This is the accepted version of the journal article: Marín Pérez, David; Bedrouni, Samir. «Geometry of certain foliations on the complex projective plane». Annali della Scuola normale superiore di Pisa - Classe di scienze, 2022. 27 pàg. Scuola Normale Superiore. This version is available at https://ddd.uab.cat/record/265181 under the terms of the license GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE by Samir B EDROUNI & David M ARÍN Abstract. — Let d ≥ 2 be an integer. The set F(d) of foliations of degree d on the complex projective plane can be identified with a Z ARISKI’s open set of a projective space of dimension d 2 + 4d + 2 on which Aut(P2C ) acts. We show that there are exactly two orbits O (F1d ) and O (F2d ) of minimal dimension 6, necessarily closed in F(d). This generalizes known results in degrees 2 and 3. We deduce that an orbit O (F ) of an element F ∈ F(d) of dimension 7 is closed in F(d) if and only if Fid 6∈ O (F ) for i = 1, 2. This allows us to show that in any degree d ≥ 3 there are closed orbits in F(d) other than the orbits O (F1d ) and O (F2d ), unlike the situation in degree 2. On the other hand, we introduce the notion of the basin of attraction B(F ) of a foliation F ∈ F(d) as the set of G ∈ F(d) such that F ∈ O (G ). We show that the basin of attraction B(F1d ), resp. B(F2d ), contains a quasi-projective subvariety of F(d) of dimension greater than or equal to dim F(d) − (d − 1), resp. dim F(d) − (d − 3). In particular, we obtain that the basin B(F23 ) contains a non-empty Z ARISKI open subset of F(3). This is an analog in degree 3 of a result on foliations of degree 2 due to C ERVEAU, D ÉSERTI, G ARBA B ELKO and M EZIANI. 2010 Mathematics Subject Classification. — 37F75, 32S65, 32M25, 32M05. Introduction The set F(d) of holomorphic foliations of degree d on P2C is identified with a Z ARISKI open subset of the 2 projective space PCd +4d+2 . We are interested here in the action of the group Aut(P2C ) = PGL3 (C) on F(d). We generalize to arbitrary degree some results known in small degrees [9, 1, 5] on this action. For F ∈ F(d), we will respectively denote by O (F ) and Iso(F ) the orbit and the isotropy group of F under the action of Aut(P2C ), i.e. O (F ) := {ϕ∗ F ∈ F(d) | ϕ ∈ Aut(P2C )} and Iso(F ) := {ϕ ∈ Aut(P2C ) | ϕ∗ F = F }. O (F ) is a Z ARISKI irreducible subset of F(d) and Iso(F ) is an algebraic subgroup of Aut(P2C ). Key words and phrases. — foliation, singularity, inflection point, orbit, isotropy group. This work has been partially funded by the Ministry of Science, Innovation and Universities of Spain through the grant PGC2018- 095998-B-I00, by the Agency for Management of University and Research Grants of Catalonia through the grant 2017SGR1725 and by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). 2 SAMIR BEDROUNI & DAVID MAR Í N Following [14] we will say that a foliation of F(d) is convex if its leaves other than straight lines have no inflection points. We will denote by FC(d) the subset of F(d) consisting of convex foliations, which is a Z ARISKI closed subset of F(d). According to [7, Proposition 2.2] every foliation of degree 0 or 1 is convex. For d ≥ 2, FC(d) is a proper closed subset of F(d) and it contains the foliation F1d defined in the affine chart (x, y) by the 1-form (see [3, page 75]) ωd1 = yd dx + xd (xdy − ydx). We know from [9, Proposition 2.3] that if F is an element of F(d) with d ≥ 2, then the dimension of O (F ) is at least 6, or equivalently, the dimension of Iso(F ) is at most 2. In addition these bounds are attained by the convex foliation F1d and the non convex foliation F2d defined by the 1-form (see [3]) ωd2 = xd dx + yd (xdy − ydx). The main result of this paper is the following. Theorem A. — Let d be an integer greater than or equal to 2 and let F be an element of F(d). Assume that the isotropy group Iso(F ) of F has dimension 2. Then F is linearly conjugated to one of the two foliations F1d and F2d defined respectively by the 1-forms 1. ωd1 = yd dx + xd (xdy − ydx); 2. ωd2 = xd dx + yd (xdy − ydx). In other words, O (F1d ) and O (F2d ) are the only orbits of dimension 6. They are closed in F(d). Moreover we have  d−1 α x αd y   Iso(F1d ) = , α ∈ C∗ , β ∈ C , 1 + βx 1 + βx  d+1 α x αd y   d ∗ Iso(F2 ) = , α∈C , β∈C ; 1 + βx 1 + βx these two groups are not conjugated. This theorem is a generalization in arbitrary degree of previous results on foliations of degrees d = 2 ([9, Proposition 2.7]) and d = 3 ([1, Theorem 10], [5, Corollary B]). We also obtain the following corollary, which generalizes [5, Corollary 3.9]: Corollary B. — Let d be an integer greater than or equal to 2 and let F be an element of F(d). If dim O (F ) ≤ 7, then O (F ) ⊂ O (F ) ∪ O (F1d ) ∪ O (F2d ). In particular, when dim O (F ) = 7, the orbit O (F ) of F is closed in F(d) if and only if Fi d 6∈ O (F ) for i = 1, 2. In the spirit of Corollary B we can ask under what condition the closure in F(d) of the orbit O (F ) of an element F of F(d) contains the foliations F1d and F2d , a question that we have already asked and studied in degree 3 in [5, Section 3]. In Section §3, we extend (Propositions 3.4 and 3.11) in arbitrary degree d our previous results in [5, Propositions 3.10, 3.12, 3.15, 3.17] concerning this question. For F ∈ F(d), we call basin of attraction of F the subset B(F ) of F(d) defined by B(F ) := {G ∈ F(d) | F ∈ O (G )}. GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 3 It follows from [9, Theorem 2.15] that in degree 2 the basin B(F12 ) contains a quasi-projective subvariety of F(2) of dimension greater than or equal to dim F(2) − 1. In Section §3, we establish an analogous result in any degree greater than 2. Theorem C (Theorem 3.10). — For any integer d ≥ 2, the basin of attraction B(F1d ) of F1d contains a quasi- projective subvariety of F(d) of dimension greater than or equal to dim F(d) − (d − 1). Notice that the non-convexity of F2d and the fact that FC(d) is closed in F(d) imply that (0.1) B(F2d ) ⊂ F(d) \ FC(d). In degree 2, according to [9, Theorem 3], inclusion (0.1) is an equality: (0.2) B(F22 ) = F(2) \ FC(2). It follows in particular from equality (0.2) that the basin B(F22 ) is a Z ARISKI open subset of F(2). For d ≥ 3 we show the following result. Theorem D (Theorem 3.18). — In any degree d ≥ 3, the basin of attraction B(F2d ) of F2d contains a quasi- projective subvariety of F(d) of dimension greater than or equal to dim F(d) − (d − 3). In particular, the basin B(F23 ) contains a non-empty Z ARISKI open subset of F(3). Along the same order of ideas, we prove the following result. Theorem E (Theorem 3.21). — For any integer d ≥ 2, the intersection B(F1d ) ∩ B(F2d ) is non-empty and it contains a quasi-projective subvariety of F(d) of dimension equal to dim F(d) − 3d. By combining equality (0.2) with the classification of C. FAVRE and J. V. P EREIRA of convex foliations of degree two (cf. [10, Proposition 7.4] or [6, Theorem A]), we see that the only closed orbits in F(2) under the action of Aut(P2C ) are those of F12 and F22 . We show in Section §4 that in any degree d ≥ 3 there are closed orbits in F(d) other than the orbits O (F1d ) and O (F2d ), unlike the situation in degree 2. More precisely, we will consider a family of elements of F(d) which has been already studied in degree d = 2 in [9, page 189], namely the family (F0d (λ))λ∈C∗ of foliations of degree d on P2C defined by the 1-form ωd0 (λ) = xdy − λydx + yd dy. We will see that, for λ = 1, F0d (1) is linearly conjugated to the foliation F1d and that,  for any λ 6= 1, dim O F0 (λ) = 7. Moreover, we will show (Proposition 4.2) that the orbit O F0d (λ) is closed for any d  1 d ≥ 3 and λ = − d−1 , resp. for any d ∈ {3, 4, 5} and any λ ∈ C∗ , and we conjecture that it is so for any d ≥ 6 ∗ and any λ ∈ C (see Conjectures 1 and 2). 