Bull Braz Math Soc, New Series (2017) 48:29–44 DOI 10.1007/s00574-016-0007-7

Stability Of Branched Pull-Back Projective Foliations W. Costa e Silva1

Received: 8 January 2015 / Accepted: 12 February 2016 / Published online: 7 October 2016 © Sociedade Brasileira de Matemática 2016

Abstract We present new irreducible components of the space of codimension one holomorphic foliations on Pn , n ≥ 3. They are associated to pull-back by branched rational maps of foliations on P2 that preserve invariant lines. Keywords Holomorphic foliations · Pull-back components · Space of foliations Mathematics Subject Classification 32S65 · 37F75

1 Introduction Let F be a holomorphic singular foliation on Pn of codimension 1, n : Cn+1 \ {0} → Pn be the natural projection and  F ∗ = ∗n (F). It is known ∗ that F can be defined by an integrable 1−form  = nj=0 A j dz j where the Aj s are homogeneous polynomials of the same degree satisfying the Euler condition: n 

z j A j ≡ 0.

(1)

j=0

To my mother. I am deeply grateful to A. Lins Neto and D. Cerveau for the discussions, suggestions and comments. This work was developed both at IMPA (Rio de Janeiro, Brazil) and (IRMAR-Université de Rennes 1 (France)). It was supported by IMPA, CNPq (process number 142250/2005-8) and CAPES-BR (process number 9814-13-2).

B 1

W. Costa e Silva [email protected] IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

123

30

W. Costa e Silva

The singular set S(F) is given by S(F) = {A0 = . . . = An = 0} with codim(S (F)) ≥ 2. The integrability condition is given by  ∧ d = 0.

(2)

The degree of F, for short deg(F), is by definition, the number of tangencies of a generic linearly embedded P1 with F. We denote the space of foliations of a fixed degree k on Pn by Fol (k, n). Due to the integrability condition and the fact that S (F) has codimension ≥ 2, we see that Fol (k, n) can be identified with a Zariski’s open set in the variety obtained by projectivizing the space of forms  of degree k + 1 which satisfy (1) and (2). It is in fact an intersection of quadrics. To obtain a satisfactory description of Fol (k; n) it would be reasonable to know the decomposition of Fol (k; n) in irreducible components. This leads us to the following: Problem: Describe and classify the irreducible components of Fol (k; n) k ≥ 0 on Pn , n ≥ 3. Up to date there are partial lists of irreducible components of Fol (k; n) which is known to be complete only when k ≤ 2. The cases k = 0, 1 were treated in Jouanolou (1979). There are known families of irreducible components in which the typical element is a pull-back of a foliation G of degree d on P2 by a rational map f : Pn P2 of degree ν ≥ 1. If f and G are generic, the pull-back foliation F = f ∗ G associated to the pair ( f, G) has degree ν(d + 2) − 2 as proved in Cerveau et al. (2001). Denote by P B(d, ν; n) the closure in Fol (ν(d + 2) − 2, n), n ≥ 3 of the set of foliations F of the form f ∗ G. The main result of Cerveau et al. (2001) can be stated as follows: Theorem 1.1 Cerveau et al. (2001) P B(d, ν; n) is a unirational irreducible component of Fol (ν(d + 2) − 2, n) for all n ≥ 3, ν ≥ 1 and d ≥ 2. The case ν = 1, of linear pull-backs, was proven in Camacho and Lins Neto (1982), whereas the case ν > 1, of nonlinear pull-backs, was proved in Cerveau et al. (2001). In this work we will focus on the following situation: Let f : Pn P2 be a rational map represented in the coordinates γ β 3 (X, Y, Z ) ∈ C and W ∈ Cn+1 by f˜ = (F0α , F1 , F2 ) where F0 , F1 and F2 ∈ C[W ] are homogeneous polynomials without common factors satisfying α.deg(F0 ) = β.deg(F1 ) = γ .deg(F2 ) = ν ≥ 2. The arithmetic conditions in α, β and γ are as follows: (1) α = 1 and β and γ are relatively prime numbers, or (2) α > 1 and α, β and γ are pairwise relatively prime numbers. If G is a foliation of degree d on P2 that preserves the thee invariant straight lines (X Y Z = 0) then G is given by the following homogeneous polynomial 1-form:  = Y Z A (X, Y, Z ) d X + X Z B (X, Y, Z ) dY + X Y C (X, Y, Z ) d Z

