψ i +1 ψ − i ∈ Dec( X ) is an elementary quadratic transformationsince ψ i , ψ i +1 share exactly two common base points and no three base points arecollinear. (cid:3) Proof of statement (C) for cubics.
Lemma 4.3.
Let τ ∈ Dec( C ) be an elementary quadratic transformation. Then we canfind ϕ, ψ ∈ Φ C,X such that ϕτ ψ − ∈ Dec( X ) can be factored into elementary quadratictransformations inside Dec( X ) .Proof. We first assume that τ is an elementary quadratic transformations which does notcontract a tangent line to C (i.e. τ has a configuration of base points as in Figure 2(i)).Let P, Q ∈ C and R / ∈ C be the base points of τ and let L be a tangent line to C passing through R . By assumption L = P R, QR .Choose a point
S / ∈ C as in Figure 5, such that no three of P, Q, R, S are collinear. If X is a nodal cubic then we choose S to avoid the tangent lines to C passing through P , Q or R . If X is a cuspidal cubic then we choose S to lie on L but avoid the tangentlines to C through P or Q . • P • Q • R • S (i) • P • Q • R • S (ii) Figure 5.
Location of the point S when X is (i) a nodal cubic and (ii)a cuspidal cubic.Since P, R, S are non-collinear there is an elementary quadratic transformation ψ ∈ Φ C,X with these base points. We let ϕ := ψτ − ∈ Φ C,X which is also an elementary quadratictransformation since ψ and τ share two base points and no three of the base points arecollinear. Thus ϕτ ψ − = id P ∈ Dec( X ) which is a decomposition into zero elementaryquadratic transformations inside Dec( X ).If τ is an arbitrary elementary quadratic transformation, then by Proposition 3.5 wecan write τ = τ n · · · τ where τ i ∈ Dec( C ) are elementary quadratic transformationswhich do not contract a tangent line to C . We can find ϕ i , ψ i ∈ Φ C,X , for i = 1 , . . . , n ,such that ϕ i τ i ψ − i ∈ Dec( X ) can be factored into elementary quadratic transformationsinside Dec( X ) and by Lemma 4.2 we can factor ψ i +1 ϕ − i ∈ Dec( X ) into elementaryquadratic transformations inside Dec( X ) for i = 1 , . . . , n −
1. Therefore, taking ϕ := ϕ n and ψ := ψ , we can factor ϕτ ψ − = ( ϕ n τ n ψ − n )( ψ n ϕ − n − )( ϕ n − τ n − ψ − n − ) · · · ( ψ ϕ − )( ϕ τ ψ − )into elementary quadratic transformations inside Dec( X ). (cid:3) An example.
Let X be a nodal (resp. cuspidal) cubic, let τ ∈ Dec( X ) and sup-pose that we conjugate τ to get τ ′ ∈ Dec( C ), for a conic C , as in the proof of Propo-sition 2.2. If τ ′ can be decomposed into n elementary quadratic transformations whichdo not contract any tangent line to C then na¨ıvely applying the proof of Theorem 3gives a decomposition of τ into at most 6( n + 1) (resp. 8( n + 1)) elementary quadratictransformations inside Dec( X ).Even in relatively simple cases this gives a very long decomposition which is far fromoptimal. For example let X be the cuspidal cubic X = V ( x − y z ) ⊂ P and considerthe de Jonqui`eres involution τ = ( xy : y : 2 x − y z ) ∈ Ine( X ). This map has oneproper base point at the cusp point P ∈ X and all other base points infinitely near to P . If C is the conic C = V ( xz − y ) then ϕ = ( x ( y + z ) : x ( x + y ) : z ( y + z )) ∈ Φ C,X and conjugating τ with ϕ gives τ ′ = ϕ − τ ϕ ∈ Dec( C ), a map of degree 3 with twoproper base points, which decomposes into four elementary quadratic transformationsin Dec( C ) not contracting any tangent line to C . Therefore we can decompose τ intoat worst 40 elementary quadratic transformations inside Dec( X ), although we expect aminimal decomposition to be much shorter.4.3. Generating sets for
Dec( X ) . Let X be the nodal cubic given by the model X = V ( x + y − xyz ) ⊂ P . We see that Aut( P , X ) is the finite group given by:Aut( P , X ) = * ω ω