1. Some definitions and notations 1.1. Singularities and local invariants. — A degree d holomorphic foliation F on P2C is defined in homo- geneous coordinates [x : y : z] by a 1-form ω = a(x, y, z)dx + b(x, y, z)dy + c(x, y, z)dz, where a, b and c are homogeneous polynomials of degree d + 1 without common factor and satisfying the ∂ ∂ ∂ E ULER condition iR ω = 0, where R = x ∂x + y ∂y + z ∂z denotes the radial vector field and iR is the interior product by R. 4 SAMIR BEDROUNI & DAVID MAR Í N Dually the foliation F can also be defined by a homogeneous vector field ∂ ∂ ∂ Z = U(x, y, z) +V (x, y, z) +W (x, y, z) , ∂x ∂y ∂z the coefficients U,V and W are homogeneous polynomials of degree d without common factor. The relation between Z and ω is given by ω = iR iZ (dx ∧ dy ∧ dz). The singular locus SingF of F is the projectivization of the singular locus of ω Sing ω = {(x, y, z) ∈ C3 | a(x, y, z) = b(x, y, z) = c(x, y, z) = 0}. Let C ⊂ P2C be an algebraic curve with homogeneous equation F(x, y, z) = 0. We say that C is an invariant curve by F if C r SingF is a union of (ordinary) leaves of the regular foliation F |P2 rSingF . In algebraic C terms, this is equivalent to require that the 2-form ω ∧ dF is divisible by F, i.e. it vanishes along each irreducible component of C . Let p be an arbitrary point of C . When each irreducible component of C passing through p is not F -invariant, we define the tangency order Tang(F , C , p) of F with C at p as follows. We fix a local chart (u, v) such that p = (0, 0); let f (u, v) = 0 be a reduced local equation of C in a neighborhood of p and let X be a vector field defining the germ of F at p. We denote by X( f ) the L IE derivative of f along X and by h f , X( f )i the ideal of C{u, v} generated by f and X( f ). Then C{u, v} Tang(F , C , p) = dimC . h f , X( f )i Notice that Tang(F , C , p) coincides with the intersection multiplicity (C .C 0 ) p at p of the two algebraic curves C = {F = 0} and C 0 = {Z(F) = 0}. Moreover, Tang(F , C , p) < +∞ by the non-invariance of the irreducible components of C passing through p. By convention, we put Tang(F , C , p) = +∞ if there is at least one irreducible component of C invariant by F and passing through p. Let us recall some local notions attached to the pair (F , s), where s ∈ SingF . The germ of F at s is defined, up to multiplication by a unity in the local ring Os at s, by a vector field X = A(u, v) ∂u∂ ∂ + B(u, v) ∂v . The algebraic multiplicity ν(F , s) of F at s is given by ν(F , s) = min{ν(A, s), ν(B, s)}, where ν(g, s) denotes the algebraic multiplicity of the function g at s. Let us denote by Ls (F ) the family of straight lines through s which are not invariant by F . For any line ` of Ls (F ), we have the inequalities 1 ≤ Tang(F , `, s) ≤ d. This allows us to associate to the pair (F , s) the following (invariant) integers τ(F , s) = min{Tang(F , `, s) | ` ∈ Ls (F )}, κ(F , s) = max{Tang(F , `, s) | ` ∈ Ls (F )}. The invariant τ(F , s) represents the tangency order of F with a generic line passing through s. It is easy to see that τ(F , s) = min{k ≥ 1 | det(Jsk X, Rs ) 6≡ 0} ≥ ν(F , s), where Jsk X denotes the k-jet of X at s and Rs is the radial vector field centered at s. The M ILNOR number of F at s is the integer µ(F , s) = dimC Os /hA, Bi, where hA, Bi denotes the ideal of Os generated by A and B. The singularity s is called radial of order n − 1, with n ∈ {2, . . . , d}, if ν(F , s) = 1 and τ(F , s) = n. GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 5 The singularity s is called non-degenerate if µ(F , s) = 1, or equivalently if the Jacobian matrix of X at s, denoted by JacX(s), possesses two nonzero eigenvalues λ, µ. In this case, the quantity tr2 (JacX(s)) λ µ BB(F , s) = = + +2 det(JacX(s)) µ λ is called the BAUM -B OTT index of F at s, see [2]. We will say that the singularity s is quasi-radial of order n − 1 if µ(F , s) = 1, BB(F , s) = 4 and κ(F , s) = n. In the sequel we will denote by QRad(F , n − 1) the set of quasi-radial singularities of F of order n − 1 and [ F , n − 1) the subset of Sing(F ) × Ls (F ) defined by by QRad( n o [ F , n − 1) := (s, `) ∈ Sing(F ) × Ls (F ) | µ(F , s) = 1, BB(F , s) = 4, Tang(F , `, s) = n . QRad( Remark 1.1. — Every radial singularity s of order n−1 of a foliation F of degree d ≥ 2 on P2C is quasi-radial of order ≥ n − 1, because κ(F , s) ≥ τ(F , s). The converse is false: for instance, for the foliation defined in the affine chart z = 1 by the 1-form (x + y)dy − ydx + (xn + yd )dx, with n ∈ {2, 3, . . . , d}, the point [0 : 0 : 1] is a quasi-radial singularity of order n − 1, but it is not radial. 1.2. Inflection points. — Let us consider a foliation F of degree d on P2C and let p be a regular point of F . Let us denote by TPp F the tangent line to the leaf of F passing through p; it is the straight line of P2C passing through p with direction Tp F . If k ∈ {2, . . . , d}, we will say that p is a (transverse) inflection point of order k − 1 of F if Tang(F , TPp F , p) = k, in which case the line TPp F is not invariant by F . When TPp F is F -invariant, the point p will be called a trivial inflection point of F . If we denote by Inv(F ) the set of invariant lines of F , then the set of trivial inflection points of F is precisely Inv(F ) \ Sing(F ). In the sequel, we will denote by Flex(F ) the set of inflection points of F and by Flex(F , k − 1) the subset of Flex(F ) consisting of transverse inflection points of F of order k − 1, i.e. n o Flex(F , k − 1) := p ∈ Flex(F ) | p 6∈ Sing(F ), Tang(F , TPp F , p) = k . Let us recall the notion of inflection divisor of F , introduced by P EREIRA [16], which allows to determine the set Flex(F ). Let Z be a homogeneous vector field of degree d on C3 defining F . The inflection divisor of F , denoted by IF , is the divisor of P2C defined by the homogeneous equation x Z(x) Z2 (x) (1.1) y Z(y) Z2 (y) = 0. z Z(z) Z2 (z) According to [16], IF satisfies the following properties: 1. The support of IF is exactly the closure of the set Flex(F ) of inflection points of F . More precisely, IF can be decomposed as IF = Iinv tr inv F + IF , where the support of IF is the set Inv(F ) of F -invariant lines and the support of IFtr is the closure of the set of transverse inflection points of F . 2. If C is an algebraic curve invariant by F , then C ⊂ IF if and only if C ⊂ Inv(F ). 3. The degree of the divisor IF is 3d. The foliation F will be called convex if its inflection divisor IF is totally invariant by F , i.e. if IF is a product of invariant lines. 6 SAMIR BEDROUNI & DAVID MAR Í N 2. Description of the foliations F of degree greater than or equal to 2 such that dim O (F ) = 6 Recall that the foliations F1d and F2d are respectively defined in the affine chart z = 1 by the 1-forms ωd1 = yd dx + xd (xdy − ydx) and ωd2 = xd dx + yd (xdy − ydx). The foliation F1d is convex with inflection divisor IF d = IFinvd = xd+1 y2d−1 and it has two singular points 1 1 s1 = [0 : 0 : 1] and s2 = [0 : 1 : 0]; the singularity s1 has maximal algebraic multiplicity d and s2 is radial of maximal order d − 1. The foliation F2d is not convex with invariant inflection divisor IFinvd = x2d+1 and 2 transverse inflection divisor IFtr d = yd−1 . The singular locus Sing(F2d ) is reduced to the point s1 = [0 : 0 : 1]; 2 ωd1 ωd2 moreover ν(F2d , s1 ) = d. We note that the 1-forms and are closed and they respectively admit as x 2 yd xd+2 first integrals  d−1 1 x 1 1  y d+1 1 + and − ; d −1 y x d +1 x x this allows to check that αd−1 x αd y αd+1 x αd y       Iso(F1d ) = , α ∈ C∗ , β ∈ C and Iso(F2d ) = , α ∈ C∗ , β ∈ C . 1 + βx 1 + βx 1 + βx 1 + βx In particular, dim Iso(Fi d ) = 2 for i = 1, 2. Thus the orbits O (F1d ) and O (F2d ) are both of dimension 6, which is the minimal dimension possible in any degree d ≥ 2 ([9, Proposition 2.3]). Theorem A announced in the Introduction shows that the orbits O (F1d ) and O (F2d ) are the only orbits having minimal dimension 6. The goal of this section is to prove this theorem. Let us denote by χ(P2C ) the L IE algebra of holomorphic vector fields on P2C : χ(P2C ) is of course the L IE algebra of the automorphism group of P2C . Let F be a foliation on P2C and let X be an element of χ(P2C ). Following [9] we will say that X is a symmetry of the foliation F if the flow exp(tX) is, for each t, in the isotropy group Iso(F ) of F . If ω defines F in an affine chart, X is a symmetry of F if and only if LX ω ∧ ω = 0, where LX ω denotes the L IE derivative of ω along X. Lemma 2.1. — Let F be a foliation of degree d on P2C and let X be a symmetry of F . Assume that there is an affine chart C2 ⊂ P2C such that the vector field X is affine (i.e. deg X ≤ 1) and let ω be a 1-form defining F in this chart. Then there is a constant λ ∈ C such that LX ω = λω. Proof. — We will use an argument similar to one in [9, Proposition 2.5]. Since LX ω ∧ ω = 0 and ω has isolated singularities, the D E R HAM-S AITO division theorem (cf. [17] or [8, Proposition 1.14]) ensures the existence of a holomorphic function g on C2 such that LX ω = gω. The 1-form ω and the vector field X being polynomials, LX ω is also polynomial; therefore g is rational and holomorphic on C2 hence polynomial. The vector field X being affine we have deg LX ω ≤ deg ω; the equality LX ω = gω implies that g is constant. If F is a foliation on P2C , we will denote by iso(F ) the L IE algebra of the algebraic group Iso(F ); iso(F ) is a L IE subalgebra of χ(P2C ) and it consists of symmetries of F . We know from [9, Proposition 2.5] that if dim iso(F ) = 2 then iso(F ) is affine, i.e. generated by two vector fields X and Y such that [X, Y] = Y. The following lemma classifies the affine L IE subalgebras of χ(P2C ) and it will be used to prove Theorem A. Lemma 2.2. — Every affine L IE subalgebra of χ(P2C ) is linearly conjugated to one of the following models ∂ (a) γ x ∂x ∂ ∂ + y ∂y , ∂y with γ ∈ C∗ ; GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 7 ∂ ∂ (b) y ∂y , ∂y ; ∂ ∂ ∂ (c) ∂x + y ∂y , ∂y ; ∂ ∂ ∂ (d) x ∂x + (x + y) ∂y , ∂y ; ∂ ∂ ∂ ∂ (e) x ∂x + 2y ∂y , ∂x + x ∂y . Proof. — Let g be an affine L IE subalgebra of χ(P2C ). Then there exist X and Y in χ(P2C ) such that g = hX, Yi and [X, Y] = Y. Fixing homogeneous coordinates [x : y : z] in P2C we have an isomorphism of L IE algebras τ : sl3 (C) → χ(P2C ) defined, for A ∈ sl3 (C), by  ∂  ∂x ∂ τ(A) = (x y z)A  .   ∂y ∂ ∂z   a11 a12 a13 Notice that if A =  a21 a22 a23  ∈ sl3 (C), then in the affine chart z = 1 the vector field τ(A) ∈ χ(P2C ) a31 a32 a33 writes as  ∂  ∂ a31 + (a11 − a33 )x + a21 y − a13 x2 − a23 xy + a32 + a12 x + (a22 − a33 )y − a13 xy − a23 y2 . ∂x ∂y Let M and N be the matrices of sl3 (C) associated to the vector fields X and Y respectively, i.e. M = τ−1 (X) and N = τ−1 (Y).  Then the L IE bracket [X, Y] corresponds to [M, N] := MN − NM and therefore [M, N] = N. −m22 − m33 m12 m13 Let us write M =  m21 m22 m23  . Taking into account the possible J ORDAN forms of a matrix m31 m32 m33 of sl3 (C), it suffices us to treat the following possibilities         −a − b 0 0 −2c 0 0 0 0 0 0 0 0 N= 0 a 0 , N =  0 c 0 , N =  0 0 0 , N =  1 0 0 . 0 0 b 0 1 c 0 1 0 0 1 0 where a, b ∈ C, c ∈ C∗ , with (a, b) 6= (0, 0).   −a − b 0 0 1. If N =  0 a 0  then the equality [M, N] = N implies that a = b = 0: contradiction. 0 0 b   −2c 0 0 2. If N =  0 c 0  then the (1, 1) coefficient of the matrix [M, N] − N is equal to 2c and is 0 1 c therefore nonzero: contradiction.   0 0 0   1 − 2m33 m12 0 3. Assume that N =  0 0 0 ; the equality [M, N] = N then leads to M =  0 m33 − 1 0 . 0 1 0 m31 m32 m33 Up to replacing M by M − m32 N we can assume that m32 = 0. Now we will distinguish several eventualities: 8 SAMIR BEDROUNI & DAVID MAR Í N   3m33 − 1 m12 0 3.1. When (3m33 − 1)(3m33 − 2) 6= 0 the matrix P =  0 3m33 − 2 0  commutes −m31 −m31 m12 3m33 − 2   1 − 2m33 0 0 with N and P−1 MP =  0 m33 − 1 0 . Thus g is linearly conjugated to 0 0 m33 τ(P−1 MP), τ(N) = (1 − 3m33 )x ∂x ∂ ∂ ∂ − y ∂y ∂ , ∂y = γ x ∂x ∂ ∂ + y ∂y , ∂y , where γ = 3m33 − 1 ∈ C∗ . 1   −m12 0δ 3.2. Assume that m33 = 13 . If δ ∈ C∗ then the matrix P =  0 1 0  commutes with N and 0 m12 m31 1  1  3 0 0 P−1 MP =  0 − 32 0 . As a result g is linearly conjugated to m31 δ 0 13 τ(P−1 MP), τ(N) = m31 ∂ δ ∂x ∂ ∂ − y ∂y , ∂y = − mδ31 ∂x ∂ ∂ ∂ + y ∂y , ∂y . The case where m31 = 0 leads to the model (b). If m31 6= 0 then by taking δ = −m31 we get the model (c).   δ 0 0 3.3. Assume that m33 = 32 . If δ ∈ C∗ then the matrix P =  0 1 0  commutes with N −δm31 −m12 m31 1  1 m12  −3 δ 0 and P−1 MP =  0 − 31 0 . As a consequence g is linearly conjugated to 0 0 23 τ(P−1 MP), τ(N) = − x ∂x ∂ + ( mδ12 x − y) ∂y ∂ ∂ ∂ , ∂y = x ∂x + (y − mδ12 x) ∂y ∂ ∂ , ∂y . The case m12 = 0 leads to the model (a) with γ = 1. If m12 6= 0 then by taking δ = −m12 we obtain the model (d).     0 0 0 −1 0 0 4. Assume that N =  1 0 0 ; then the equality [M, N] = N implies that M =  m32 0 0 . 0 1 0 m31 m32 1   1 0 0 Up to replacing M by M − m32 N we can assume that m32 = 0. The matrix P =  0 1 0  commutes m31 − 2 0 1   −1 0 0 with N and P−1 MP =  0 0 0 . Therefore g is linearly conjugated to 0 0 1 τ(P−1 MP), τ(N) = − 2x ∂x ∂ ∂ − y ∂y ∂ , y ∂x ∂ + ∂y ∂ = 2x ∂x ∂ + y ∂y ∂ , y ∂x ∂ + ∂y . By permuting the coordinates x and y we obtain the model (e). Proof of Theorem A. — Since dim iso(F ) = dim Iso(F ) = 2, [9, Proposition 2.5] implies that iso(F ) is affine. Therefore, up to linear conjugation, iso(F ) is one of the models (a)–(e) of Lemma 2.2. Let ω be a 1-form defining F in the affine chart z = 1 ω = A(x, y)dx + B(x, y)dy, A, B ∈ C[x, y], gcd(A, B) = 1. GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 9 We will study the five possible models (a)–(e) of the L IE algebra iso(F ) and show that ω is linearly conju- gated to one of the two 1-forms ωd1 or ωd2 . 1. Assume that iso(F ) is of one of the types (a)–(d), i.e. that iso(F ) = X, Y where X ∈ {γ x ∂x ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∗ y ∂y , ε ∂x + y ∂y , x ∂x + (x + y) ∂y }, Y = ∂y with ε ∈ {0, 1} and γ ∈ C . By Lemma 2.1 there exist λ, µ ∈ C such that LX ω = λω and LY ω = µω. Since LY dx = dLY x = 0 and LY dy = dLY y = 0, we have LY ω = Y(A)dx + Y(B)dy = ∂A ∂B ∂A ∂B ∂y dx + ∂y dy. Therefore LY ω = µω if and only if ∂y = µA and ∂y = µB. Since A, B ∈ C[x, y] and µ ∈ C, it follows that µ = 0, A(x, y) = A(x) and B(x, y) = B(x). Thus ω = A(x)dx + B(x)dy, A, B ∈ C[x], gcd(A, B) = 1. ∂ 1.1. Let us consider the case where X = γ x ∂x ∂ + y ∂y with γ ∈ C∗ . We have LX ω = X(A)dx + AdX(x) + X(B)dy + BdX(y) = γ xA0 + γ A dx + γ xB0 + B dy,   so that LX ω = λω if and only if γ xA0 = (λ − γ)A and γ xB0 = (λ − 1)B. By putting κ = λ−γ γ and ν = γ , λ−1 0 0 the last two equations can be rewritten as xA = κ A and xB = νB and can be immediately integrated to give A(x) = α xκ and B(x) = β xν , where α, β ∈ C. Since A, B ∈ C[x] and gcd(A, B) = 1, we deduce that α, β ∈ C∗ , κ, ν ∈ N and κ ν = 0. The equality deg F = d then implies that – either κ = 0 and ν = d, in which case ω = αdx + β xd dy; – or ν = 0 and κ = d, in which case ω = α xd dx + βdy.   If ω = αdx + β xd dy, resp. ω = α xd dx + βdy, by making the change of coordinates (x, y) 7→ xy , − βαx , we reduce ourselves to ω = ωd1 = yd dx + xd (xdy − ydx), resp. ω = ωd2 = xd dx + yd (xdy − ydx). ∂ ∂ 1.2. Let us examine the case where X = ε ∂x + y ∂y with ε ∈ {0, 1}. Since LX dx = dLX x = 0 and LX dy = dLX y = dy, we have LX ω = X(A)dx + X(B)dy + Bdy = ε A0 dx + (εB0 + B)dy. Therefore LX ω = λω if and only if ε A0 = λ A and εB0 = (λ − 1)B. Since A, B ∈ C[x] and λ ∈ C, it follows that AB = 0: contradiction with gcd(A, B) = 1. ∂ ∂ 1.3. Let us study the eventuality: X = x ∂x + (x + y) ∂y . We have dX(x) = dx and dX(y) = dx + dy, so that LX ω = X(A)dx + Adx + X(B)dy + B(dx + dy) = (xA0 + A + B)dx + (xB0 + B)dy. The condition LX ω = λω is then equivalent to the system of differential equations xA0 + B = (λ − 1)A and xB0 = (λ − 1)B, which can be easily integrated to yield A(x) = (a − b ln x)xλ−1 and B(x) = bxλ−1 , where a, b ∈ C. Since A ∈ C[x], we deduce that b = 0 and therefore B ≡ 0: contradiction with gcd(A, B) = 1. 2. Assume that iso(F ) is of type (e), i.e. iso(F ) = X, Y where X = x ∂x ∂ ∂ + 2y ∂y ∂ , Y = ∂x ∂ + x ∂y . As before by writing explicitly that LX ω = λω and LY ω = µω, with λ, µ ∈ C (Lemma 2.1), we obtain the system of partial differential equations x ∂A ∂A ∂x + 2y ∂y = (λ − 1)A, x ∂B ∂B ∂x + 2y ∂y = (λ − 2)B, ∂A ∂x + x ∂A ∂y = µ A − B, ∂B ∂x + x ∂B ∂y = µB. It follows in particular that (x2 − 2y) ∂B ∂y = (µ x + 2 − λ)B and (x2 − 2y) ∂A ∂y = (µ x + 1 − λ)A − xB. Elementary integrations then lead to λ−2−µ x  p  λ−2−µ x B(x, y) = b(x)(x2 − 2y) 2 and A(x, y) = a(x) x2 − 2y − xb(x) (x2 − 2y) 2 , 10 SAMIR BEDROUNI & DAVID MAR Í N for some functions a and b of the coordinate x. Since A, B ∈ C[x, y] and gcd(A, B) = 1, we deduce that λ − 2 − µ x = 0 and a(x) = 0 for any x ∈ C, hence λ = 2, µ = 0 and a ≡ 0. Therefore B(x, y) = b(x) and A(x, y) = −xb(x) = −xB(x, y): contradiction with gcd(A, B) = 1. 3. Foliations of F(d) degenerating onto F1d and F2d In this section we will study the problem of knowing whether the closure of the orbit of a foliation of F(d) contains the foliations F1d and F2d . The following definition will be useful. Definition 3.1 ([9]). — Let F and F 0 be two foliations of F(d). We say that F degenerates onto F 0 if the closure O (F ) (in F(d)) of the orbit O (F ) contains O (F 0 ) and O (F ) 6= O (F 0 ). Remarks 3.2. — Let F and F 0 be two foliations such that F degenerates onto F 0 . Then: (i) dim O (F 0 ) < dim O (F ); (ii) deg Iinv inv tr tr 0 F ≤ deg IF 0 , which is equivalent to deg IF ≥ deg IF 0 . In particular, if F is convex, F is also convex. Corollary B is an immediate consequence of Theorem A and Remark 3.2 (i). Remark 3.3. — Corollary B applies particularly to any foliation of F(d) which is homogeneous, i.e. which is given, for a suitable choice of affine coordinates (x, y), by a homogeneous 1-form ω = A(x, y)dx +B(x, y)dy, where A, B ∈ C[x, y]d and gcd(A, B) = 1. Indeed, for such a foliation H ∈ F(d), we have (cf. [4]) Iso(H ) = {(α x, α y) | α ∈ C∗ }; in particular, dim O (H ) = 7 and consequently O (H ) ⊂ O (H ) ∪ O (F1d ) ∪ O (F2d ). Assertion 1. (resp. 2.) of the following proposition gives a necessary (resp. sufficient) condition for a foliation of F(d) to degenerate onto the foliation F1d . Proposition 3.4. — Let F be an element of F(d) such that F1d 6∈ O (F ). The following assertions hold: 1. If F degenerates onto F1d , then F possesses a non-degenerate singularity m satisfying BB(F , m) = 4. 