CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE

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1 Journal of Singularities Volume ), 52-7 received: 7 April 215 in revised form: 9 March 216 DOI: /jsing d CLASSIFICATION OF FOLIATIONS ON CP 2 OF DEGREE WITH DEGENERATE SINGULARITIES CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE Abstract. The aim of this work is to classify foliations on CP 2 of degree with degenerate singular points. For that we construct a stratification of the space of holomorphic foliations by locally closed, irreducible, non-singular algebraic subvarieties which parametrize foliations with a special degenerate singularity. We also prove that there are only two foliations with isolated singularities with automorphism group of dimension two, the maximum possible dimension. Finally we obtain the unstable foliations with only one singular point, that is, a singular point with Milnor number Introduction The aim of this work is to classify holomorphic foliations on CP 2 of degree with certain degenerate singular point using Geometric Invariant Theory GIT). This theory was developed principally by David Hilbert and David Mumford see [6]). We obtain locally closed, irreducible, non-singular algebraic subvarieties which parametrize foliations of degree with a special degenerate singularity. We also get the dimension and explicit generators for each stratum. Similar results for degree 2 are given in [2] and in [], we have some general results for degree d. Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties. More specifically, we have a linear action by a reductive group on a projective variety and we can construct a good quotient if we remove the closed set of unstable points. When the projective variety parametrizes geometric objects, the unstable points are in some sense degenerate objects. For example, the unstable plane algebraic curves with respect to the action by projective transformations are curves with non-ordinary singularities with order greater than 2. In this article the projective variety F is the space of holomorphic foliations on CP 2 of degree and the action is given by change of coordinates. For this action we obtain the closed set of unstable foliations. We will prove that a foliation is unstable if and only if it has a special degenerate singular point see Theorem 8). In this closed set we construct the stratification studied by Kirwan in [12]), Hesselink in [9]) and Kempf in [11]). The strata are locally closed, non-singular, irreducible algebraic subvarieties of F. We characterize the generic foliation on every stratum according to the Milnor number and multiplicity of their singularities. We also obtain the dimension of the strata see Theorem 7). 2 Mathematics Subject Classification. Primary 7F75, 14L24. Key words and phrases. holomorphic foliation, unstable point, degenerate singularity, stratification. The first author was partially supported by CONACyT Grant The second author was partially supported by DAIP-UG Grant 191/21.

2 FOLIATIONS ON CP 2 OF DEGREE 5 As a corollary we describe the irreducible components of the closed set of unstable foliations. We find, up to change of coordinates, the only two foliations with isolated singularities with automorphism group of dimension 2 see Theorem 1). Finally we classify unstable foliations on CP 2 of degree with only one singular point, that is with Milnor number 1 see Theorem 11). This result is important because the classification of foliations on CP 2 with only one singular point is known only for degree 2 see [5] and [2]). In sections 2 and we recall the basic results about Geometric Invariant Theory and foliations that we need in the sequel. We compute in section 4 the unstable foliations of degree using the numerical criterion of one parameter subgroups. The construction of the stratification of the space of foliations and the characterization of the generic foliation on every stratum is included in section 5. The last section is devoted to give some important corollaries of the construction. 2. Geometric Invariant Theory In this section we recall basic facts about Geometric Invariant Theory. All the definitions and results can be found in [14] and [11]. Let V be a projective variety in CP n, and consider a reductive group G acting linearly on V. Definition 1. Let x V CP n, and consider x C n+1 such that x x. Denote by Ox) the orbit of x in the affine cone of V and by Ox) the orbit of x. Then i) x is unstable if Ox). ii) x is semi-stable if / Ox). The set of semi-stable points will be denoted by V ss. iii) x is stable if it is semi-stable, Ox) is closed in V ss and dim Ox) = dim G. The set of stable points will be denoted by V s. The main result in GIT is the following: Theorem 1. see page 74 in [14]) i) There exists a projective variety Y and a morphism φ : V ss Y, which is a good quotient. ii) There exists an open set Y s Y such that φ 1 Y s ) = V s and the morphism φ : V s Y s is a good quotient and an orbit space. It is very often difficult to find the unstable points for a given action, but there exists a very useful criterion due to Hilbert and Mumford. Let us describe it. A 1-parameter subgroup 1-PS) of the group G is an algebraic morphism λ : C G. Since the action on V is linear, this induces a diagonal representation of C : C GLn + 1, C) t λt) : C n+1 C n+1 v λt)v. Therefore there exists a basis {v,..., v n } of C n+1 such that λt)v i = t ri v i, where r i Z. Definition 2. Let x X and let λ : C G be a 1-PS of G. If x x and x = n i= a iv i, then λt) x = n i= tri a i v i. We define the following function The numerical criterion can now be stated. µx, λ) := min{r i : a i }.

3 54 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE Theorem 2. see Theorem 4.9 of [14]) i) x is stable if and only if µx, λ) < for every 1-PS, λ, of G. ii) x is unstable if and only if there exists a 1-PS, λ, of G such that µx, λ) >. Definition. If µx, λ) > we will say that x is λ-unstable. The following is a useful tool for applying the criterion of 1-PS when G = SLn, C). We formulate the result for the case n =. Lemma 1. see [14]) Every 1-parameter subgroup of SL, C) has the form t k1 λt) = g t k2 g 1, t k for some g SL, C) and some integers k 1, k 2, k such that k 1 k 2 k and k 1 + k 2 + k =.. Foliations on CP 2 of degree d This section provides the definitions and results that we need to know about holomorphic foliations on CP 2 for the development of the paper. Definition 4. A holomorphic foliation X of CP 2 of degree d is a non-trivial morphism of vector bundles: X : O1 d) T CP 2, modulo multiplication by a nonzero scalar. The space of foliations of degree d is where d. F d := PH CP 2, T CP 2 d 1)), Take homogeneous coordinates x : y : z) on CP 2. Up to multiplication by a nonzero scalar there are two equivalent ways to describe a foliation of degree d see [8]): 1) By a homogeneous vector field: X = P x, y, z) x + Qx, y, z) y + Rx, y, z) z = P x, y, z) Qx, y, z) Rx, y, z) where P, Q, R C[x, y, z] are homogeneous of degree d. And if we consider the radial foliation E = x x + y y + z z, then X and X + F x, y, z)e represent the same foliation for all F C[x, y, z] homogeneous of degree d 1. 2) By a homogeneous 1-form: Ω = Lx, y, z)dx + Mx, y, z)dy + Nx, y, z)dz, such that L, M, N C[x, y, z] are homogeneous of degree d + 1 and these satisfy the Euler s condition xl + ym + zn =. With this we can see that the space of foliations on CP 2 of degree d is a projective space of dimension d 2 + 4d + 2. We will use the description 1 for the rest of the paper. We now define the notion of singular point for a foliation and two important invariants for this.