123

(3)

Stability of Branched Pull-Back Projective Foliations

31

where A, B and C are homogeneous polynomials of degree d −1 satisfying the relation A + B + C = 0, according to Eq. (1). If f and G are inside of these new classes of rational maps and foliations and are ∗ also generic, then the  pull-back foliation F = f G associated to the pair ( f, G) has α,β,γ α,β,γ degree ν,d = ν (d − 1) + α1 + β1 + γ1 −2. Denote by P B(ν,d , n) the closure   α,β,γ in Fol ν,d , n , n ≥ 3 of the set of foliations F of the form f ∗ G. If α, β and γ are in the situations (1) and (2) as described previously we are able to prove the following: α,β,γ

Theorem A P B(ν,d , n) is a unirational   α,β,γ Fol ν,d , n for all n ≥ 3, d ≥ 2 and ν ≥ 2.

irreducible

component

of

Let us indicate some differences between the work Cerveau et al. (2001) and the current situation. With respect to comparisons on the geometry of a generic element, in the set P B(d, ν; n) a generic point has no algebraic invariant hypersurfaces since a generic degree d ≥ 2 holomorphic foliation on P2 also does not have. Unlike the α,β,γ previous case, a generic element of P B(ν,d , n) has three algebraic invariant hypersurfaces smooth and transverse. This exemplifies that these pull-back components are of a different nature from those that were known so far. Moreover, the greatest part of both jobs consist in make a deep analysis of the local structure of the singular set of a generic element of each subset. In Cerveau et al. (2001), to prove that a pull-back generic foliation is stable under holomorphic deformations the techniques developed in Camacho and Lins Neto (1982) concerning the stability of homogeneous integrable 1-forms are sufficient what does not happen in the branched case. Once the results contained in Camacho and Lins Neto (1982) can not be used in present situation, to surpass this difficulty, we had to import techniques which were developed in Calvo-Andrade et al. (2004) and Lins Neto (1981) about the stability of quasi-homogeneous singularities. Furthermore, in the present work, due to the appearance of such singularities, and also of quasi-homogeneous 1-forms, it is necessary the use of techniques of weighted blow-ups and weighted projective spaces to overpass the technical difficulties.

2 Branched Rational Maps Let f : Pn P2 be a rational map and f˜ : Cn+1 → C3 its natural lifting in homogeneous The indeterminacy locus of f is, by definition, the set  coordinates.  −1 ˜ I ( f ) = n f (0) . Observe that the restriction f |Pn \I ( f ) is holomorphic. Definition 2.1 e denote by B R M (n, ν, α,β, γ ), the set of  branched rational maps,  γ β α n 2 f :P P of degree ν given by f = F0 : F1 : F2 where F0 , F1 and F2 are homogeneous polynomials without common factors, with deg (F0 ) .α = deg (F1 ) .β = deg (F2 ) .γ = ν, ν ≥ 2, where α, β and γ satisfy: (1) If α = 1 then β and γ are relatively prime numbers (2) If 1 < α we require that α, β and γ pairwise relatively prime numbers.

123

32

W. Costa e Silva

Henceforth we will always deal with α, β and γ satisfying these hypothesis. Let us note that the indeterminacy locus I ( f ) is the intersection of the 3 hypersurfaces (F0 = 0), (F1 = 0) and (F2 = 0). Definition 2.2 We say that f ∈ B R M (n, ν, α, β, γ ) is generic if for all p ∈ f˜−1 (0) \ {0} we have d F0 ( p) ∧ d F1 ( p) ∧ d F2 ( p) = 0. If f ∈ B R M (n, ν, α, β, γ ) is generic then I ( f ) is the transverse intersection of the 3 hypersurfaces (F0 = 0), (F1 = 0) and (F2 = 0). As a consequence, I ( f ) is smooth. For instance, if n = 3, f is generic and deg( f ) = ν, then by Bezout’s theorem ν3 I ( f ) consists of αβγ distinct points with multiplicity αβγ . If n = 4, then I ( f ) is a ν3 αβγ Pn of

smooth connected algebraic curve in P4 of degree

. In general, for n ≥ 4, I ( f )

ν is a smooth connected algebraic submanifold of degree αβγ and codimension three. The set of generic maps will be denoted by Gen (n, ν, α, β, γ ). 3

Proposition 2.3 Gen (n, ν, α, β, γ ) is a Zariski dense subset of B R M (n, ν, α, β, γ ).