2. If F possesses a quasi-radial singularity of maximal order d − 1, i.e. if QRad(F , d − 1) 6= 0,/ then F d degenerates onto F1 . Proof. — 1. Assume that F degenerates onto F1d . Then there is an analytic family of foliations (Fε ) defined by a family of 1-forms (ωε ) such that Fε belongs to O (F ) for ε 6= 0 and Fε=0 = F1d . The non-degenerate singular point of F1d , denoted by m0 , is “stable” in the sense that there is an analytic family (mε ) of non- degenerate singular points of Fε such that mε=0 = m0 . The Fε ’s being conjugated to F for ε 6= 0, the foliation F admits a non-degenerate singular point m such that ∀ ε ∈ C∗ , BB(Fε , mε ) = BB(F , m). Since µ(Fε , mε ) = 1 for any ε ∈ C, the function ε 7→ BB(Fε , mε ) is continuous, hence constant on C. As a consequence BB(F , m) = BB(Fε=0 , mε=0 ) = BB(F1d , m0 ) = 4. GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 11 2. Assume that F has a quasi-radial singularity m of order d − 1. Then µ(F , m) = 1, BB(F , m) = 4 and κ(F , m) = d. This last equality ensures the existence of a line `m passing through m, not invariant by F and such that Tang(F , `m , m) = d. Let us choose an affine coordinate system (x, y) such that m = (0, 0) and `m = {x = 0}. The foliation F is defined in these coordinates by a 1-form ω of type d  ω = Cd (x, y)(xdy − ydx) + ∑ Ai (x, y)dx + Bi (x, y)dy , where Ai , Bi ∈ C[x, y]i , Cd ∈ C[x, y]d . i=1 We have d d ω ∧ dx = ∑ Bi (0, y)dy ∧ dx = ∑ Bi (0, 1)yi dy ∧ dx. x=0 i=1 i=1 Then the equality Tang(F , `m , m) = d translates into Bi (0, 1) = 0 for i ∈ {1, 2, . . . , d − 1} and Bd (0, 1) 6= 0. This allows to write B1 (x, y) = αx, Bd (x, y) = xBbd−1 (x, y) + γ yd , Bi (x, y) = xBei−1 (x, y) for i ∈ {2, 3, . . . , d − 1}, where Bei−1 ∈ C[x, y]i−1 , Bbd−1 ∈ C[x, y]d−1 , γ ∈ C∗ , α ∈ C. The equalities µ(F , m) = 1 and BB(F , m) = 4 imply that α 6= 0 and A1 (x, y) = δx − αy for some δ ∈ C. Thus ω is of type  d d−1 ω = δxdx + xBbd−1 (x, y) + γ yd dy + Cd (x, y) + α xdy − ydx + ∑ Ai (x, y)dx + x ∑ Bei−1 (x, y)dy,   i=2 i=2 where Ai ∈ C[x, y]i , Bei−1 ∈ C[x, y]i−1 , Bbd−1 ∈ C[x, y]d−1 , δ ∈ C, α, γ ∈ C∗ . By putting ϕ = εd x, εy and θ = α(xdy − ydx) + γ yd dy, we obtain  d d−1 1   ϕ∗ ω = θ + εd−1 δxdx + xBbd−1 (εd−1 x, y)dy + εd Cd (εd−1 x, y) (xdy − ydx) + ∑ εi−1 Ai (εd−1 x, y)dx + x ∑ εi−1 Bei−1 (εd−1 x, y)dy εd+1 i=2 i=2   x γ x which tends to θ as ε tends to 0. By making the change of coordinates (x, y) 7→ y − αy , y , we reduce ourselves to θ = ωd1 = yd dx + xd (xdy − ydx). As a result F degenerates onto F1d . Example 3.5. — Let us consider the homogeneous foliation H1d defined in the affine chart z = 1 by the 1-form ω d1 = yd dx − xd dy. We know from [4, Proposition 4.1] that H1d is convex and admits the points [1 : 0 : 0] and [0 : 1 : 0] as radial singularities of maximal order d − 1. Therefore H1d degenerates onto F1d (Proposition 3.4) and it does not degenerate onto F2d , because F2d is not convex. Thus, according to Remark 3.3, we have O (H1d ) = O (H1d ) ∪ O (F1d ). Example 3.6. — Let us consider the family (G d (γ))γ∈C of foliations of degree d on P2C defined in the affine chart z = 1 by ηd (γ) = (x − γ y)dy − ydx + xd dx − yd dy. We remark that the point m = [0 : 0 : 1] is a non-degenerate singularity of G d (γ) with BAUM -B OTT index 4. Moreover, along the line ` = {y = 0} we have ηd (γ) ∧ dy = xd dx ∧ dy, so that Tang(G d (γ), `, m) = d. y=0 It follows that the singularity m of G d (γ) is quasi-radial of maximal order d − 1. As a consequence G d (γ) degenerates onto F1d (Proposition 3.4). 12 SAMIR BEDROUNI & DAVID MAR Í N The converse of assertion 2. of Proposition 3.4 is false as the following example shows. Example 3.7. — Let F be the foliation of degree d ≥ 2 on P2C defined in the affine chart z = 1 by ω = xdy − ydx + P(y)dy, where P is a polynomial of C[y] of degree d admitting 0 as a root of multiplicity ≤ d − 1, i.e. P is of the form P(y) = yν (a0 + a1 y + · · · + ad−ν yd−ν ), where ν ∈ {1, 2, . . . , d − 1}, ai ∈ C, a0 ad−ν 6= 0. The singular locus of F consists of the two points m = [0 : 0 : 1] and m0 = [1 : 0 : 0]; moreover µ(F , m) = 1, BB(F , m) = 4, κ(F , m) = ν < d, µ(F , m0 ) > 1. It follows that F has no quasi-radial singularity of maximalorder d − 1, i.e. QRad(F , d − 1) = 0. / However, ad−ν 1 F degenerates onto F1d . Indeed, by putting ϕ = x, y , we see that εd ε εd+1 ∗ lim ϕ ω = xdy − ydx + yd dy. ε→0 ad−ν Question 1. — Let F be a foliation of degree d ≥ 2 on P2C . Is it true that if F degenerates onto F1d then – either F admits a quasi-radial singularity of maximal order d − 1, – or F is conjugated to Example 3.7, i.e. up to linear conjugation F is given by a 1-form of type xdy − ydx + P(y)dy with P ∈ C[y], deg P = d and P(0) = 0? Proposition 3.8. — Let d be an integer greater than or equal to 2. Let us denote by U1 (d) the subset of F(d) defined by n o U1 (d) := F ∈ F(d) | ∀s ∈ Sing(F ), µ(F , s) = 1, τ(F , s) = 1 . Then: (i) U1 (d) isa non-empty Z ARISKI open subset of F(d); in particular, for any γ ∈ C, G d (γ) ∈ U1 (d) if and (d + 1)d+1  only if γ γd+1 + 6= 0. dd (ii) Let F be an element of U1 (d). For any singular point s ∈ Sing(F ), the set n o Λ(F , s) := `s ∈ Ls (F ) | Tang(F , `s , s) > 1 d has at most 2 elements. In particular, the set S [ F , n − 1) is finite. QRad( n=2 To prove this proposition, we need the following lemma. Lemma 3.9. — Let F be a foliation of degree d ≥ 2 on P2C , s a singular point of F , `s a line passing through s and not invariant by F and X = A(x, y) ∂x ∂ + B(x, y) ∂y ∂ a polynomial vector field defining F in an affine chart (x, y) containing s. Let us denote by (x0 , y0 ) the coordinates of s and let a(x − x0 ) + b(y − y0 ) = 0 be an equation of the line `s . Then, for any n ∈ {2, 3, . . . , d}, Tang(F , `s , s) ≥ n if and only if dj   aA(x0 + bt, y0 − at) + bB(x0 + bt, y0 − at) = 0, ∀ j ∈ {1, 2, . . . , n − 1}. dt j t=0 n o In particular, the set Λ(F , s) := `s ∈ Ls (F ) | Tang(F , `s , s) > τ(F , s) is finite and its cardinality is at most τ(F , s) + 1. GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 13 Proof. — The 1-form ω = A(x, y)dy − B(x, y)dx also defines the foliation F because iX ω = 0. We have  ω ∧ d a(x − x0 ) + b(y − y0 ) = P(t)dy ∧ dx, (x,y)=(x0 +bt,y0 −at) where P(t) = aA(x0 + bt, y0 − at) + bB(x0 + bt, y0 − at). Thus Tang(F , `s , s) = ν(P(t), 0). Notice that P(0) = 0 because the point s being singular for F , we have A(x0 , y0 ) = B(x0 , y0 ) = 0. Then Tang(F , `s , s) ≥ n if and only if the root t = 0 of the polynomial P has multiplicity at least n, that is if and only if P0 (0) = P00 (0) = · · · = P(n−1) (0) = 0, hence the announced equivalence holds. By conjugating ω by the translation (x + x0 , y + y0 ), we can assume that s = (0, 0). Let us denote τ(F , s) simply by τ. Then the vector field X decomposes in the form d+1 X = Cτ−2 (x, y)R + ∑ Xi , i=τ ∂ ∂ ∂ ∂ where R = x ∂x + y ∂y is a polynomial of degree ≤ τ − 2, Xi = Ai (x, y) ∂x , Cτ−2 + Bi (x, y) ∂y is a homogeneous vector field of degree i, with det(Xτ , R) 6≡ 0. Thus, we have   d+1   d+1  aA(bt, −at) + bB(bt, −at) = a xCτ−2 (x, y) + ∑ Ai (x, y) + b yCτ−2 (x, y) + ∑ Bi (x, y) i=τ i=τ (x,y)=(bt,−at) d+1   = ∑ aAi (bt, −at) + bBi (bt, −at) i=τ d+1 = ∑ t i Qi+1 (a, b), i=τ where Qi+1 (a, b) := aAi (b, −a)+bBi (b, −a) is a homogeneous polynomial of degree i+1 in (a, b). From this, we deduce that Tang(F , `s , s) > τ if and only if Qτ+1 (a, b) = 0. As a result n o Λ(F , s) = `s = {ax + by = 0} ∈ Ls (F ) | Qτ+1 (a, b) = 0 . Now, the polynomial Qτ+1 is not identically zero because Qτ+1 (a, b) = − det(Xτ , R) (x,y)=(b,−a) 6≡ 0. It follows that Λ(F , s) has cardinality at most τ + 1. Proof of Proposition 3.8. — We have n o U1 (d) = F ∈ F(d) | ∀s ∈ Sing(F ), det(JacX(s)) 6= 0, det(Js1 X, Rs ) 6≡ 0 , where X denotes a polynomial vector field defining F in an affine chart containing s and Rs is the radial vector field centered at s. It follows that U1 (d) is a Z ARISKI open subset  of F(d). To establish assertion (i), d+1 d it remains to show that for any γ ∈ C, G (γ) ∈ U1 (d) if and only if γ γ d+1 + (d+1) dd 6= 0. In homogeneous coordinates, the foliation G d (γ) is defined by the 1-form Ωd (γ) = z xd − yzd−1 dx − z yd + γ yzd−1 − xzd−1 dy + yd+1 − xd+1 + γ y2 zd−1 dz .    