4 FOLIATIONS ON CP 2 OF DEGREE 55 Definition 5. A point p = a : b : c) CP 2 is singular for the above foliation X if P a, b, c), Qa, b, c), Ra, b, c)) = ka, kb, kc) for some k C. The set of singular points of X will be denoted by SingX). Definition 6. Let fy, z) gy, z) be a local generator of X in p = 1 : b : c). Then the Milnor number of p is µ p X) := dim C O C 2,p <f,g>, the multiplicity of p is m p X) := min{ord p f), ord p g)}. Proposition 1. see [4]) Let X be a foliation of degree d with isolated singularities then d 2 + d + 1 = From Lemma 1.2 in [7] we can deduce that ) p CP 2 µ p X). {X F d : there exists p CP 2 such that µ p X) 2} is a divisor in F d, therefore we have the following: Theorem. The set {X F d : every singular point for X has Milnor number 1} is open and non-empty in F d. Finally we give the definition of algebraic leaf for a foliation. Definition 7. A plane curve defined by a polynomial F x, y, z) is an algebraic leaf for X or invariant by X if and only if there exists a polynomial Hx, y, z) such that: F x, y, z) F x, y, z) F x, y, z) P x, y, z) + Qx, y, z) + Rx, y, z) = F H. x y z Theorem 4. see Theorem 1.1, p.158 in [1] and [1]) The set is open and non-empty in F d. {X F d : X has no algebraic leaves} Generically a foliation on CP 2 of degree d does not have degenerate singularities and does not have algebraic leaves. So it is important to classify foliations in the complement of these sets. In this article we say something about that for degree. The group P GL, C) of automorphisms of CP 2 is a reductive group that acts linearly on F d by change of coordinates: P GL, C) F d F d g, X) gx = DgX g 1 ). In the computations we will use SL, C) instead of P GL, C), we will get the same results.

5 56 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE 4. Unstable Foliations on CP 2 of degree As we saw before the space of foliations F is a projective space of dimension 2. In this section we apply the numerical criterion of one parameter subgroups to obtain the closed set of unstable foliations of degree. Remember that X F d is unstable with respect to the action by change of coordinates if and only if there exists λ a 1-PS of SL, C) such that µx, λ) > see Theorem 2). For all λ a 1-PS of SL, C) there exists g SL, C) such that Dt) := gλt)g 1 is a diagonal 1-PS, with the form: D : C SL, C), t t k1 t k2 t k, for some intergers k 1, k 2, k such that k 1 k 2 k and k 1 + k 2 + k =. Since µgx, D) = µx, g 1 Dg) = µx, λ) see remark 4.1 of [14]), every unstable foliation is in the orbit of an unstable point with respect to a diagonal 1-PS. Therefore, we will find the unstable foliations with respect to a diagonal one parameter subgroup and then we will take the set of orbits of these points. Let us consider the basis for the vector space H CP 2, T CP 2 2)) given by {M x, M y, x z, x2 y z, xy2 z, y : M C[x, y, z] is a monic monomial of degree }. z This basis diagonalizes the action of SL, C). Let X = P x + Q y + R z be a foliation on CP2 of degree where P x, y, z) = a α,β x α y β z α β Qx, y, z) = b α,β x α y β z α β Rx, y, z) = c α,β x α y β z α β. Then we are looking for the points X F such that there exist k 1, k 2, k Z with k 1 k 2 k and k 1 + k 2 + k = and such that max{ E P, E Q, E R } <, where E P = min{ k 1 α 1) k 2 β k γ : a α,β } E Q = min{ k 1 α k 2 β 1) k γ : b α,β } E R = min{ k 1 α k 2 β k γ 1) : c α,β }. From definition 2, µx, D) = max{ E P, E Q, E R }, where D is the diagonal 1-PS defined above. Since k 1 > and k <, then we can define q i := ki k 1. Therefore q 1 + q 2 + q =, 1 q 2 q and q 2 [ 1 2, 1] Q. We must find the conditions in the rational numbers q i to have non-zero coefficients for the monomials of P, Q and R. It is easy to obtain the following conclusion.

6 FOLIATIONS ON CP 2 OF DEGREE 57 Coefficients a α,β α 1)k 1 + βk 2 + γk < α = 2, β =, γ = 1, q 2, 1] α = 1, β = 2, γ =, q 2 [ 1 2, ) α = 1, β = 1, γ = 1, q 2 [ 1 2, 1] α = 1, β =, γ = 2, q 2 [ 1 2, 1] α =, β =, γ =, q 2 [ 1 2, 1 ) α =, β = 2, γ = 1, q 2 [ 1 2, 1] α =, β = 1, γ = 2, q 2 [ 1 2, 1] α =, β =, γ =, q 2 [ 1 2, 1] Coefficients b α,β αk 1 + β 1)k 2 + γk < α = 2, β =, γ = 1, q 2 1 2, 1] α = 1, β = 1, γ = 1, q 2, 1] α = 1, β =, γ = 2, q 2 1, 1] α =, β =, γ =, q 2 [ 1 2, ) α =, β = 2, γ = 1, q 2 [ 1 2, 1] α =, β = 1, γ = 2, q 2 [ 1 2, 1] α =, β =, γ =, q 2 [ 1 2, 1] Coefficients c α,β αk 1 + βk 2 + γ 1)k < α =, β =, γ =, q 2 [ 1 2, 1 4 ) From this, we see that a,, a 2,1, b,, b 2,1, b 1,2, c,, c 2,1, c 1,2 = and we can have a 1,1, a 1,, a,2, a,1, a,, b,2, b,1, b,. Now we do a partition of [ 1 2, 1] to have the subspaces of unstable foliations with respect to a diagonal 1-PS. [ q 2 1 ] 2, 1 a 2,, b 2,, b 1,1, b 1, = q 2 1 ), 1 a 2,, b 2,, b 1,1 = 4 q 2 [ 14 ), a 2,, b 2,, b 1,1, c, = q 2 = a 2,, a 1,2, b 2,, b 1,1, b,, c, = q 2, 1 ) a 1,2, b 2,, b,, c, = [ 1 q 2, 1 ] a 1,2, a,, b 2,, b,, c, = 2 ] 1 q 2 2, 1 a 1,2, a,, b,, c, =. Consider the seven subspaces with the corresponding coefficients equal to zero. In these sets we have maximal subspaces of H CP 2, T CP 2 2)). That we describe below: V 1 := xy 2 x, xyz x, xz2 x, y x, y2 z x, yz2 x, z x, xz 2 y, y y, y2 z y, yz2 y, z y, y z C V 2 := x 2 z x, xyz x, xz2 x, y2 z x, yz2 x, z x, x2 z y, xyz y, xz2 y, y 2 z y, yz2 y, z y C