3 Foliations with Three Invariant Straight Lines In this section we will describe a class of foliations that will be used throughout the text. Definition 3.1 Let G ∈ Fol(d, 2). We say that G is hyperbolic if its singularities are of hyperbolic type, that is, at each singular point of G the eigenvalues λ1 and λ2 of a / R. local vector field defining G are non null and λλ21 ∈ We observe that at each singularity q the foliation has two local smooth transverse invariant curves. The set H(d, 2), of the hyperbolic foliations of degree d is an open and dense subset of Fol(d, 2), for all d ≥ 2. It is also known that G has exactly d 2 + d + 1 singularities. Denote by I l3 (d, 2) the set of the holomorphic foliations on P2 of degree d ≥ 2 leaving invariant the lines X = 0, Y = 0 and Z = 0. Set by A(d) = I l3 (d, 2) ∩ H(d, 2). Observe that A(d) is a Zariski dense subset of I l3 (d, 2). The next proposition will be useful to prove our main result. Proposition 3.2 Let d ≥ 2. There exists an open and dense subset M (d) ⊂ A (d), such that if G ∈ M (d) then the only algebraic invariant curves of G are the three lines (X Y Z = 0). Proof Let us first show that such a foliation exists in A(d). Let U0 = {C2 , (x, y)} be an affine chart of P2 . Let H be a holomorphic foliation in P2 , which on U0 is given by the polynomial vector field X 0 (x, y) = x(i x + (1 − i)y + 1)

123

∂ ∂ + y(x + y + i) . ∂x ∂y

Stability of Branched Pull-Back Projective Foliations

33

It is not difficult to see that all its singularities are hyperbolic. Using a known result [Thm.1 pp 891 from Cerveau and Lins-Neto (1991)] we have that if S is an algebraic invariant curve for the foliation H, its degree must satisfy deg(S) ≤ 4. Hence the only possible remaining algebraic invariant would be a line. On the other hand, this line would have to pass through 3 singularities. But this is impossible according to Camacho-Sad’s Index Theorem Lins Neto (1988) for the foliation H. Now take the holomorphic ramified map T : C2 → C2 given by (x, y) → (x d−1 , y d−1 ), d ≥ 2. The vector field T∗ X 0 defines a holomorphic foliation G on P2 having the three invariant lines (X Y Z = 0). The map T can be extended to a mapping T : P2 → P2 , and we have that G = T ∗ H. Remark that all algebraic invariant curves of G are the three lines. To finish the argument, we perceive that for d ≥ 2 fixed, the subsets of foliations with algebraic invariant curves different from the 3 lines is a union of algebraic subsets  whose complement in I l3 (d, 2) is an open and dense subset in the usual topology. For each fixed d we denote this set by M(d).

4 Description of Generic Ramified Pull-Back Foliations on Pn Definition 4.1 Let f ∈ Gen (n, ν, α, β, γ ) and G ∈ M(d); we say that ( f, G) is a generic pair if [Sing (G) ∩ ( f )] = ∅, where 

 ( f ) := 2 f˜ W ∈ Cn+1 |d F0 (W ) ∧ d F1 (W ) ∧ d F2 (W ) = 0 . Proposition 4.2 If F = f ∗ G where ( f, g) is a generic pair, then the degree of F is α,β,γ

ν,d

1 1 1 − 2. = ν (d − 1) + + + α β γ

Set W = {F; F = f ∗ (G), where ( f, G) is a generic pair } of generic pull-back α,β,γ foliations. We remark that it is an open (not Zariski) and dense subset of P B(ν,d , n). 4.1 The Singular Set of a Pull-Back Generic Foliation 4.1.1 Central Points of Quasi-Homogeneity Let us describe F = f ∗ (G) in a neighborhood of a point p ∈ I ( f ). It is easy to show that there exists a local chart (U, (x0 , x1 , x2 , y) ∈ C3 × Cn−3 ) around p˜ ∈ −1 n ( p) γ β α 3 ˜ ˜ such that the lifting f of f is of the form f |U = (x0 , x1 , x2 ) : U → C . In particular the lifting F ∗ |U ( p) ˜ is represented by the 1-form  η(x0 , x1 , x2 , y) = α.x1. x2 .A