The singular locus Sing G d (γ) consists of the points  s0 = [0 : 0 : 1], sk = [xk : xkd : 1], s0l = [1 : ξl : 0], k ∈ {1, 2, . . . , d 2 − 1}, l ∈ {0, 1, . . . , d}, 2iπ  2 −1 where ξ = exp d+1 and the xk ’s are the roots of the polynomial P(x) = xd + γ xd−1 − 1. 14 SAMIR BEDROUNI & DAVID MAR Í N In the affine chart z = 1, resp. x = 1, G d (γ) is given by the vector field ∂ ∂ ∂ ∂ Y = yd + γ y − x + xd − y , resp. Z = yd+1 + γ y2 zd−1 − 1 + z yd + γ yzd−1 − zd−1 . ∂x ∂y ∂y ∂z A direct computation show that det(JacY(s0 )) = 1 6= 0, det(Js10 Y, Rs0 ) = γ y2 and 2 det(JacZ(s0l )) = (d + 1)ξ−2l = det(JacY(sk )) = 1 − dγ xkd−1 − d 2 xkd −1 = (d − 1) dγ xkd−1 − d − 1 , because P(xk ) = 0,  6 0, 2 2 2 1 −l l det(Js1k Y, Rsk ) = dxkd −d + γ y − xkd − dxkd−1 x − xk 6≡ 0, because xk 6= 0.   det(Js0 Z, Rs0l ) = dξ y − ξ z 6≡ 0, l From these we deduce that G d (γ) ∈ U1 (d) if and only if γ 6= 0 and dγ xkd−1 − d − 1 6= 0, i.e. if and only if γ 6= 0 and xkd−1 6= d+1 dγ . Now, by putting Q(t) = t d+1 + γt − 1, we have P(x) = Q(xd−1 ) so that t ∈ C is a root of the 0 polynomial Q(t) if and only if there exists k ∈ {1, 2, . . . , d 2 − 1} such that t0 = xkd−1 . It follows that (d + 1)d+1     d d +1 d+1 G (γ) ∈ U1 (d) ⇐⇒ γ Q 6= 0 ⇐⇒ γ γ + 6= 0. dγ dd Assertion (ii) is an immediate consequence of Lemma 3.9. Theorem 3.10. — Let d be an integer greater than or equal to 2. Let us denote by Σ1 (d) the subset of F(d) defined by n o Σ1 (d) := F ∈ F(d) | QRad(F , d − 1) 6= 0/ . Then (a) 0/ 6= Σ1 (d) B(F1d ); (b) Σ1 (d) is a constructible subset of F(d) of dimension greater than or equal to dim F(d) − (d − 1). Proof. — (a) Σ1 (d) contains the foliations H1d and G d (γ), γ ∈ C (Examples 3.5 and 3.6) and is therefore non- empty. Assertion 2. of Proposition 3.4 ensures that Σ1 (d) ⊂ B(F1d ); this inclusion is strict as Example 3.7 shows. (b) Let us denote by P̌2C the dual projective plane of P2C . Let π : F(d) × P2C × P̌2C → F(d) be the projection onto the first factor; we have Σ1 (d) = π(W1 (d)), where [ W1 (d) : = {F } × QRad( [ F , d − 1) F ∈Σ1 (d) n o = (F , s, `) ∈ F(d) × P2C × P̌2C | s ∈ Sing(F ), ` ∈ Ls (F ), µ(F , s) = 1, BB(F , s) = 4, Tang(F , `, s) = d . According to Lemma 3.9, W1 (d) can be rewritten as     s = (x0 , y0 ) ∈ ` = {a(x − x0 ) + b(y − y0 ) = 0}       tr2 (JacX(s))     A(x0 , y0 ) = 0, B(x0 , y0 ) = 0, det(JacX(s)) 6= 0,     =4    det(JacX(s))  (3.1) W1 (d) = (F , s, `) ∈ F(d) × P2C × P̌2C ,    aA(x0 + bt, y0 − at) + bB(x0 + bt, y0 − at) 6≡ 0        dj         aA(x0 + bt, y0 − at) + bB(x0 + bt, y0 − at) = 0, j = 1, . . . , d − 1     dt j t=0 where X = ∂ A(x, y) ∂x ∂ + B(x, y) ∂y is a polynomial vector field defining F in an affine chart (x, y) containing s. It follows that W1 (d) is a quasi-projective subvariety of F(d) × P2C × P̌2C . Thus, by C HEVALLEY’s Theorem [11, Exercise II.3.19], the set Σ1 (d) = π(W1 (d)) is constructible. GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 15 According to the above  discussion  Proposition 3.8 (i), the intersection U1 (d) ∩ Σ1 (d) contains the folia- and d d+1 (d+1)d+1 tions G (γ), with γ γ + dd 6= 0, and is therefore non-empty (U1 (d) being the set of F ∈ F(d) such that for any s ∈ SingF , µ(F , s) = 1 and τ(F , s) = 1). Then there exists an irreducible component Σ01 (d) of k Σ1 (d) such that U1 (d) ∩ Σ01 (d) 6= 0. W1i (d) be the decomposition of W1 (d) into its irreducible S / Let W1 (d) = i=1 components. Let us denote by π0 : W1 (d) → F(d) the restriction of π to W1 (d). Then, there is n ∈ {1, . . . , k} k such that π0 (W1n (d)) = Σ01 (d). Indeed, since Σ1 (d) = π0 (W1 (d)), we have Σ01 (d) ⊂ Σ1 (d) = π0 (W1i (d)). The S i=1 irreducibility of Σ01 (d) therefore ensures the existence of n ∈ {1, . . . , k} such that Σ01 (d) ⊂ π0 (W1n (d)) ⊂ Σ1 (d). Since Σ01 (d) is an irreducible component of Σ1 (d) and since π0 (W1n (d)) is irreducible by continuity of π0 , we deduce that π0 (W1n (d)) = Σ01 (d). Thus, since U1 (d) is a Z ARISKI open subset of F(d) (Proposition 3.8 (i)), the morphism π0 induces by restriction a dominant morphism of quasi-projective varieties πn0 : W1n (d) ∩ π−1 0 0 (U1 (d)) → Σ1 (d) ∩U1 (d). Notice that all the fibers of π0 over the elements of U1 (d) ∩ Σ1 (d) are finite and non-empty. Indeed, if F ∈ U1 (d) ∩ Σ1 (d) then, by Proposition 3.8 (ii), the set QRad( [ F , d − 1) is finite and non-empty; therefore so is π0 (F ) = {F } × QRad(F , d − 1). Since π0 (W1 (d) ∩ π−1 −1 [ n 0 (U1 (d))) ⊂ U1 (d) ∩ Σ1 (d), it follows that all the non-empty fibers of πn0 are finite and therefore zero-dimensional. The fiber dimension theorem (cf. [15, Theorem 3, page 49]) then implies that dim(W1n (d) ∩ π−1 0 0 (U1 (d))) = dim(Σ1 (d) ∩ U1 (d)); since W1n (d) ∩ π−1 0 0 (U1 (d)) and Σ1 (d) ∩ U1 (d) are non-empty open subsets of the irreducible varieties W1 (d) n and Σ01 (d) respectively, we have dim Σ01 (d) = dim(Σ01 (d) ∩U1 (d)) = dim(W1n (d) ∩ π−1 n 0 (U1 (d))) = dimW1 (d). Now, from (3.1) we deduce that each irreducible component W1i (d) of W1 (d) has dimension dimW1i (d) ≥ dim(F(d) × P2C × P̌2C ) − 4 − (d − 1) = dim F(d) − (d − 1), hence dim Σ1 (d) = dim Σ1 (d) ≥ dim Σ01 (d) = dimW1n (d) ≥ dim F(d) − (d − 1). Assertion 1. (resp. 2.) of the following proposition gives a necessary (resp. sufficient) condition for a foliation of F(d) to degenerate onto the foliation F2d . Proposition 3.11. — Let F be an element of F(d) such that F2d 6∈ O (F ). The following assertions hold: 1. If F degenerates onto F2d , then deg IFtr ≥ d − 1. 2. If F admits an inflection point of maximal order d − 1, i.e. if Flex(F , d − 1) 6= 0, / then F degenerates onto F2d . Proof. — 1. If F degenerates onto F2d , then deg IFtr ≥ deg IFtr d . An immediate computation shows that 2 IFtr d = yd−1 so that deg IFtr d = d − 1, hence the announced inequality holds. 2 2 2. Assume that F possesses such a point. We choose an affine coordinate system (x, y) such that p = (0, 0) is an inflection point of order d − 1 of F and x = 0 is the tangent line to the leaf of F passing through p. 16 SAMIR BEDROUNI & DAVID MAR Í N Let ω be a 1-form defining F in these coordinates. Since TPp F = {x = 0}, ω is of type d ω = Cd (x, y)(xdy − ydx) + αdx + ∑ Ai (x, y)dx + Bi (x, y)dy , where Ai , Bi ∈ C[x, y]i , Cd ∈ C[x, y]d , α ∈ C∗ .  i=1 We have d d ω ∧ dx = ∑ Bi (0, y)dy ∧ dx = ∑ Bi (0, 1)yi dy ∧ dx. x=0 i=1 i=1 Therefore the hypothesis that (0, 0) is an inflection point of order d − 1 of F translates into Bi (0, 1) = 0 for i ∈ {1, 2, . . . , d − 1} and Bd (0, 1) 6= 0. Then we can write Bd (x, y) = xBbd−1 (x, y) + βyd , Bi (x, y) = xBei−1 (x, y) for i ∈ {1, 2, . . . , d − 1}, where Bei−1 ∈ C[x, y]i−1 , Bbd−1 ∈ C[x, y]d−1 , β ∈ C∗ . Thus ω is of type  d d−1 ω = αdx + xBbd−1 (x, y) + βyd dy +Cd (x, y) xdy − ydx + ∑ Ai (x, y)dx + x ∑ Bei−1 (x, y)dy,  i=1 i=1 where Ai ∈ C[x, y]i , Bei−1 ∈ C[x, y]i−1 , Bbd−1 ∈ C[x, y]d−1 , α, β ∈ C∗ . Let us consider the family of automorphisms ϕ = ϕε = (εd+1 x, εy). We have d d−1 1 ∗  d b d d  d+1 d i d ε d+1 ϕ ω = αdx + ε x Bd−1 (ε x, y) + βy dy + ε Cd (ε x, y) (xdy − ydx) + ∑ ε Ai (ε x, y)dx + x ∑ εi Bei−1 (εd x, y)dy i=1 i=1 which tends to αdx + βyd dy as ε tends to 0. Clearly αdx + βyd dy defines a foliation conjugated to F2d ; as a d result F degenerates onto F2 . Example 3.12. — Let us consider the homogeneous foliation H2d defined in the affine chart z = 1 by the 1-form ω d2 = xd dx − yd dy. We know from [4, Proposition 4.1] that H2d has no non-degenerate singularity with BAUM -B OTT index 4 and that Flex(H2d , d − 1) = {xy = 0} \ {[0 : 0 : 1]} 6= 0. / d d d Thus H2 degenerates onto F2 (Proposition 3.11) and it does not degenerate onto F1 (Proposition 3.4). Consequently, according to Remark 3.3, we have O (H2d ) = O (H2d ) ∪ O (F2d ). Example 3.13 (J OUANOLOU’s foliation). — Let us consider the foliation FJd of degree d ≥ 2 on P2C de- fined, in the affine chart z = 1, by ωdJ = (xd y − 1)dx + (yd − xd+1 )dy. This example is due to J OUANOLOU and is historically the first explicit example of foliation without invariant algebraic curve ([12]). The point p = (0, 0) is an inflection point of maximal order d − 1 of FJd because TPp FJd = {x = 0} and ωdJ ∧ dx = yd dy ∧ dx. As a result FJd degenerates onto F2d (Proposition 3.11). x=0 However, we know from [13, Section 3] that every singularity s of FJd is non-degenerate with BAUM -B OTT index (d + 2)2 BB(FJd , s) = 2 6= 4, d +d +1 GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 17 so that FJd does not degenerate onto F1d (Proposition 3.4). The converse of assertion 2. of Proposition 3.11 is false as the following example shows. Example 3.14. — Let F be the foliation of degree d ≥ 2 on P2C defined in the affine chart z = 1 by ω = dx + P(y)dy, where P ∈ C[y], deg P = d. It is