00 0 1 , + ≃ S where ω ∈ k is a primitive cube root of unity. If k is an uncountable field then Dec( X ) isan uncountable group and therefore cannot be generated by Aut( P , X ) and any finite(or countable) collection of elementary quadratic transformations.Now suppose X is the cuspidal cubic given by the model X = V ( x − y z ) ⊂ P . In thiscase Aut( P , X ) is infinite:Aut( P , X ) = * a a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ∈ k × + ≃ G m . We do not know whether or not Dec( X ) can be generated by Aut( P , X ) and anycountable collection of elementary quadratic transformations.5. Rational curves of higher degree
We provide a family of plane rational curves X d ⊂ P , birationally equivalent to a lineand of degree d ≥
4, to show that we cannot expect Theorems 1, 2 & 3 to be true forcurves of higher degree.Let X d denote the rational curve given by X d = V ( x d − y d − z ) ⊂ P which has a uniquesingular point P = (0 : 0 : 1), a cusp of multiplicity d −
1, and a unique inflectionpoint Q = (0 : 1 : 0). Let L Q = ( z = 0) be the tangent line intersecting X d at Q with multiplicity d and let L P = ( y = 0) be the tangent line to the cusp P . Any deJonqui`eres transformation of degree d with major base point at P and all other basepoints on X d \ P sends X d onto a line.A map in Aut( P , X d ) has to fix P and Q and preserve L P and L Q . It is straightforwardto check that: Aut( P , X d ) = (cid:8) ( ax : y : a d z ) (cid:12)(cid:12) a ∈ k × (cid:9) ≃ G m . HE DECOMPOSITION GROUPS OF CONICS AND RATIONAL CUBICS 11
Lemma 5.1.
The standard involution σ = ( yz : zx : xy ) ∈ Bir( P ) is the only elemen-tary quadratic map that preserves X d , up to composition with an element of Aut( P , X d ) .Proof. It is easy to check that σ ∈ Dec( X d ). Any other elementary quadratic transfor-mation τ ∈ Dec( X d ) must have one base point at P ∈ X d , one base point in the smoothlocus of X d and one base point not contained in X d . In particular τ − also has a basepoint at P . Since the line τ − ( P ) is tangent to a point of X d with multiplicity ≥ d − τ − ( P ) = L Q . As the line L Q is contracted, both τ and τ − must havetwo base points on L Q , one of which is L Q ∩ X d = Q . Now the line τ − ( Q ) is tangentto the cusp P so we must have τ − ( Q ) = L P , as in Figure 6.Since the lines L P and L Q are contracted, the base points of τ are P = (0 : 0 : 1), Q = (0 : 1 : 0) and L P ∩ L Q = (1 : 0 : 0). Hence, up to an element of Aut( P , X d ), wemust have τ = σ . (cid:3) • P •• Q L Q L P Q L Q L P P • • L Q • L P P Q
Figure 6.
Resolution of the standard involution σ ∈ Dec( X d ). Proposition 5.2. If d ≥ , the group Dec( X d ) cannot be generated by linear maps andelementary quadratic transformations.Proof. By Lemma 5.1, the subgroup of Dec( X d ) generated by linear maps and ele-mentary quadratic transformations is given by h Aut( P , X d ) , σ i . Since σ = id P and σλ = λ − σ for any λ ∈ Aut( P , X d ), all elements of this subgroup are of the form λ or λσ and are either linear or quadratic. But there are many elements in Dec( X d ) of degree >
2; for example the de Jonqui`eres transformation τ a = ( xy d − : y d : (1 − a ) x d + ay d − z )for a ∈ k × . (cid:3) Remark 5.3.
The family of maps { τ a | a ∈ k × } , appearing at the end of the proof ofProposition 5.2, form a subgroup of Ine( X d ) isomorphic to G m since τ b τ a = τ ab for all a, b ∈ k × . References [1]
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Tom Ducat, Research Institute for Mathematical Sciences, Kyoto University, Kyoto606-8502 Japan
E-mail address : [email protected] Isac Hed´en, Research Institute for Mathematical Sciences, Kyoto University, Kyoto606-8502 Japan
E-mail address : [email protected]
Susanna Zimmermann, Universit´e Toulouse Paul Sabatier, Institut de Math´ematiques,118 route de Narbonne, 31062 Toulouse Cedex 9
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