γ β x0α , x1 , x2 β

γ



 d x0 + β.x0. x2 .B

+ γ .x0. x1 .C(x0α , x1 , x2 )d x2 ,

γ β x0α , x1 , x2

 d x1 (4)

123

34

W. Costa e Silva

which is the pull-back of , the 1-form defining G (see Eq. 3). A 1−form like this is called a quasi-homogeneous singularity of type (βγ , αγ , αβ; αβγ (d − 1)). In particular, note that η is a quasi-homogeneous 1−form invariant by the C∗ - action   (x0 , x1 , x2 , y) → s βγ x0 , s αγ x1 , s αβ x2 , y .

(5)

The set of points, (0, 0, 0, y) in C3 × Cn−3 , corresponding to (η = 0) will be denoted by C(η). It will be called the set of central points of quasi-homogeneity of η. The union of all central points of quasi-homogeneity of the foliation F will be denoted by C(F). According to [Calvo-Andrade et al. (2004), proposition 1, p. 992], C(F) is stable by holomorphic perturbations in the following sense: if Ft is a deformation of F0 the set C(Ft ) of central points of Ft is a deformation of C(F0 ). Recall that a singular point p is in the Kupka set of the foliation F if there exists a local 1-form ω defining F at p such that dω( p) = 0. The Kupka set of F is the union of all Kupka points of F. It is known that this kind of singularities are stable under small holomorphic perturbations of F, (see Gómez-Mont and Lins Neto 1991; Kupka 1964). 4.1.2 The Kupka Set of F = f ∗ (G) Let τ be a singularity of G and Vτ = f −1 (τ ). If ( f, G) is a generic pair then Vτ \I ( f ) = K τ is contained in the Kupka set, K (F), of F. This can be seen using the normal forms obtained in Eq. 4. 4.2 Deformations of the Singular Set of F0 = f0∗ (G0 ), Where ( f0 , G0 ) is a Generic Pair We will state two lemmas which say that for any pull-back of a generic foliation F0 and any germ of a holomorphic family of foliations (Ft )t∈(C,0) such that F0 = Ft=0 , then (Ft )t∈(C,0) has a singular set with the same properties that F0 for all t ∈ (C, 0). Recall that C(F0 ) coincides with the indeterminacy set of f 0 , I ( f 0 ). Lemma 4.3 [Lins Neto (2007), p. 81] There exists a germ of isotopy of class C ∞ , (I (t))t∈(C,0) having the following property: I (0) = I ( f 0 ) = C(F0 ) and I (t) = C(Ft ) is a codimension 3 smooth algebraic subvariety of Pn for all t ∈ (C, 0). In the case n > 3, I (t) is connected since I ( f 0 ) is connected. On the other hand, when n = 3 we have I (t) = p1 (t), . . . , p j (t), . . . , p ν 3 (t). αβγ

Remark that K (F0 ) = ∪τ ∈Sing(G0 ) K τ . Let us consider a representative of the germ of family (Ft ), defined on a disc Dδ := (|t| < δ). Lemma 4.4 [Lins Neto (2007), p. 81] There exists  > 0 and smooth isotopies φτ : D × Vτ → Pn , τ ∈ Sing(G0 ), such that Vτ (t) = φτ ({t} × Vτ ) satisfies: (a) Vτ (t) is an algebraic subvariety of codimension two of Pn and Vτ (0) = Vτ for all τ ∈ Sing(G0 ) and for all t ∈ D .

123

Stability of Branched Pull-Back Projective Foliations

35

(b) Vτ (t)\I (t) is contained in the Kupka set of Ft for all τ ∈ Sing(G0 ) and for all t ∈ D . In particular, the transversal type of Ft is constant along Vτ (t)\I (t). (c) I (t) ⊂ Vτ (t) for all τ ∈ Sing(G0 ) and for all t ∈ D . Moreover, if τ = τ  , and τ, τ  ∈ Sing(G0 ), we have Vτ (t) ∩ Vτ  (t) = I (t) for all t ∈ D and the intersection is transversal.