7 58 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE Then, V := x 2 z x, xyz x, xz2 x, y x, y2 z x, yz2 x, z x, xyz y, xz2 y, y 2 z y, yz2 y, z, y C {X H CP 2, T CP 2 2)) : there exists D a 1-PS diagonal such that µx, D) > } = V 1 V 2 V. Therefore we can state: Theorem 5. The closed set of unstable foliations on CP 2 of degree is F un = SL, C)PV 1 SL, C)PV 2 SL, C)PV. 5. The Stratification of F In the previous section we exhibit the closed set of unstable foliations on CP 2 of degree. In this section we will use properties of the singularities of the foliations to construct locally closed, non-singular subvarieties of F un. Firstly we will explain the stratification described in the following Theorem by F. Kirwan and then apply it to F. Theorem 6. see Theorem 1.5 in [12]) Let V be a non-singular projective variety with a linear action by a reductive group G. Then there exists a stratification {S β : β B} of V such that the unique open stratum is V ss and every stratum S β in the set of unstable points is non-singular, locally closed and isomorphic to G Pβ Yβ ss ss, where Yβ is a non-singular locally-closed subvariety of V and P β is a parabolic subgroup of G. Throughout the text we will use the same notation as in 12 of [12]. Definition 8. Let Y G) be the set of one parameter subgroups λ : C G. Define in Y G) N the equivalence relation: λ 1, n 1 ) is related with λ 2, n 2 ) if and only if λ 1 t n2 ) = λ 2 t n1 ) for all t C. A virtual one parameter subgroups of G is an equivalence class of this relation, the set of these classes will be denoted by MG). The indexing set B of the stratification is a finite subset of MG) and this may be described in terms of the weights of the representation of G which defines the action. For the construction we must consider on MG) a norm q which is the square of an inner product,. This norm gives the partial order > on B. On the other hand, the representation of D on C n+1, where D is a maximal torus of G, splits as a sum of scalar representations given by characters α,..., α n. These characters are elements of the dual of MD) but we can identify them with elements of MD) using,. Definition 9. Once we have the indexing set B we can describe the objects that appear in Theorem 6. Let β B, we define: Z β = {x :... : x n ) V : x j = if α j, β qβ)}, Y β = {x :... : x n ) V : x j = if α j, β < qβ) and x j for some j with α j, β = qβ)},

8 FOLIATIONS ON CP 2 OF DEGREE 59 the map p β : Y β Z β, x,..., x n ) x,..., x n) as x j = x j if α j, β = qβ) and x j = otherwise. Consider Stabβ), the stabilizer of β under the adjoint action of G. There exists a unique connected reductive subgroup G β of Stab β such that MG β ) = {λ MStab β ) : λ, β = } see in [12]). With this group we can define Z ss β = {x Z β : x is semistable under the action of G β on Z β } and Y ss β = p 1 β Zss β ). Finally the parabolic group of β is: if x Y ss β then P β = {g G : gx Y ss β }. Remark 1. Since S β is isomorphic to G Pβ Y ss β ss, it has dimension dim Yβ + dim G dim P β The representation of F. Norbert A Campo and Vladimir Popov give in [15] a computer program such that given a reductive group and one of its representation, the output is the finite subset B of virtual 1-parameter subgroups for the above stratification. For a more detailed construction of the virtual 1-parameter subgroups in the case of the action by change of coordinates of SL, C) in F d we refer to section of []. For F the virtual 1-parameter subgroups for the stratification are: 5 β 1 :=, 2 ) ) ) ) 5 5, 7, β 2 :=, 1, 4, β := 2,,, β 4 := 2, 5 6, 5, 6 ) ) β 5 := 42, 11 42, 22, β 6 := 21 6, 1, 5, β 7 := 6, 2 ), 4, β 8 := 1,, 1), 2 β 9 := 21, 4 ) 2 21, 16, β 1 := 21, 1 ) ) 1 6, 5, β 11 := 6 2,, 1, 2 ) 2 1 β 12 :=, 1, 1, β 1 := 6, 1 ) 5 6, 1, β 14 := 21, 1 21, 4 ), 21 2 β 15 := 21, 1 42, 5 ) 7, β 16 := 42 78, 1 9, 5 ). 78 Now we consider the induced representation H CP 2, T CP 2 2)) of the Lie algebra sl C). The weight diagrama for this irreducible representation is the following the number i denoted the virtual 1-PS β i ):

9 6 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE From this information we can easily obtain the sets Z i and Y i described in definition 9. In Y i with i {1, 2,, 4, 5, 7, 8, 1, 1} every foliation has a curve of singularities, we can study these foliations as foliations of degree 2, so we are going to discard these strata. To obtain the strata with foliations with isolated singularities we must find Z ss i := {x Z i : µx, λ) λ, β i for all λ MStabβ i ))}. See definition 12.1 of [12]. For this we will use the following results. Lemma 2. see [2, p. 4]) Let X Z i such that the virtual one parameter subgroup n, n 1, n 2 ) corresponding to β i satisfies n > n 1 > n 2. Then X Zi ss if and only if β i is the closest point to zero in C X with respect to D, where C X is the convex hull formed with the weights of X. The only virtual 1-PS where n 1 = n 2 in β 12, for finding Z ss 12 we need further analysis. We must recall that Stabβ 12 ) is the stabilizer of β 12 under the adjoint action of SL, C) on MSL, C)) see in [12]), i.e., 2 Stabβ 12 ) = g SL, C) : g 1 g 1 = 1 = a 11 a 22 a 2 SL, C). a 2 a We know that if λ MStabβ 12 )) then there exists g Stabβ 12 ) such that gλg 1 has the form Diagt k1, t k2, t k ), where k 1 k 2 k ; therefore:

10 FOLIATIONS ON CP 2 OF DEGREE 61 Z ss 12 = {x Z 12 : µgx, λ) λ, β 12, for all λ = Diagt k1, t k2, t k ), where k 1 k 2 k and g Stabβ 12 )}. Where, from the weight diagram: Z 12 = P xy 2 x, xyz x, xz2 x, y y, y2 z y, yz2 y, z y, y, z C and we have λt) = Diagt k1, t k2, t k ), β 12 = 2 k 1 1 k 2 1 k. For X Z 12 we obtain a 1,2 t 2k2 xy 2 + a 1,1 t k2 k xyz + a 1, t 2k xz 2 λt) X = b, t 2k2 y + b,2 t k2 k y 2 z + b,1 t 2k yz 2 + b, t k2 k z, c, t k k2 y therefore µx, λ) = min{ 2k 2, k 2 k, 2k, k 2 k, k k 2 }. With the conditions 2k 2 2 k 1 1 k 2 1 k k 2 k k 2 k 2 k 1 1 k 2 1 k 2k 2 k 1 1 k 2 1 k k 2 k k 2 k 2 k 1 1 k 2 1 k k 2 k k k 2 2 k 1 1 k 2 1 k k 2 k. we conclude that Z ss 12 = {X Z 12 : a 1,2, a 1,1, b,, b,2, c, ), a 1,1, a 1,, b,2, b,1, b, ) }. Now we can give the full list of linear subspaces of F for the construction of the strata. a, y 6 = P b, z Z 6 : a,, b, a 1, xz 2 + a, y 9 = P b,1 yz 2 : a,, a 1,, b,1 ) a 1,1 xyz + a, y 11 = P b 1, xz 2 + b,2 y 2 z : b 1,, a 1,1, a,, b,2 ) Z ss Z ss Z ss 12 = P a 1,2xy 2 + a 1,1 xyz + a 1, xz 2 b, y + b,2 y 2 z + b,1 yz 2 + b, z : c, y } a 1,2, a 1,1, b,, b,2, c, ), a 1,1, a 1,, b,2, b,1, b, ) a 1,2 xy 2 14 = P b 1, xz 2 + b, y : b 1,, a 1,2, b, ) 15 = P a 2,x 2 z + a, y b 1,1 xyz : a,, a 2,, b 1,1 ) Z ss Z ss Z ss

11 62 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE 16 = P b 1, xz 2 : b 1,, c, c, y, Z ss Y ss and a, y + a,2 y 2 z + a,1 yz 2 + a, z 6 = P b, z : a,, b, 9 = P a 1,xz 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z b,1 yz 2 + b, z : a,, a 1,, b,1 ) Y ss a 1,1 xyz + a 1, xz 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z 11 = P b 1, xz 2 + b,2 y 2 z + b,1 yz 2 + b, z : } b 1,, a 1,1, a,, b,2 ) Y ss 12 = P Y ss 2 j= a 1,jxy j z 2 j + j= a,jy j z j j= b,jy j z j c, y : } a 1,2, a 1,1, b,, b,2, c, ), a 1,1, a 1,, b,2, b,1, b, ) a 1,2 xy 2 + a 1,1 xyz + a 1, xz 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z 14 = P b 1, xz 2 + b, y + b,2 y 2 z + b,1 yz 2 + b, z : } b 1,, a 1,2, b, ) Y ss 15 = P a 2,x 2 z + a 1,1 xyz + a 1, xz 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z b 1,1 xyz + b 1, xz 2 + b,2 y 2 z + b,1 yz 2 + b, z : } a,, a 2,, b 1,1 ) Y ss a 1,2 xy 2 + a 1,1 xyz + a 1, xz 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z 16 = P b 1, xz 2 + b, y + b,2 y 2 z + b,1 yz 2 + b, z : c, y Y ss } b 1,, c,.

12 FOLIATIONS ON CP 2 OF DEGREE 6 6. Strata of the space of foliations of degree and its singularities In this section we calculate the Milnor number and the multiplicity of a common singularity in the generic foliation in every stratum. We also obtain the dimension of the strata. Note that the point p = 1 : : ) is a singularity for every foliation in Yi ss for all i = 6, 9, 11, 12, 14, 15, 16. Along this section we use the following notation: given a foliation, X = P x, y, z) Qx, y, z) Rx, y, z) Y ss i, we consider the corresponding local polynomial vector field around, ): X = Q1, y, z) yp 1, y, z)) y + R1, y, z) zp 1, y, z)) z. We define f i y, z) := Q1, y, z) yp 1, y, z), g i y, z) := R1, y, z) zp 1, y, z) and I f, g) will be the intersection index of f and g at, ) Stratum 6. As we saw before if X Y ss 6 then a, and b, are different from zero and f 6 y, z) = Q1, y, z) yp 1, y, z) = b, z a, y 4 a,2 y z a,1 y 2 z 2 a, yz g 6 y, z) = zp 1, y, z) = a, y z a,2 y 2 z 2 a,1 yz a, z 4. Note that b, z and P 1, y, z) does not have common tangent lines, therefore µ p X) = I f 6 y, z), g 6 y, z)) = I f 6 y, z), z) + I f 6 y, z), P 1, y, z)) = I a, y 4, z) + 9 = 1 m p X) =. ) ) f6 z Finally, the 2-jet of is trivial and the -jet is, if we suppose b g 6, = 1. On the other ) z hand, if X is a foliation of degree with m 1::) X) =, µ 1::) X) = 1 and with -jet, it is easy to see that X Y6 ss. In this case the corresponding parabolic subgroup P 6 is the subgroup of upper triangular matrices, therefore dim S 6 = dim Y6 ss + dim SL, C) dim P 6 = 7 see Remark 1) Stratum 9. If X Y ss 9 then a, and a 1,, b,1 ), ); therefore and f 9 y, z) = Q1, y, z) yp 1, y, z) = b,1 a 1, )yz 2 + b, z a, y 4 a,2 y z a,1 y 2 z 2 a, yz g 9 y, z) = zp 1, y, z) = a 1, z a, y z a,2 y 2 z 2 a,1 yz a, z 4, µ p X) = I f 9 y, z), g 9 y, z)) = I f 9 y, z), z) + I f 9 y, z), P 1, y, z)) = I a, y 4, z) + 2I z, a, y ) + I b,1 y + b, z, P 1, y, z)) = 1 + I b,1 y + b, z, P 1, y, z)). Note that I b,1 y + b, z, P 1, y, z)) is