easy to check that Sing(F ) = [1 : 0 : 0] and IF = P (y). If the derivative P0 has a single root, i.e if P is tr 0  of the form P(y) = a(y − α)d + b, where α, a, b ∈ C, a 6= 0, then F is conjugated to F2d ; indeed, we have 1 ∗ ϕ ω = dx + yd dy, where ϕ = (ax − by, y + α). a We assume that the derivative P0 has at least two distinct roots; this implies that d ≥ 3. A straightforward computation shows that F has no inflection point of maximal order d − 1, i.e. Flex(F , d − 1) = 0. / However, F degenerates onto F d . Indeed, by writing P(y) = a + a y + · · · + a yd , a ∈ C, a 6= 0, and by putting   2 0 1 d i d ad 1 ψ= d+1 x, y , we obtain that ε ε εd+1 ∗ lim ψ ω = dx + yd dy. ε→0 ad Question 2. — Let F be a foliation of degree d ≥ 3 on P2C . Is it true that if F degenerates onto F2d then – either F possesses an inflection point of maximal order d − 1, – or F is conjugated to Example 3.14, i.e. up to linear conjugation F is given by a 1-form of type dx + P(y)dy with P ∈ C[y], deg P = d? Proposition 3.15. — Let d be an integer greater than or equal to 2. Let us denote by U2 (d) the set of foliations F ∈ F(d) whose inflection divisor IF is transverse (i.e. IF = IFtr ) and reduced. Then (i) U2 (d) contains the J OUANOLOU’s foliation FJd and it is a (non-empty) Z ARISKI open subset of F(d); (ii) for any d ≥ 3, every foliation F ∈ U2 (d) has a finite number (possibly zero) of transverse inflection d Flex(F , k − 1) is finite. S points of order greater than or equal to 2; in other words, the set k=3 To establish this proposition, let us first prove the following lemma. Lemma 3.16. — Let F be a foliation of degree d ≥ 2 on P2C , p a regular point of F and X a polynomial vec- tor field defining F in an affine chart (x, y) containing p. Then, for any k ∈ {2, 3, . . . , d}, Tang(F , TPp F , p) ≥ k X(x) X2 (x) · · · Xk (x)   if and only if the matrix has rank 1. X(y) X2 (y) · · · Xk (y) p n ∂ Remark 3.17. — If X = ∑ Xi (z1 , . . . , zn )is a holomorphic vector field on Cn and if t 7→ α(t) is an integral i=1 ∂zi curve of X, then we have the following formula which can be easily proved by induction on j : (3.2) d j α(t) = (X j (z ), . . . , X j (z )) ◦ α(t). 1 n dt j Proof. — Let t 7→ α(t) be the integral curve of X passing through p at t = 0. The point p being regular for F , we have Tp F 3 α0 (0) = X(p) 6= 0. Up tolinear conjugation,  we can assume that i i p = (0, 0) and TPp F = {y = 0}. We can then write α(t) = ∑ xi ti! , ∑ yi ti! with y1 = 0 and x1 6= 0. i≥1 i≥1 18 SAMIR BEDROUNI & DAVID MAR Í N i Thus, Tang(F , TPp F , p) = ν(g(t), 0), where g(t) = ∑ yi ti! . As a result, Tang(F , TPp F , p) ≥ k if and only i≥2   x1 x2 · · · xk if y2 = y3 = · · · = yk = 0, or equivalently if and only if the matrix has rank 1. 0 y2 · · · yk Now, by using formula (3.2), we see that X(x) X2 (x) · · · Xk (x)     x1 x2 · · · xk = , 0 y2 · · · yk X(y) X2 (y) · · · Xk (y) (x,y)=(0,0) hence the lemma follows. Proof of Proposition 3.15. — (i) For F ∈ F(d), to say that IF is transverse and reduced means that F has no invariant line and that IF has no multiple component, which shows that U2 (d) is a Z ARISKI open subset of F(d). As we have already mentioned in Example 3.13, the J OUANOLOU’s foliation FJd has no invariant algebraic curve [12]; in particular, it has no invariant line and consequently IF d = IFtr d . To establish the first announced J J assertion, it remains to prove that IF d is reduced. In homogeneous coordinates, the foliation FJd is defined J by the vector field yd ∂x ∂ + zd ∂y ∂ + xd ∂z ∂ ; an immediate computation, using formula (1.1), shows that IF d has J equation F(x, y, z) = 0, where F(x, y, z) = x2d+1 zd−1 + y2d+1 xd−1 + z2d+1 yd−1 − 3xd yd zd . We must show that F has no multiple factor in C[x, y, z]. Since F ∈ Z[x, y, z], it suffices to show that F has no multiple factor in F2 [x, y, z]. Indeed, if F had a multiple factor in C[x, y, z], then one of the resultants Resx (F, ∂F ∂F ∂F ∂x ) ∈ Z[y, z] or Resy (F, ∂y ) ∈ Z[x, z] or Resz (F, ∂z ) ∈ Z[x, y] would be identically zero and therefore so would be its reduction modulo 2; so that F would also have a multiple factor in F2 [x, y, z]. We have to show that gcd(F, ∂F ∂F ∂F ∂x , ∂y , ∂z ) = 1 in F2 [x, y, z], or equivalently that gcd(F, ∂F ∂x ) = 1 in F2 (y, z)[x], gcd(F, ∂F ∂y ) = 1 in F2 (x, z)[y], gcd(F, ∂F ∂z ) = 1 in F2 (x, y)[z]. The coordinates x, y, z playing a symmetric role, it suffices again to show that gcd(F, ∂F ∂x ) = 1 in F2 (y, z)[x]. In F2 [x, y, z] we have   F = x2d+1 zd−1 + y2d+1 xd−1 + z2d+1 yd−1 + xd yd zd and ∂F ∂x = xd−2 xd+2 zd−1 + dxyd zd + (d + 1)y2d+1 . Then x = 0 is not a root of F ∈ F2 (y, z)[x] and consequently y2d+1 F2 (y, z)[x] 3 gcd(F, ∂F ∂x ) = gcd(F, ϕ), where ϕ = xd+2 + dxzyd + (d + 1) . zd−1 Moreover, a straightforward computation shows that yd+1 yd+1 y2d+2      x3 F = xd+2 zd−1 − (d − 1)xyd zd − dy2d+1 ϕ + yd−1 z2d+1 x + d x2 + (d 2 − d − 1) d x + d(d + 1) 2d , z z z GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 19 so that yd+1  yd+1    F2 (y, z)[x] 3 gcd(F, ϕ) = gcd x + d x − d , ϕ , because d 2 − d ≡ d(d + 1) ≡ 0 mod 2 z z d+1 y2d+1   y d+2 d = gcd x − d , x + dxzy + (d + 1) d−1 z z d+1 2d+1   y y = gcd x − d , xd+2 − d−1 z z = 1,  d+1 d+2 y y2d+1 because 6 = in the field F2 (y, z). As a result F2 (y, z)[x] 3 gcd(F, ∂F ∂x ) = 1. zd zd−1 (ii) Let F be a foliation of degree d ≥ 3 on P2C with reduced and transverse inflection divisor IF , i.e. d F ∈ U2 (d). We want to show that the set Γ(F ) := Flex(F , k − 1) is finite. By definition of Γ(F ) we have S k=3 n o (3.3) Γ(F ) ⊂ p ∈ P2C | p 6∈ Sing(F ), Tang(F , TPp F , p) ≥ 3 . Let X be a vector field defining F in an affine chart C2 = {(x, y)} ⊂ P2C . Lemma 3.16 and inclusion (3.3) imply that Γ(F ) ∩ C2 is contained in the set of points p ∈ C2 such that X(x) X2 (x) X(x) X3 (x)     X(x) 0 (p) 6= , IX (p) := 2 (p) = 0, X(IX )(p) = (p) = 0. X(y) 0 X(y) X (y) X(y) X3 (y) Now, the affine chart C2 = {(x, y)} ⊂ P2C being arbitrary, Γ(F ) is finite if and only if Γ(F ) ∩ C2 is finite. It suffices therefore to show that the algebraic curves IF ∩ C2 = {IX (x, y) = 0} and C := {X(IX )(x, y) = 0} intersect at a finite number of points, i.e. that they have no common component. Let us argue by contradiction and assume that there exist K, L, L0 ∈ C[x, y], with deg K > 0, such that IX = KL and X(IX ) = KL0 . Then KL0 = X(KL) = X(K)L + KX(L) and therefore X(K)L = K(L0 − X(L)). Moreover, the hypothesis that IF is reduced implies that gcd(K, L) = 1. It follows that there is L00 ∈ C[x, y] such that X(K) = KL00 , which means that the algebraic curve C 0 := {K(x, y) = 0}, contained in IF , is invariant by F , contradicting the hypothesis that IF is transverse. Theorem 3.18. — Let d be an integer greater than or equal to 2. Let us denote by Σ2 (d) the subset of F(d) defined by n o Σ2 (d) := F ∈ F(d) | Flex(F , d − 1) 6= 0/ . Then (a) B(F22 ) = F(2) \ FC(2) = Σ2 (2) and, for any d ≥ 3, we have 0/ 6= Σ2 (d) B(F2d ); (b) Σ2 (d) is a constructible subset of F(d); (c) for any d ≥ 3, we have dim Σ2 (d) ≥ dim F(d) − (d − 3). In particular, the set Σ2 (3), and therefore B(F23 ), contains a non-empty Z ARISKI open subset of F(3). Proof. — (a) As we have already said in Introduction, the first equality B(F22 ) = F(2) \ FC(2) follows from [9, Theorem 3]. The second equality F(2) \ FC(2) = Σ2 (2) is a consequence of the following obvious remark: if F ∈ F(2) \ FC(2) then every transverse inflection point of F is of order 1. 