4.3 End of the Proof of Theorem A 4.3.1 Part 1 Set Va = f 0−1 (a), Vb = f 0−1 (b), Vc = f 0−1 (c), where a = [0 : 0 : 1], b = [0 : 1 : 0]

and c = [1 : 0 : 0] and denote by Vτ ∗ = f 0−1 (τ ∗ ), where τ ∗ ∈ Sing(G0 )\{a, b, c}. Using the previous notation, we get from lemma 4.4, Va (t), Vb (t) and Vc (t) for all t ∈ D . We will use them to define a family of rational maps ( f t )t∈D , a deformation of f 0 in Gen (n, ν, α, β, γ ). Proposition 4.5 Let (Ft )t∈D be a deformation of F0 = f 0∗ (G0 ), where ( f 0 , G0 ) is a generic pair, with G0 ∈ M(d), f 0 ∈ Gen (n, ν, α, β, γ ) and deg( f 0 ) = ν ≥ 2. Then there exists a deformation ( f t )t∈D of f 0 in Gen (n, ν, α, β, γ ) such that: (i) Va (t), Vb (t) and Vc (t) are fibers of ( f t )t∈D  . (ii) I (t) = I ( f t ), ∀t ∈ D  . β

γ

Proof Let f˜0 = (F0α , F1 , F2 ) : Cn+1 → C3 be the homogeneous expression of f 0 . Then Vc , Vb , and Va appear as the complete intersections {F1 = F2 = 0}, {F0 = F2 = 0}, and {F0 = F1 = 0} respectively. Hence I ( f 0 ) = Va ∩ Vb = Va ∩ Vc = Vb ∩ Vc . Using the stability criteria (see [Sernesi (2006), Sect. 4.6, p. 235–236]), it follows that Va (t) is a complete intersection, say Va (t) = {F0 (t) = F1 (t) = 0}, where {F0 (t)}t∈D  and {F1 (t)}t∈D  are deformations of F0 and F1 and D  is a possibly smaller neighborhood of 0. Moreover, F0 (t) = 0 and F1 (t) = 0 meet transversely along Va (t). In the same way, it is possible to define Vc (t) and Vb (t) as complete intersections, say { Fˆ1 (t) = F2 (t) = 0} and { Fˆ0 (t) = Fˆ2 (t) = 0} respectively, where {F j (t)}t∈D  and { Fˆ j (t)}t∈D  are deformations of F j , 0 ≤ j ≤ 2. Let us find polynomials P0 (t), P1 (t) and P2 (t) such that Vc (t) = {P1 (t) = P2 (t) = 0}, Vb (t) = {P0 (t) = P2 (t) = 0} and Va (t) = {P0 (t) = P1 (t) = 0}. Observe first that since F0 (t), F1 (t) and F2 (t) are near F0 , F1 and F2 respectively, they meet as a regular complete intersection at: J (t) = {F0 (t) = F1 (t) = F2 (t) = 0} = Va (t) ∩ {F2 (t) = 0}. Hence J (t)∩{ Fˆ1 (t) = 0} = Vc (t)∩ Va (t) = I (t), which implies that I (t) ⊂ J (t). ν3 In the case n = 3, once I (t) and J (t) have αβγ points, we have that I (t) = J (t) for all t ∈ D  . In the case n ≥ 4, both sets are codimension-three smooth and connected submanifolds of Pn (Lefschetz theorem), implying again that I (t) = J (t). In particular, we obtain that   I (t) = {F0 (t) = F1 (t) = F2 (t) = 0} ⊂ Fˆ j (t) = 0 , 0 ≤ j ≤ 2.