13 64 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE 2 a 1,, b,1 b,1 =, b, ) or a 1, = and b,1 y + b, z is not tangent for P 1, y, z)) b 1,, b, ) = or a 1, = and b,1 y + b, z is tangent for P 1, y, z)) If a 1, it is clear that the multiplicity of the singular point is. If a 1, = ) then b,1 b,1 a and also the multiplicity is. Finally, the -jet is 1, )yz 2 + b, z a 1, z. Then in the open set where a 1,, b,1 every foliation has a singular point with multiplicity and Milnor number 12. In this case the corresponding parabolic subgroup is the subgroup of upper triangular matrices, therefore dim S 9 = Stratum 11. Remember that if X Y ss 11 then b 1, and a 1,1, a,, b,2 ) are different from zero, and f 11 y, z) = Q1, y, z) yp 1, y, z) = b 1, z 2 + b,2 a 1,1 )y 2 z + b,1 a 1, )yz 2 + b, z a, y 4 a,2 y z a,1 y 2 z 2 a, yz, g 11 y, z) = zp 1, y, z) = a 1,1 yz 2 a 1, z a, y z a,2 y 2 z 2 a,1 yz a, z 4. If a, =, then z = is a curve of singularities. Suppose a,. Note that And I f 11, g 11 ) = I a, y 4, z) + I z, a, y )+ I b 1, z + b,2 y 2 + b,1 yz + b, z 2, P 1, y, z)) = 7 + I b 1, z + b,2 y 2 + b,1 yz + b, z 2, P 1, y, z)). I b 1, z + b,2 y 2 + b,1 yz + b, z 2, a 1,1 yz + a 1, z 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ) = I b 1, z + b,2 y 2 + b,1 yz + b, z 2, a, a ) 1,1 b,2 y b 1, + a,2 a 1, b,2 a ) 1,1 b,1 y 2 z + a,1 a 1, b,1 a ) 1,1 b, yz 2 b 1, b 1, b 1, b 1, + a, a ) ) 1, b, z b 1, if a, b 1, a 1,1 b,2 4 if a, b 1, = a 1,1 b,2 and a,2 b 1, a 1,1 b,1 + a 1, b,2 = 5 if [...] and a,1 b 1, a 1,1 b, + a 1, b,1 6 if [...], a,1 b 1, = a 1,1 b, + a 1, b,1 and a, b 1, a 1, b, if [...], a,1 b 1, = a 1,1 b, + a 1, b,1 and a, b 1, = a 1, b, where [...] is a, b 1, = a 1,1 b,2, a,2 b 1, = a 1,1 b,1 + a 1, b,2. We conclude that in the open set of Y 11 where a, b 1, a 1,1 b,2 every foliation has a singularity with Milnor number 1. Since ) b 1, then the multiplicity for the singular point 1 : : ) z 2 is equal to 2 and the 2-jet is. The corresponding parabolic subgroup is the subgroup of upper triangular matrices, therefore dim S 11 = 12.

14 FOLIATIONS ON CP 2 OF DEGREE Stratum 12. If X Y ss 12 then we have that are different from zero and a 1,2, a 1,1, b,2, b,, c, ) and a 1,1, a 1,, b,, b,1, b,2 ) f 12 y, z) = Q1, y, z) yp 1, y, z) = b, a 1,2 )y + b,2 a 1,1 )y 2 z + b,1 a 1, )yz 2 + b, z a, y 4 a,2 y z a,1 y 2 z 2 a, yz g 12 y, z) = c, y zp 1, y, z) = c, y a 1,2 y 2 z a 1,1 yz 2 a 1, z a, y z a,2 y 2 z 2 a,1 yz a, z 4. These polynomials are homogenous in two variables, then generically we have I p f 12, g 12 ) = 9. If a 1,2, a 1,1, c, ) then g 12 and m p X) =. If a 1,2, a 1,1, c, ) = then b,, b,2 ) and we have also m p X) =. In this case the parabolic subgroup is P 12 = α 11 α 12 α 1 α 22 α 2 α 2 α SL, C), therefore dim S 12 = 1. Moreover, the set {X F : there exists p such that m p X) =, µ p X) = 9}, is an open set in S 12 because a foliation with these properties for the point 1 : : ) is unstable and it does not be in another stratum Stratum 14. If X Y ss 14 then b 1, and a 1,2, b, ) are different from zero, and Note that f 14 y, z) = Q1, y, z) yp 1, y, z) = b 1, z 2 + b, a 1,2 )y + b,2 a 1,1 )y 2 z + b,1 a 1, )yz 2 + b, z a, y 4 a,2 y z a,1 y 2 z 2 a, yz g 14 y, z) = zp 1, y, z) = a 1,2 y 2 z a 1,1 yz 2 a 1, z a, y z a,2 y 2 z 2 a,1 yz a, z 4. I f 14, g 14 ) = I b, a 1,2 )y a, y 4, z) + I f 14, a 1,2 y 2 a 1,1 yz a 1, z 2 a, y a,2 y 2 z a,1 yz 2 a, z ). If we suppose that a 1,2 and b, a 1,2 then the Milnor number of 1 : : ) is 7. If a 1,2, b, = a 1,2 and a, we have µ 1::) X) = 8. On the other hand, a 1,2, b, = a 1,2