20 SAMIR BEDROUNI & DAVID MAR Í N The set Σ2 (d) contains the foliations H2d and FJd (Examples 3.12 and 3.13) and is therefore non-empty. According to assertion 2. of Proposition 3.11, we have Σ2 (d) ⊂ B(F2d ); this inclusion is strict for any d ≥ 3 as Example 3.14 shows. (b) Let π : F(d) × P2C → F(d) be the projection onto the first factor; notice that Σ2 (d) = π(W2 (d)), where [ W2 (d) : = {F } × Flex(F , d − 1) F ∈Σ2 (d) n o = (F , p) ∈ F(d) × P2C | p 6∈ Sing(F ), Tang(F , TPp F , p) = d . By Lemma 3.16, W2 (d) can be rewritten as X(x) X j (x)     X(x) 0 n o (3.4) W2 (d) = (F , p) ∈ F(d) × P2C | (p) 6= , (p) = 0, j = 2, . . . , d , X(y) 0 X(y) X j (y) where X denotes a polynomial vector field defining F in an affine chart (x, y) containing p. It follows that W2 (d) is a quasi-projective subvariety of F(d) × P2C . Therefore, by C HEVALLEY’s theorem [11, Ex- ercise II.3.19], the set Σ2 (d) = π(W2 (d)) is constructible. (c) From the above discussion and Proposition 3.15 (i), we have FJd ∈ U2 (d) ∩ Σ2 (d) 6= 0/ (U2 (d) being the set of foliations of F(d) with reduced and transverse inflection divisor). Therefore there exists an irreducible component Σ02 (d) of Σ2 (d) such that U2 (d) ∩ Σ02 (d) 6= 0. / We denote by π0 : W2 (d) → F(d) the restriction of π n W2i (d) be the decomposition of W2 (d) into its irreducible components. Then, by S to W2 (d). Let W2 (d) = i=1 arguing as in the proof of Theorem 3.10, we see that there is k ∈ {1, . . . , n} such that π0 (W2k (d)) = Σ02 (d). Since U2 (d) is a Z ARISKI open subset of F(d) (Proposition 3.15 (i)), the morphism π0 therefore induces by restriction a dominant morphism of quasi-projective varieties πk0 : W2k (d) ∩ π−1 0 0 (U2 (d)) → Σ2 (d) ∩U2 (d). Notice that, for any F ∈ U2 (d) ∩ Σ2 (d), the fiber π0 (F ) is finite and non-empty, because π−1 −1 0 (F ) = {F } × Flex(F , d − 1) and Flex(F , d − 1) is finite and non-empty by assertion (ii) of Proposition 3.15. Since π0 (W2k (d) ∩ π−1 k 0 (U2 (d))) ⊂ U2 (d) ∩ Σ2 (d), we deduce that all the non-empty fibers of π0 are finite and therefore zero-dimensional. The fiber dimension theorem (cf. [15, Theorem 3, page 49]) then ensures that dim(W2k (d) ∩ π−1 0 k −1 0 0 (U2 (d))) = dim(Σ2 (d) ∩ U2 (d)); since W2 (d) ∩ π0 (U2 (d)) and Σ2 (d) ∩ U2 (d) are non-empty open subsets of the irreducible varieties W2k (d) and Σ02 (d) respectively, we have dim Σ02 (d) = dim(Σ02 (d) ∩U2 (d)) = dim(W2k (d) ∩ π−1 k 0 (U2 (d))) = dimW2 (d). Now, it follows from (3.4) that each irreducible component W2i (d) of W2 (d) has dimension dimW2i (d) ≥ dim(F(d) × P2C ) − (d − 1) = dim F(d) − (d − 3), hence dim Σ2 (d) = dim Σ2 (d) ≥ dim Σ02 (d) = dimW2k (d) ≥ dim F(d) − (d − 3). The subset Σ2 (d) ⊂ F(d) being constructible, it contains a dense open subset of its closure Σ2 (d). In degree d = 3 we have dim Σ2 (3) ≥ dim F(3) and therefore dim Σ2 (3) = dim F(3), so that Σ2 (3) = F(3) because F(3) is irreducible. It follows that Σ2 (3) contains a dense open subset of F(3). This ends the proof of the theorem. GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 21 Remark 3.19. — The set F(d) contains elements which degenerate onto both F1d and F2d , e.g. the family of foliations G d (γ), γ ∈ C. Indeed, on the one hand, we have seen (Example 3.6) that G d (γ) degenerates onto F1d . On the other hand, by putting ϕ = ( xε , yε ) we obtain that lim εd+1 ϕ∗ ηd (γ) = ω d2 , which shows that G d (γ) ε→0 degenerates onto the homogeneous foliation H2d (Example 3.12) and therefore, by transitivity, onto F2d . Example 3.20. — Let us consider the homogeneous foliation H1,2 d defined in the affine chart z = 1 by the 1-form ω d1,2 = (xd + yd )dx + xd dy. This foliation degenerates onto both F1d and F2d . Indeed, on the one hand, H1,2 d is given in the affine chart y = 1 by d θ 1,2 = xdz − zdx + xd dz + xd (xdz − zdx); d . Thus, by Proposition 3.4, we see that the point [0 : 1 : 0] is a radial singularity of maximal order d − 1 of H1,2 d degenerates onto F d . On the other hand, a straightforward computation shows that H1,2 1 d Flex(H1,2 , d − 1) = {y = 0} \ {[0 : 0 : 1]} 6= 0; / d also degenerates onto F d (Proposition 3.11). consequently, H1,2 2 d ) ⊂ O (H d ) ∪ O (F d ) ∪ O (F d ) (Remark 3.3), we deduce that in fact Since O (H1,2 1,2 1 2 d ) = O (H d ) ∪ O (F d ) ∪ O (F d ). O (H1,2 1,2 1 2 Theorem 3.21. — Let d be an integer greater than or equal to 2. Then (a) 0/ 6= Σ1 (d) ∩ Σ2 (d) ⊂ B(F1d ) ∩ B(F2d ) ⊃ B(H1,2 d ); d ) contains a quasi-projective subvariety of F(d) of dimension equal to dim F(d) − 3d. (b) B(H1,2 d (Example 3.20) and is Proof. — (a) The intersection Σ1 (d) ∩ Σ2 (d) contains the homogeneous foliation H1,2 therefore non-empty. The inclusion Σ1 (d) ∩ Σ2 (d) ⊂ B(F1d ) ∩ B(F2d ) follows from Theorems 3.10 and 3.18. d ) ⊂ B(F d )∩ B(F d ). Let F ∈ B(H d ), i.e. F ∈ F(d) such that H d ∈ O (F ). Let us show the inclusion B(H1,2 1 2 1,2 1,2 d degenerates onto F d , i = 1, 2, it follows that F d ∈ O (H d ) ⊂ O (F ), hence F ∈ B(F d ) ∩ B(F d ). Since H1,2 i i 1,2 1 2 (b) Let us denote by Σ(H1,2 d ) the subset of F(d) defined as follows: an element F of F(d) belongs to d ) if and only if Σ(H1,2 (1) F admits an invariant line `; (2) there is a system of homogeneous coordinates [x : y : z] ∈ P2C in which ` = {z = 0} and F is defined in the affine chart z = 1 by a 1-form ω of type d−1 d−1   ω = ∑ ωi + λω d1,2 = ∑ ωi + λ (xd + yd )dx + xd dy , i=0 i=0 where λ ∈ C∗ and the ωi ’s are homogeneous 1-forms of degree i. d Notice that Σ(H1,2 ) ⊂ B(H1,2 d ). Indeed, by putting ϕ = ( x , y ) and by writing ω = P (x, y)dx + Q (x, y)dy, ε ε i i i where Pi , Qi ∈ C[x, y]i , we obtain d−1 εd+1 ϕ∗ ω = ∑ (εd−i Pi (x, y)dx + εd−i Qi (x, y)dy) + λωd1,2 i=0 22 SAMIR BEDROUNI & DAVID MAR Í N which tends to λω d1,2 as ε tends to 0. It follows that H1,2 d ∈ O (F ) for any F ∈ Σ(H d ), hence the inclusion 1,2 d ) ⊂ B(H d ) holds. Σ(H1,2 1,2 Moreover, every foliation F ∈ F(d) is given in the affine chart z = 1 by a 1-form of type d  ∑ Ai (x, y)dx + Bi (x, y)dy +Cd (x, y)(xdy − ydx), i=0 d d  where Ai , Bi ∈ C[x, y]i ,Cd ∈ C[x, y]d with gcd yCd − ∑ Ai , xCd + ∑ Bi = 1. Condition (2) is then equivalent i=0 i=0 to taking Cd ≡ 0, Ad (x, y) = λ(xd + yd ), Bd (x, y) = λ xd . Since the set of foliations of F(d) admitting an invari- ant line is a Z ARISKI closed subset of F(d), we deduce that Σ(H1,2 d ) is a quasi-projective subvariety of F(d). Since ω and µω define the same foliation if µ 6= 0, and the choice of a line ` ⊂ P2C is equivalent to the choice of a point in P̌2C , conditions (1) and (2) imply that d−1 d dim Σ(H1,2 ) = 2 + 2 ∑ (i + 1) = d 2 + d + 2 = dim F(d) − 3d. i=0 4. A family of foliations of F(d) with orbits of dimension less than or equal to 7 In this section we will establish some properties of the family (F0d (λ))λ∈C∗ of foliations of degree d on P2C defined in the affine chart z = 1 by ωd0 (λ) = xdy − λydx + yd dy. In homogeneous coordinates, F0d (λ) is given by     Ωd0 (λ) = −λyzd dx + z xzd−1 + yd dy + y (λ − 1)xzd−1 − yd dz. Thus, the singular locus of F0d (λ) consists of the two points s1 = [0 : 0 : 1] and s2 = [1 : 0 : 0]. The singularity s1 is non-degenerate with BAUM-B OTT index BB(F0d (λ), s1 ) = 2 + λ + λ1 and the singular point s2 has maximal algebraic multiplicity d. We see that for λ = 1 the 1-form Ωd0 (1) writes in the affine chart x = 1 as zd dy + yd (zdy − ydz); we deduce that F0d (1) is conjugated to the foliation F1d and is therefore convex. In the sequel we assume that λ ∈ C \ {0, 1}. A direct computation, using formula (1.1), leads to IFinvd (λ) = yz2d−1 IFtr d (λ) = (λ − 1)x − (d − 1)λ + 1 yd ;  (4.1) and 0 0 it follows that, for any λ ∈ C \ {0, 1}, F0d (λ) is not convex. A straightforward computation shows that the algebraic curve (1 − λd)x + yd = 0 is invariant by F0d (λ). ωd0 (λ) What is more, the rational 1-form ηd0 (λ) = is closed. For λ = d1 we note that ηd0 ( d1 ) = y ((1 − λd) x + yd ) ωd0 (λ) x has as first integral d − ln y; this allows to see that Iso(F0d ( d1 )) is the group {(αd x, αy) | α ∈ C∗ }. yd+1 dy When λ ∈ C \ {0, 1, d1 } a straightforward computation shows that ηd0 (λ) integrates into   λ ln (1 − λd) x + yd − ln y, GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 23 which allows to verify that the isotropy group is here again Iso(F0d (λ)) = {(αd x, αy) | α ∈ C∗ }. It follows in particular that, for any λ ∈ C \ {0, 1}, O (F0d (λ)) has dimension 7. Notice that two foliations F0d (λ) and F0d (λ0 ) are conjugated if and only if λ = λ0 . Proposition 4.1. — Let λ be a nonzero complex number. Let F be an element of F(d) such that F0d (λ) 6∈ O (F ). 1. If F degenerates onto F0d (λ), then F admits a non-degenerate singular point m satisfying BB(F , m) = 2 + λ + λ1 . 2. If F possesses a non-degenerate singular point m such that 1 BB(F , m) = 2 + λ + and κ(F , m) = d, λ then F degenerates onto F0d (λ). Proof. — It suffices to argue as in the proof of Proposition 3.4, replacing the foliation F1d by F0d (λ) and the equality BB(F , m) = 4 by BB(F , m) = 2 + λ + λ1 . Proposition 4.2. — The orbit O F0d (λ) is closed in F(d) in the following two cases:  1 (i) d ≥ 3 and λ = − ; d −1 (ii) d ∈ {3, 4, 5} and λ ∈ C∗ . The proof of this proposition uses the following lemma. Lemma 4.3. — Let λ be a nonzero complex number. Then, the orbit O F0d (λ) is closed in F(d) if and only  if F0d (λ) does not degenerate onto F2d . Proof. — The direct implication is obvious. Let us prove the converse. From the above discussion, F0d (1) is conjugated to the convex foliation F1d ; therefore its orbit O F0d (1)) is closed in F(d). For any λ ∈ C\{0, 1}, the unique non-degenerate singular point s1 = [0 : 0 : 1] of F0d (λ) has BAUM-B OTT index BB(F0d (λ), s1 ) = 2 + λ + λ1 6= 4; this implies, according to assertion 1. of Proposition 3.4, that F0d (λ) does not degenerate onto F1d . Moreover, for any λ ∈ C\{0, 1}, O (F0d (λ)) has dimension 7. The converse implication then follows immediately from Corollary B. 1 Proof of Proposition 4.2. — (i) Let us put λ0 = − d−1 ; according to (4.1) we have IFtr d (λ ) = (λ0 − 1)x, hence 0 0 deg IFtr d (λ ) = 1 < d − 1 for any d ≥ 3. According to the first assertion of Proposition 3.11, it follows that, 0 0 for any d ≥ 3, the foliation F0d (λ0 ) does not degenerate onto F2d , so that its orbit O F0d (λ0 ) is closed  in F(d) (Lemma 4.3). (ii) Let [x : y : z] be homogeneous coordinates in P2C . For n ∈ N, let us denote by Λ1n the C-vector space of 1-forms in the variables x, y, z, whose coefficients are homogeneous polynomials of degree n. Let us put α = ydz − zdy, β = zdx − xdz and γ = xdy − ydx. We have the identification F(d) = [Ω] ∈ P(Λ1d+1 ) | Ω = pdx + qdy + rdz, p, q, r ∈ C[x, y, z]d+1 , xp + yq + zr = 0, gcd(p, q, r) = 1  = [Ω] ∈ P(Λ1d+1 ) | Ω = Aα + Bβ +Cγ, A, B ∈ C[x, y, z]d , C ∈ C[x, y]d , gcd yA − xB, zB − yC, xC − zA = 1 .   