123

36

W. Costa e Silva

Using Noether’s Theorem (see Cerveau and Lins-Neto 1994; [Lins Neto 2007, p. 86]) and the fact that all polynomials involved are homogeneous, we have Fˆ1 (t) ∈ < F0 (t), F1 (t), F2 (t) >. Since deg{F0 (t)} > deg{F1 (t)} > deg{F2 (t)}, we conclude that Fˆ1 (t) = F1 (t) + g(t)F2 (t), where g(t) is a homogeneous polynomial of degree deg{F1 (t)} − deg{F2 (t)}. Moreover note that Vc (t) = V { Fˆ1 (t), F2 (t)} = V {F1 (t), F2 (t)}, where V {H1 , H2 } denotes the projective algebraic variety defined by {H1 = H2 = 0}. Similarly for Vb (t) we have that Fˆ2 (t) ∈ < F0 (t), F1 (t), F2 (t) >. On the other hand, since Fˆ2 (t) has the lowest degree, we can assume that Fˆ2 (t) = F2 (t). In an analogous way we have that Fˆ0 (t) = F0 (t) + m(t)F1 (t) + n(t)F2 (t) for the polynomial Fˆ0 (t). Now observe that V { Fˆ0 (t), Fˆ2 (t)} = V {F0 (t)+m(t)F1 (t), F2 (t)}. γ β Hence we can define f t = {P0α (t), P1 (t), P2 (t)} where P0 (t) = F0 (t) + m(t)F1 (t), P1 (t) = F1 (t) and P2 (t) = F2 (t). This defines a family of mappings ( f t )t∈D  : P3 P2 , and Va (t), Vb (t) and Vc (t) are fibers of f t for fixed t. Observe that, for   sufficiently small, ( f t )t∈D  is generic in the sense of Definition 2.2, and its indeterminacy locus I ( f t ) is precisely I (t). Furthermore, since Gen(n, ν, α, β, γ ) is open, we can suppose that this family ( f t )t∈D  is in it. Henceforth we will only consider the situation n = 3. In the local coordinates X (t) = (x0 (t), x1 (t), x2 (t)) near some point of I (t) the local expression of the map f t can be written as: P0 (t) = u 0t x0 (t) + x1 (t)x2 (t)h 0t , P1 (t) = u 1t x1 (t) + x0 (t)x2 (t)h 1t and P2 (t) = u 2t x2 (t) + x0 (t)x1 (t)h 2t , where the functions u it ∈ O∗ (C3 , 0) and h it ∈ O(C3 , 0) for 0 ≤ i ≤ 2. Note that ∀i, limt→0 h it = 0. We want to show that an orbit of the C∗ -action   βγ αγ αβ (6) (x0 , x1 , x2 ) → s x0 , s x1 , s x2 that extends globally as a singular curve of the foliation Ft is a fiber of f t . An orbit that is not contained in any coordinate plane will called generic orbit. Recall that the condition α < β < γ implies that αγ < β(α + γ ) and αβ < γ (α + β). Lemma 4.6 If βγ ≤ α(β + γ ) then any generic orbit of the C∗ -action given by the expression 6 that extends globally as singular curve of the foliation Ft is also a fiber of f t for fixed t. Proof In order to simplify we will omit the parameter t. A generic orbit δ(s) can be parametrized as s → (m 0 s βγ , m 1 s αγ , m 2 s αβ ); m 0 m 1 m 2 = 0. Without loss of generality, we can suppose that m 0 = m 1 = m 2 = 1. We have α  β  βγ α(β+γ ) αγ β(α+γ ) h0 : s u1 + s h1 : f t (δ(s)) = s u 0 + s γ  . s αβ u 2 + s γ (α+β) h 2 The condition βγ ≤ α(β + γ ) enables us to extract s αβγ from f t (δ(s)). Hence we obtain (7) f t (δ(s)) = [(u 0 + s k h 0 )α : (u 1 + s l h 1 )β : (u 2 + s m h 2 )γ ]

123

Stability of Branched Pull-Back Projective Foliations

37

where k = α(β + γ ) − βγ , l = β(γ + α) − αγ and m = γ (α + β) − αβ. Since Vτ is a fiber of f , f 0 (Vτ ) = [d : e : f ] ∈ P2 with d.e. f = 0. If we take a covering ν3 of I ( f ) = { p1 , . . . , p j , . . . , p ν 3 } by small open balls B j ( p j ), 1 ≤ j ≤ αβγ , αβγ

the set Vτ \ ∪ j B j ( p j ) is compact. For a small deformation f t of f 0 we have that f t [Vτ (t)\ ∪ j B j ( p j )(t)] stays near f [Vτ \ ∪ j B j ( p j )]. Hence for t sufficiently small the components of expression 7 do not vanish both inside as well as outside of the neighborhood ∪ j B j ( p j )(t). This is possible only if f t is constant along these curves. In fact, f t (Vτ (t)) is either a curve or a point. If it is a curve then it cuts all lines of P2 and therefore the components should be zero somewhere. Hence f t (Vτ (t)) is constant 

and we conclude that Vτ (t) is a fiber. When βγ > α(β + γ ) the situation requires more detail. Let us suppose that the orbits contained in the coordinate planes that extends globally as singular curves of the foliation Ft are fibers of f t . Without loss of generality, we can suppose that the orbit that we will consider is contained in x0 (t) = 0. In this case it can be written as γ β (x0 = x1 − cx2 = 0). We have that the germ of f 0,t at the point p j (t) belongs to the γ β ideal generated by x0 (t) and (x1 − cx2 )(t). Hence we can write the function h 0t from the expression P0 (t) = u 0t x0 (t) + x1 (t)x2 (t)h 0t as γ