15 66 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE and a, = implies that we have a curve of singularities. Supposing a 1,2 =, we obtain b, a 1,2,, and with this we have I f 14, a 1,1 yz + a 1, z 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ) = I b 1, z 2 + b, y + b,2 y 2 z + b,1 yz 2 + b, z, a 1,1 yz + a 1, a, + a,2 a ), b,2 y 2 z + a,1 a ), b,1 yz 2 + a, a, b, b, = + I b 1, z 2 + b, y + b,2 y 2 z + b,1 yz 2 + b, z, a 1,1 y + ) z 2 b 1, b, ) ) z b, b, a 1, a, ) b 1, z b, ) ) z 2. + a,2 a ), b,2 y 2 + a,1 a ), b,1 yz + a, a, b, b, b, b, We can verify that if a 1,1 then the last expression is equal to 5. When a 1,1 = we can see that I f 14, a 1,1 yz + a 1, z 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ) = I b 1, z 2 + b, y + b,2 y 2 z + b,1 yz 2 + b, z, a,y + a,2y 2 z + a,1yz 2 + a,z ), where a i,j denotes a i,j a1, b 1, b i,j. In conclusion, I b 1, z 2 + b, y + b,2 y 2 z + b,1 yz 2 + b, z, a,y + a,2y 2 + a,1yz + a,z 2 ) 6 if a, 7 if a, = and a,2 = 8 if a,, a,2) = and a,1 9 if a,, a,2, a,1) = and a, if a,, a,2, a,1, a,) =. Therefore, 7 if b, a 1,2 8 if b, = a 1,2 and a, ) or if a 1,2 = and a 1,1 ) 9 if a 1,2, a 1,1 ) = and a, µ 1::) X) = 1 if a 1,2, a 1,1, a,) = and a,2 11 if a 1,2, a 1,1, a,, a,2) = and a,1 12 if a 1,2, a 1,1, a,, a,2, a,1) = and a, if a 1,2, a 1,1, a,, a,2, a,1, a,) = ) z 2 Since b 1, we get m 1::) X) = 2 with 2-jet. Since the parabolic subgroup is the subgroup of upper triangular matrices we obtain dim S 14 = 14. Moreover, the set ) z 2 {X F : there exists p such that m p X) = 2, µ p X) = 7 and 2-jet linearly equivalent to },

16 FOLIATIONS ON CP 2 OF DEGREE 67 is an open set in S 14, because a foliation with these properties for the point 1 : : ) is unstable and it does not be in another stratum Stratum 15. If X Y ss 15 we have that a, and a 2,, b 1,1 ) are different from zero, and f 15 y, z) = Q1, y, z) yp 1, y, z) = b 1,1 a 2, )yz + b 1, z 2 + b,2 a 1,1 )y 2 z + b,1 a 1, )yz 2 + b, z g 15 y, z) = zp 1, y, z) a, y 4 a,2 y z a,1 y 2 z 2 a, yz = a 2, z 2 a 1,1 yz 2 a 1, z a, y z a,2 y 2 z 2 a,1 yz a, z 4. Note that I f 15, g 15 ) = I a, y 4, z) + I f 15, a 1,2 y a 1,1 z a 1, z 2 a, y a,2 y 2 z a,1 yz 2 a, z ) = 4 + I b 1,1 yz + b 1, z 2 + b,2 y 2 z + b,1 yz 2 + b, z, a 2, z + a 1,1 yz + a 1, z 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ) = 4 + I z, a, y ) + I b 1,1 y + b 1, z + b,2 y 2 + b,1 yz + b, z 2, a 2, z + a 1,1 yz + a 1, z 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ). It is clear that if b 1,1 and a 2, are different from zero then the intersection index of f 15 and g 15 is 8. Suppose that a 2, and b 1,1 =, then I b 1, z + b,2 y 2 + b,1 yz + b, z 2, a 2, z + a 1,1 yz + a 1, z 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ) 2 if b,2 if b,2 = and b 1, = 4 if b,2, b 1, ) = and b,1 6 if b,2, b 1,, b,1 ) = and b, if b,2, b 1,, b,1, b, ) = Now suppose a 2, = and b 1,1. We define Ly, z) = b 1,1 y + b 1, z My, z) = a 1,1 yz + a 1, z 2 Ny, z) = b,2 y 2 + b,1 yz + b, z 2 F y, z) = a, y + a,2 y 2 z + a,1 yz 2 + a, z. We have I L + N, M + F ) = 2 if L M.

17 68 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE Now suppose L M, we have two cases: if L N, I L + N, M + F ) = I L + N, F M L N ) if L 2 LF MN) 4 if L 2 LF MN) and L LF MN) = 5 if L LF MN) and L 4. LF MN) 6 if L 4 LF MN) On the other hand, if L N: I L + N, F M L N ) = I L, F M L N ) + I 1 + N L, F M L N ) = I L, F M L N ) = { if L LF MN) if L LF MN) We conclude that in S 15 we have a nonempty open set which consists of foliations with a singularity with multiplicity 2 and Milnor number 8. But we can have in this stratum foliations with a singularity with multiplicity 2 and Milnor number 9, 1, 11, 12 and 1 or with a curve ) of singularities. The 2-jet around the singular point 1 : : ) of X is 2, )yz + b 1, z 2 b1,1 a a 2, z 2. Since the parabolic subgroup is the subgroup of upper triangular matrices we obtain dim S 15 = Stratum 16. If X Y ss 16 then b 1, and c,. We have f 15 y, z) = Q1, y, z) yp 1, y, z) = b 1, z 2 + b, a 1,2 )y + b,2 a 1,1 )y 2 z + b,1 a 1, )yz 2 + b, z a, y 4 a,2 y z a,1 y 2 z 2 a, yz g 15 y, z) = c, y zp 1, y, z) = c, y a 1,2 y 2 z a 1,1 yz 2 a 1, z a, y z a,2 y 2 z 2 a,1 yz a, z 4, and the polynomials c, y a 1,2 y 2 z a 1,1 yz 2 a 1, z and b 1, z 2 do not have common factors. As a result ) the Milnor number of 1 : : ) is 6. And the multiplicity of this point is 2 with z 2 2-jet. The corresponding parabolic subgroup is the subgroup of upper triangular matrices, therefore dim S 16 = 15. In the following theorem we summarize the above. Theorem 7. The spaces S i = SL, C)Yi ss for i {1,..., 16}, are locally closed, irreducible non-singular algebraic subvarieties of F. They form a stratification of the closed set of unstable foliations F un, and S i j i S j. Moreover, these varieties satisfy the following:

18 Stratum S 1, S 2, S, S 4, S 5, S 7, S 8, S 1, S 1 S 6 dim S 6 = 7 S 9 dim S 9 = 9 S 11 dim S 11 = 12 S 12 dim S 12 = 1 S 14 dim S 14 = 14 S 15 dim S 15 = 14 S 16 dim S 16 = 15 FOLIATIONS ON CP 2 OF DEGREE 69 Characterization of the generic foliation Every foliation has a curve of singularities { X F : p with m p X) =, µ p X) ) = 1 and -jet linearly equivalent z } to = S6 {X F : p with m p X) =, µ p X) = 12} S 9 is open in S 9 {X F : p with m p X) = ) 2, µ p X) = 1 and 2-jet linearly equivalent to z 2 } S 11 is open in S 11 Contains {X F : p with m p X) =, µ p X) = 9} as an open set Contains { X F : p with m p X) ) = 2, µ p X) = 7 and 2-jet linearly z 2 } equivalent to as an open set {X F : p with m p X) = 2, µ p X) = 8} S 15 is open in S 15 {X F : p with m p X) = 2, µ p X) ) = 6 and 2-jet linearly equivalent to z 2 } = S 16 In [] we also have studied the strata S 6, S 9 and S 16. As a consequence of the above we can mention the following general result. Theorem 8. Let X F with isolated singularities. Then X is unstable if and only if: 1) X has a singular point with multiplicity or 2) X has a singular point with multiplicity 2 and 2-jet linearly equivalent to z 2 y or ) X S 15. Moreover, the irreducible components of F un are the closure of the locally closed subvarieties S 15 and S 16. The first one has dimension 14 and the second one has dimension 15. Proof. The first affirmation is consequence of the results in the table. For the second one we use proposition 4.2 of [9], it says that S j is irreducible and S j = SL, C)Y j for all j. Since Y 16 j 15 Y j and S 15 S 16, we have that F un = S 15 S 16 is the decomposition in irreducible components. Remark 2. Note that V = Y 15 and V 1 = Y 16 in theorem Semistable non-stable foliations on CP 2 of degree. In this subsection we describe the semistable non-stable foliations on CP 2 of degree. Theorem 9. The set of semistable non-stable foliations on CP 2 of degree with isolated singularities is P y, z) SL, C)P b 1,1 xyz + b 1, xz 2 + j= b,jy j z j : c 1, xz 2 + c,2 y 2 z + c,1 yz 2 + b, z } P y, z) C [y, z], b,, c,2 ), b 1,1, c 1, ).

19 7 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE Proof. Let X F ss F s then X is not in any strata and it satisfies one of the following properties: 1) dim OX) < 8: in this case, by Theorem 1.2 of [1], the foliation is λ-invariant for some 1-PS λ and it is not in any Z j. Then the foliation is, up to change of coordinates, such that the line with its weights pass through zero, if we see the representation we conclude that X is: b 1,1 xyz + b, y, c 1, xz 2 + c,2 y 2 z where b,, c,2 ), b 1,1, c 1, ), since X has isolated singularities we can suppose c 1, = 1. 2) OX) is not closed in F ss : then there exists Y F ss OX) OX)). Since dim OY ) < 8, we conclude that Y is the above foliation, therefore X has its weights in one hyperplane given by the weights of Y, therefore X is, up to change of coordinates, in P y, z) P b 1,1 xyz + b 1, xz 2 + j= b,jy j z j : P y, z) C [y, z], b,, c,2 ), b 1,1, c 1, ) c 1, xz 2 + c,2 y 2 z + c,1 yz 2 + b, z. 7. Corollaries 7.1. The dimension of the orbits. Generically the orbit of a foliation on CP 2 has dimension 8, for example, a stable foliation satisfies this property. It theorem 1.2 of [1] we classify foliations with isolated singular points such that the dimension of the orbit is less than or equal to 7. We can see in proposition 2. of [5] that the dimension of an orbit of a foliation with isolated singularities of degree d is greater than or equal to 6. In the same paper the authors describe the two unique foliations of degree 2, up to change of coordinates, such that the orbit has dimension 6. For the case of foliations of degree we have the same situation. Theorem 1. There are, up to change of coordinates, two foliations on CP 2 of degree with isolated singularities with automorphism group of dimension 2: y x + z y and y x + z z. Proof. In proposition 2.5 of [5] we can see that if AutX) has dimension 2 then it is isomorphic to the group of affine transformations of the line. Therefore by theorem 1.2 of [1], X is λ-invariant for some 1-PS λ, and X is also invariant by C, +). This last affirmation implies, by the same theorem, that X is unstable with a singular point with Milnor number 12. An unstable foliation invariant by a 1-PS is, up to change of coordinates, in Z 15 Z 14 Z 12 Z 11 Z 9 Z 6. It is easy to see that the unique foliations with a singular point with Milnor number 12 are in Z 9 and Z 6, they are y x + z y and y x + z z Foliations on CP 2 of degree with one singular point. The classification of foliations on CP 2 with one singular point is known only for degree 2 see [5] and [2]). In this section we describe all the unstable foliations on CP 2 of degree with one singular point, that means with a singular point with Milnor number 1. To obtain the result we need the following lemma. Lemma. Let X be a foliation on CP 2 of degree d. If X has a singular point p with multiplicity d and Milnor number greater than d 2, then X has an invariant line that passes through p.

20 FOLIATIONS ON CP 2 OF DEGREE 71 Proof. We can suppose that X is a foliation on CP 2 of degree d such that m 1::) X) = d and µ 1::) X) > d 2. Then xp d 1 + P d X = Q d R d where P k, Q k, R k C[y, z] are homogeneous of degree k. In the chart U, the foliation is Q d yp d 1 ) y + R d zp d 1 ) z, since µ 1::) X) > d 2 then there exists a line L = αy βz such that Q d yp d 1 = LF and R d zp d 1 = LG for some F, G C[y, z]. Therefore αq d βr d = LP d 1 + αf βg), and this means that L is invariant for X and it passes through 1 : : ). Theorem 11. The unstable foliations on CP 2 of degree with one singular point are: 1) The stratum S 6, which has dimension 7. 2) The subspace of S 9 : { SL, C) a 1,xz 2 + y + a,2 y 2 z + a,1 yz 2 + a, z b,1 yz 2 + b, z : b,1 =, a 1, b, or a 1, =, b 1 and b,1 y + b, z y + a,2 y 2 z + a,1 yz 2 + a, z }, of dimension 8. ) The subspace of S 11 : SL, C){ a 1,1 xyz + a 1, xz 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z xz 2 + b,2 y 2 z + b,1 yz 2 + b, z : a 1,1, a,, b,2 ), a, a 1, b, ; a 1, = a 1,1 b, + a 1, b,1 ; a,2 = a 1,1 b,1 + a 1, b,2 ; a, = a 1,1 b,2 }, of dimension 9. 4) The subspace of S 12 : SL, C){ xα 1 y β 1 z)α 2 y β 2 z) + y + a,2 y 2 z + a,1 yz 2 + a, z αα 1 y β 1 z) 2 α 2 y β 2 z) : It has isolated singularities, α 1, α 2 ) and α 1 α 2 = ; or αα 1 = 1 and β 1, β 2 C }, of dimension 1. 5) The subspace of S 15 : SL, C){ x 2 z + a 1,1 xyz + a 1, xz 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z } b, z : a,, of dimension 1.