24 SAMIR BEDROUNI & DAVID MAR Í N By writting     A = ξ1 xd + ξ3 xd−1 y + · · · + ξ2d+1 yd + ξ2d+3 xd−1 + ξ2d+5 xd−2 y + · · · + ξ4d+1 yd−1 z + ξ4d+3 xd−2 + ξ4d+5 xd−3 y + · · · + ξ6d−1 yd−2 z2 + · · · + ξd 2 +3d+1 zd ,     B = ξ2 xd + ξ4 xd−1 y + · · · + ξ2d+2 yd + ξ2d+4 xd−1 + ξ2d+6 xd−2 y + · · · + ξ4d+2 yd−1 z + ξ4d+4 xd−2 + ξ4d+6 xd−3 y + · · · + ξ6d yd−2 z2 + · · · + ξd 2 +3d+2 zd , C = ξd 2 +3d+3 xd + ξd 2 +3d+4 xd−1 y + ξd 2 +3d+5 xd−2 y2 + · · · + ξd 2 +4d+2 xyd−1 + ξd 2 +4d+3 yd , 2 we can identify the class [Ω] of Ω = Aα + Bβ +Cγ to the element [ξ1 : ξ2 : · · · : ξd 2 +4d+3 ] ∈ PCd +4d+2 . Thus, we can identify F(d) with the Z ARISKI open set: A = ξ1 xd + ξ3 xd−1 y + · · · + ξ2d+1 yd + ξ2d+3 xd−1 + ξ2d+5 xd−2 y + · · · + ξ4d+1 yd−1 z + · · · + ξd 2 +3d+1 zd          B = ξ2 xd + ξ4 xd−1 y + · · · + ξ2d+2 yd + ξ2d+4 xd−1 + ξ2d+6 xd−2 y + · · · + ξ4d+2 yd−1 z + · · · + ξd 2 +3d+2 zd       2  [ξ1 : ξ2 : · · · : ξd 2 +4d+3 ] ∈ PCd +4d+2 .    C = ξd 2 +3d+3 xd + ξd 2 +3d+4 xd−1 y + ξd 2 +3d+5 xd−2 y2 + · · · + ξd 2 +4d+2 xyd−1 + ξd 2 +4d+3 yd           gcd yA − xB, zB − yC, xC − zA = 1 Then, via this identification, we have F2d = Ωd2 = xd β + yd γ] = [0 : 1 : 0 : 0 : · · · : 0 : 0 : 1     and F0d (λ) = Ωd0 (λ) = (yd + xzd−1 )α + λ yzd−1 β = 0| : 0 {z       : · · · : 0} : 1 : |0 : 0 {z : · · · : 0} . : · · · : 0} : 1 : 0 : 0 : λ : 0| : 0 {z 2d d 2 +d−5 d+3 In addition, the orbit of a foliation F = [Ω] ∈ F(d) is n o ∗ 2 O (F ) = [ϕ Ω] ϕ = [a1 x + a2 y + a3 z : a4 x + a5 y + a6 z : a7 x + a8 y + a9 z] ∈ Aut(PC ) . 2 Let [x1 : x2 : · · · : xd 2 +4d+3 ] be a system of homogeneous coordinates in PCd +4d+2 . For d = 3, let us consider the following homogeneous polynomial in x1 , x2 , . . . , x24 of degree 5:   P3 = −90x2 x1 (294x1 − 269x4 ) + 10x2 (29x3 + 4x6 ) + 86x42 x22 x24 − 1125x22 (21x1 − 23x4 ) x23 x24   + 45x2 2x3 (294x1 + 13x4 ) − x6 (552x1 − 271x4 ) + 1125x2 x5 x21 x24 + 28125x2 x10 x21 x23 x24   2 2 + 25 108 (x9 − 2x12 ) (3x1 − 4x4 ) + 9x10 (112x3 − 93x6 ) + 675x2 x11 x21 x24 − 6000x2 x10 x22 x24   3 3 − 5625x5 x11 x21 + 20 (2x1 − x4 ) (41x9 − 7x12 ) + 30x10 (2x3 − 3x6 ) + 50x2 x11 x22 − 50625x23 x24 2   − 5 2x9 (207x1 − 116x4 ) − x12 (153x1 − 314x4 ) + 5x10 (356x3 − 359x6 ) + 1350x2 x11 x21 x22 x23     2 + 1875 x11 (2x3 − x6 ) + x5 (2x9 − x12 ) x21 x22 − 375x2 2x1 (3x1 − 7x4 ) − x2 (3x3 − 2x6 ) + 8x42 x23 2     2 2 + 50 5x10 (39x1 − 38x4 ) − 3x2 (x9 − 32x12 ) x21 x23 − 50 x10 (14x1 − 37x4 ) − 3x2 (7x9 + x12 ) x22 x23   2 2 + 15 5x11 (21x1 + 22x4 ) − 8x3 (14x9 − 43x12 ) + 6x6 (13x9 − 56x12 ) − 350x5 x10 x21 x23 + R x21   2 − 5 20x11 (24x1 − 7x4 ) + 4x9 (97x3 − 43x6 ) + x12 (94x3 − 211x6 ) − 600x5 x10 x21 x22 + S x21 x22   2 2 − 75 2x10 (78x1 − 29x4 ) − 15x2 (2x9 − 19x12 ) x21 x22 x24 + 125x2 x10 x22 x23 + T x22 +Ux21 x23 +V x22 x23 , GEOMETRY OF CERTAIN FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 25 where R = 5568x6 x5 (3x1 − 4x4 ) − 18x3 x5 (1612x1 − 1941x4 ) + 6x32 (1952x3 − 4389x6 ) + 3x62 (7057x3 − 2136x6 ) − 11250x2 x52 + 2700x7 (3x1 − 4x4 )2 + 54x8 (3x1 − 4x4 ) (106x3 − 89x6 ) , S = 27000x2 x7 (3x1 − 4x4 ) − 24x32 (658x1 − 249x4 ) + 1512x4 x8 (11x1 − 4x4 ) + 252x12 (83x5 − 36x8 ) − 90x2 x3 (329x5 − 318x8 ) − 2x4 x5 (17073x1 − 6047x4 ) + 3x1 x6 (8712x3 − 3599x6 ) − x4 x6 (11658x3 − 6041x6 ) + 90x2 x6 (226x5 − 267x8 ) , T = 20x1 x3 (294x1 − 253x4 ) − 40x1 x6 (159x1 − 152x4 ) + 1900x2 x3 (x3 − x6 ) + 20x42 (68x3 − 95x6 ) − 25x2 x6 (40x3 − 33x6 ) + 60x1 x2 (361x5 − 252x8 ) − 10x2 x4 (983x5 − 756x8 ) + 67500x22 x7 , U = 90x1 x3 (98x1 − 117x4 ) − 30x1 x6 (171x1 − 284x4 ) − 150x2 x6 (68x3 − 35x6 ) − 30x2 x4 (167x5 + 396x8 ) + 7050x2 x32 + 20x42 (73x3 − 157x6 ) + 270x1 x2 (41x5 + 33x8 ) , V = 5x2 x4 (1604x3 − 611x6 ) − 30x12 (294x1 − 563x4 ) − 30x42 (355x1 − 86x4 ) − 30x1 x2 (463x3 − 242x6 ) − 75x22 (109x5 − 198x8 ) . A computation carried out with Maple shows that evaluatingP3 at an arbitrary element [ξ1 : ξ2 : · · · : ξ24 ] of O F03 (λ) , we find P3 [ξ1 : ξ2 : · · · : ξ24 ] = 0, i.e. O F03 (λ) is contained in the zero locus of P3   Zeros(P3 ) := [x1 : x2 : · · · : x24 ] ∈ P23   C | P3 [x1 : x2 : · · · : x24 ] = 0 , which is a Z ARISKI closed subset of P23 3 ∗  C . Therefore we have O F0 (λ) ⊂ Zeros(P3 ) for any λ ∈ C . More- over, we have P3 (0, 1, 0, 0, · · · , 0, 0, 1) = −50625 6= 0, hence F23 6∈ Zeros(P3 ). It follows that, for any λ ∈ C∗ , we have F23 6∈ O F03 (λ) , so that F03 (λ) does not  3 3  degenerate onto F2 . Consequently, according to Lemma 4.3, the orbit O F0 (λ) is closed in F(3). 4 5  To show that the orbit O F0 (λ) , resp. O F0 (λ) , is closed in F(4), resp. F(5), it suffices to argue as in degree d = 3, replacing the polynomial P3 by the following polynomial P4 , resp. P5 :   P4 = 3x3 (129x3 − 212x6 ) + 3x4 (178x5 + 15x8 ) + 12x1 (22x5 − 3x8 ) + 5184x2 x7 − 20x62 x31 + 1728x15 x31 2 2 − 432 (2x13 − x16 ) x31 x32 + 48 (42x11 − 31x14 ) x31 x33 − 18 (24x11 − 19x14 ) x32 − 162x2 (4x1 − 15x4 ) x34   − 18 2x1 (27x3 − 20x6 ) − x4 (15x3 − x6 ) + x2 (170x5 − 69x8 ) x32 + 4212x12 x31 x34 − 486x12 x32 x33   + 36 3 (x1 − x4 ) (12x1 − x4 ) + 22x2 (3x3 − 2x6 ) x33 − 10368x22 x35 ,  resp. P5 = 50x7 (4906x1 − 4749x4 ) − 27040x10 (5x1 − 6x4 ) − 5x5 (10596x3 − 13469x6 ) + 20x8 (1019x3 − 2028x6 )  2 + 569100x2 x9 x43 + 142275x19 x43 − 11690x17 x43 x44 + 98140x14 x43 x47 − 140x2 (2180x1 − 1691x4 ) x47  + 35 (1564x13 − 1645x16 ) x43 x46 + 8620x8 (2x1 − x4 ) − 50x5 (141x1 − 11x4 ) + 10x3 (513x3 − 1580x6 )    + 70x2 (2779x7 − 2704x10 ) + 9875x62 x44 − 35 (x1 − x4 ) (295x1 + 683x4 ) − x2 (3776x3 − 4427x6 ) x46 2 2 + 70 (323x18 − 253x15 ) x43 x45 + 7 (686x13 − 293x16 ) x44 x45 − 2975x15 x44 − 15946x14 x45 − 1422750x22 x48   + 14x3 (15x1 + 1124x4 ) − 14x6 (10x1 + 1129x4 ) − 595x2 (221x5 − 250x8 ) x45 + 49210x14 x44 x46 . 26 SAMIR BEDROUNI & DAVID MAR Í N For d ≥ 6, we propose: Conjecture 1. — Let d be an integer greater than or equal to 6 and λ a nonzero complex number. A homo- 2 geneous coordinate system [x1 : x2 : · · · : xd 2 +4d+3 ] being fixed in PCd +4d+2 , there exists a homogeneous poly- nomial Qd ∈ C[x1 , x2 , · · · , xd 2 +4d+3 ] of degree 3, not depending on λ, which vanishes on the orbit O F0d (λ) and does not vanish at the point F2d = [0 : 1 : 0 : 0 : · · · : 0 : 0 : 1 . Computations made with Maple by the first author show the validity of this conjecture for d small (d ≤ 30) by taking the polynomial Qd in the following form:   xd 2 +4d+3 ! d−1 4  xd 2 +4d+2  Qd = xd 2 +3d+3 ∑ αi x2d+2i+1 xd +4d+2−i + ∑ βi x2d+2i+4 xd +4d+2−i + (x1 x2 · · · xd+1 )M    2 2 ..  i=1 i=0  .  xd 2 +3d+3 ! d−3 + xd 2 +3d+4 δ0 x2d+4 xd 2 +4d+1 + δ1 x2d+6 xd 2 +4d + ∑ γi x2d+2i+1 xd +4d+1−i 2 , i=1   L1  L2  where M =   is a square matrix of order d + 1 whose lines are of the form:   ..  .  Ld+1   L1 = 0 0 a1,3 x1 + b1,3 x4 a1,4 x3 + b1,4 x6 a1,5 x5 + b1,5 x8 · · · a1, d+1 x2d−3 + b1, d+1 x2d   L2 = b2,1 x2 a2,2 x1 + b2,2 x4 a2,3 x3 + b2,3 x6 a2,4 x5 + b2,4 x8 a2,5 x7 + b2,5 x10 · · · a2, d+1 x2d−1 + b2, d+1 x2d+2 .. .   | 0 {z· · · 0} a2k−1, 2k+1 x2k−1 + b2k−1, 2k+1 x2k+2 a2k−1, 2k+2 x2k+1 + b2k−1, 2k+2 x2k+4 · · · a2k−1, d+1 x2d−2k−1 + b2k−1, d+1 x2d−2k+2 L2k−1 = 0  min(2k, d+1)   | 0 {z· · · 0} b2k, 2k−1 x2k a2k, 2k x2k−1 + b2k, 2k x2k+2 a2k, 2k+1 x2k+1 + b2k, 2k+1 x2k+4 · · · a2k, d+1 x2d−2k+1 + b2k, d+1 x2d−2k+4 , L2k = 0  2k−2 where αi , βi , γi , δi , ai, j , bi, j ∈ C with b2,1 6= 0. 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September 8, 2022 S AMIR B EDROUNI, Faculté de Mathématiques, USTHB, BP 32, El-Alia, 16111 Bab-Ezzouar, Alger, Algérie E-mail : sbedrouni@usthb.dz, sbedrouni@mat.uab.es DAVID M ARÍN, Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès (Barcelona), Spain. Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès (Barcelona), Spain • E-mail : davidmp@mat.uab.es