β

h 0t = x0 (t)h 01t + (x1 (t) − cx2 (t))h 02t where h 01t , h 02t ∈ O2 . In this way, we can repeat the argument of Lemma 4.6 and extract the factor s αβγ and the result follows. Therefore to complete the proof for the situation βγ > α(β + γ ) we have to prove the following: Lemma 4.7 If βγ > α(β +γ ) then any orbit of the C∗ -action given by the expression 6 which is contained in some coordinate plane at p j (t) and that extends globally as a singular curve of the foliation Ft is a fiber of the mapping f t for fixed t. To simplify the notation we will omit the index t in some expressions. Proof We can suppose that such a orbit can be parametrized as s → (0, s γ , s β ). After evaluating the mapping f t on this orbit we get: β

γ

f t (δ(s)) = [s α(β+γ ) h α0 : s βγ u 1 : s βγ u 2 ]. This can be written as γ β [s α(β+γ ) h˜0 : s βγ u 1 : s βγ u 2 ] = [X (s) : Y (s) : Z (s)].

(8)

Firstly we prove that f t (Vτ (t)) is contained in a line of the form (Y − λZ = 0) of P2 . Let us consider the meromorphic function with values in P1 given by gt (s) = Z (s) Y (s)

=

γ

u2

β

u1

. When s → 0 this function goes to a constant λ = 0, λ = ∞. Observe

that for small t the function

β

P1 γ P2

(t) : Vτ (t)\ ∪ j B j ( p j (t)) → P1 stays near

β

P1 γ P2

(0) :

123

38

W. Costa e Silva

Vτ (0)\ ∪ j B j ( p j (0)) → P1 . Note that since Vτ (0) is a fiber We conclude that f t (Vτ (t)) ⊂ (Y − λZ = 0)  If βγ > α(β + γ ) we can write Eq. 8 as

P1 .

β

P1 γ P2

(0) does not vanish.

γ

β

[h˜ 0 (s) : s m u 1 : s m u 2 ] where m = βγ − α(β + γ ). Observe that when s = 0 the function h˜ 0 (s) could vanish; in this case such a point corresponds to a indeterminacy point p j (t) of f t for some j. At p j (t) we can write the first component of Eq. 8 as h˜ 0 (s) = s ρ j h˜ j (s) where either h˜ j (s) ∈ O∗ (C, 0) or h˜ 0 ≡ 0. However, in the second case we are done, that is, Vτ (t) is a fiber of f t . At p j (t) we have two possibilities: First case: ρ j < m. In this case we can write Eq. 8 as γ β (9) [h˜ j (s) : s m−ρ j u 1 : s m−ρ j u 2 ]. If s → 0 the image goes to [1 : 0 : 0], hence f t |Vτ (t) ( p j (t)) = [1 : 0 : 0]. Second case: ρ j ≥ m. We can write Eq. 9 as β

γ

[s ρ j −m h˜ j (s) : u 1 : u 2 ].

(10)

If s → 0 the image goes to [a : λ : 1] where a ∈ C. This is due to the fact that the image of such a point belongs to the curve (Y − λZ = 0)  P1 and hence we can write it as [a : λ : 1]. Suppose that f t |Vτ (t) is not constant and consider the mapping f t |Vτ (t) : Vτ (t) → f t (Vτ (t)) ⊂ (Y −λZ = 0) for fixed t. Denote by Q = { j|ρ j < m}. Note that p ∈ Vτ (t) and f t |Vτ (t) ( p) = [1 : 0 : 0] imply that p = p j (t) for some j ∈ Q; that is, ( f t |Vτ (t) )−1 [1 : 0 : 0] = { p j (t), j ∈ Q}. Moreover, by Eq. 10 we have mult ( f t |Vτ (t) , p j (t)) = m − ρ j . In particular, the degree of f t |Vτ (t) is deg( f t |Vτ (t) ) =