21 72 CLAUDIA R. ALCÁNTARA AND RAMÓN RONZÓN-LAVIE In S 6, S 9 and S 12 the singular point has multiplicity and in S 11, S 15 has multiplicity 2. Proof. It remains only to find the foliations with one singular point in S 12, the other cases were studied in the construction of the strata. To obtain the dimension of these spaces, we must observe that if X is a foliation in any of the described linear subspaces of Y9 ss, Y11 ss, or Y15 ss then for g SL, C) we have that gx is in the same linear subspace if and only if g is in the corresponding parabolic subgroup. Therefore the dimension of the space is the dimension of the linear subspace plus. Now let X Y12 ss such that 1 : : ) is the unique singularity. By the above lemma, X has an invariant line αy βz. There exists g P 12 such that z is invariant for gx Y12 ss. For that we can suppose: xl 1 y, z)l 2 y, z) + a, y + a,2 y 2 z + a,1 yz 2 + a, z X = L y, z)l 4 y, z)l 5 y, z) where L k y, z) = α k y β k z and α k, β k, a,j, C for k = 1,..., 5 and j =, 1, 2,. The Milnor number of 1 : : ) is I L L 4 L 5 yl 1 L 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ), zl 1 L 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ) ) = Iz, L L 4 L 5 yl 1 L 2 + a, y )) + IL L 4 L 5, L 1 L 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ), and this is 1 if and only if Iz, L L 4 L 5 yl 1 L 2 + a, y )) = 4, IL L 4 L 5, L 1 L 2 + a, y + a,2 y 2 z + a,1 yz 2 + a, z ) = 9, and this happens if and only if a,, z L L 4 L 5 yl 1 L 2 ), L L 4 L 5 = αl 2 1L 2 for some α, and L 2 1L 2 a, y +a,2 y 2 z+a,1 yz 2 +a, z ). Since z L 1 L 2 αl 1 y) then we have the following cases: α 1 =, α 2 = or αα 1 = 1. The condition for X to be in Y12 ss says that if α 1 = then α 2 and if α 2 = then α 1. If αα 1 = 1 then β 1, β 2 C. We get xα 1 y β 1 z)α 2 y β 2 z) + y + a,2 y 2 z + a,1 yz 2 + a, z X = αα 1 y β 1 z) 2 α 2 y β 2 z). Then the dimension of the projectivization of the linear space where X lives is 7. When we move the invariant line through 1 : : ) we obtain a family of foliations of dimension 8 and when we take the action by SL, C) module the parabolic subgroup P 12 we obtain a space of dimension 1. To finish the classification of foliations on CP 2 of degree with one singularity with Milnor number 1 remains to find the semistable foliations with this property. For foliations on CP 2 of degree 2 we know that there exists only one semistable foliation, up to change of coordinates see Theorem 5.9 of [2]), with only one singular point. In this case the singularity is a saddle-node, that means multiplicity 1 non-nilpotent. For degree the situation is different, for example, we have the following foliations:

22 FOLIATIONS ON CP 2 OF DEGREE 7 X 1 = z x + x2 z + xy 2 ) y xyz + y ) z X 2 = y 2 z x + xyz + z ) y y z. Both are semistable foliations of degree with only one singularity, the first one has a nilpotent singularity and in second one the singularity has multiplicity 2. In general is very difficult to find foliations on CP 2 of degree d with one singular point. It is clear that using this stratification we can get all the unstable foliations. We think that using recursively this construction it is possible to find also the semi-stable foliations of degree d with a singularity with Milnor number d 2 +d+1. Acknowledgments. The authors would like to thank the reviewers for their comments that help to improve this paper. References [1] Alcántara, C. R. Singular points and automorphisms of unstable foliations of CP2 Bol. Soc. Mat. Mexicana ) 16 21), no. 1, [2] Alcántara, C. R. Foliations on CP 2 of degree 2 with degenerate singularities. Bull. Braz. Math. Soc. N.S.) 44 21), no., [] Alcántara, C. R. Stratification of the space of foliations on CP2. J. Symbolic Comput ), DOI: 1.116/j.jsc [4] Brunella, M. Birational Geometry of Foliations. IMPA Monographs, 1. Springer, Cham, 215. [5] Cerveau, D.; Déserti, J.; Garba Belko, D.; Meziani, R. Géométrie classique de certains feuilletages de degré deux. Bull Braz Math Soc, New Series 412), 21), [6] Fogarty, J.; Kirwan, F.; Mumford, D. Geometric Invariant Theory. Springer-Verlag, [7] Gómez-Mont, X.; Kempf, G. Stability of Meromorphic Vector Fields in Projective Spaces. Comment. Math. Helvetici ) DOI: 1.17/BF [8] Gómez-Mont, X.; Ortiz-Bobadilla, L. Sistemas dinámicos holomorfos en superficies. Aportaciones Matemáticas: Notas de Investigación [Mathematical Contributions: Research Notes],. Sociedad Matemática Mexicana, México, [9] Hesselink, W. H. Desingularizations of varieties of nullforms. Invent. Math ), no. 2, [1] Jouanolou, J.P. Equations de Pfaff Algébriques. Springer-Verlag, [11] Kempf, G. Instability in Invariant Theory. Annals of Mathematics ) DOI: 1.27/ [12] Kirwan, F. Cohomology of Quotients in Symplectic and Algebraic Geometry. Mathematical Notes ), [1] Lins Neto, A.; Soares, M.G. Algebraic Solutions of one-dimensional foliations. J. Differ. Geom ), [14] Newstead, P. Introduction to Moduli Problems and Orbit Spaces. Springer-Verlag, [15] Popov, V. The Cone of Hilbert Nullforms. arχiv: v1 Departamento de Matemáticas, Universidad de Guanajuato, Callejón Jalisco s/n, A.P. 42, C.P. 6, Guanajuato, Gto. México. address: claudiagto@gmail.com, claudia@cimat.mx; ronzon@cimat.mx