 (m − ρ j ). j

On the other hand, if p ∈ ( f t |Vτ (t) )−1 [0 : λ : 1] then (P0α ( p) = 0) and so mult ( f t |Vτ (t) , p) is equal to the intersection number of (P0α (t) = 0) and Vτ (t) at p. Hence deg( f t |Vτ (t) ) = Vτ (t).P0α (t) = deg(Vτ (t)) × deg(P0α (t)) =

 ν3 = (m − ρ j ). α j

But (m − ρ j ) ≤ m = βγ − α(β + γ ) and so 

(m − ρ j ) ≤ #Q × m ≤

j∈Q

  1 1 1 ν3 × (βγ − α(β + γ )) = ν 3 − − αβγ α β γ

which implies that α1 ≤ α1 − β1 − γ1 and we arrive to a contradiction. Therefore, Q = ∅,

 f t |Vτ (t) is a constant and Vτ (t) is a fiber of f t .

123

Stability of Branched Pull-Back Projective Foliations

39

Remark 4.8 The arguments of lemma 4.7 hold only in dimension 3, besides this fact, they are sufficient to prove our main theorem, since the general case follows from the case n = 3. 4.3.2 Part 2 Let us now define a family of foliations (Gt )t∈D , Gt ∈ M(d) (see Proposition 3.2) such that Ft = f t∗ (Gt ) for all t ∈ D . Firstly we consider the case n = 3. Let Mw (t) be the family of complex algebraic threefolds obtained from P3 by blowing-up with weights ν3 w := (βγ , αγ , αβ) at the αβγ points p1 (t), . . . , p j (t), . . . , p ν 3 (t) corresponding to αβγ

I (t) of Ft ; and denote by πw (t) : Mw (t) → P3

the blowing-up map. The exceptional divisor of πw (t) consists of (t)−1 ( p

ν3 αβγ

ν3 αβγ

orbifolds

E j (t) = πw , which are weighted projective planes of type j (t)), 1 ≤ j ≤ 2 Pw (for more detail see Martín-Morales (2013) ex. 3.6 p 957). More precisely, if we blow-up Ft at the point p j (t), then the restriction of the strict transform πw∗ Ft to the exceptional divisor E j (t) = P2w is the same quasi-homogeneous 1-form that defines Ft at the point p j (t)(see Eq. 4). On the other hand, [Iano-Fletcher (2000), Lemma 5.7, p. 106] ensures that each exceptional divisor E j (t) = P2w is isomorphic to P2 , provided we impose the following arithmetical conditions in α, β and γ : (1) α = 1 and β and γ are relatively prime numbers, or (2) α > 1 and α, β and γ are pairwise relatively prime numbers. We denote these equivalences as E j (t) = P2w  P2 . Now we will prove a lemma which ensures that the deformations ηt of η0 also have invariant hyperplanes. Lemma 4.9 In case (2) above the deformation ηt of η0 leaves invariant the coordinate hyperplanes (x0 x1 x2 = 0). In case (1) above the deformation ηt of η0 leaves invariant the coordinate hyperplanes (x1 x2 = 0). Proof Let us consider case (2). The holonomy map of the x2 -axis at x2 = 1 is  H (x0 , x1 ) = e

2iπ γα

.x0 , e

2iπ γβ

 .x1 .

We will prove that this holonomy map leaves invariant the foliation ηt |(x2 =1) . Let us write the foliation ηt |(x2 =1) as a vector field Yt , ηt |(x2 =1) = i Yt (d x0 ∧ d x1 ). β β When t = 0 we have that η0 |(x2 =1) = αx0 A(x0α , x1 , 1)d x0 + βx1 A(x0α , x1 , 1)d x1 β β and so Y0 = βx1 A(x0α , x1 , 1) ∂∂x0 −αx0 A(x0α , x1 , 1) ∂∂x1 , so that H ∗ Y0 = Y0 . It follows ∗ that H Yt = Yt . Let us prove that the axis (x0 = 0) is Yt -invariant. If not, then the first component of Yt has a monomial of the type x1n ∂∂x0 , for which H ∗ (x1n ∂∂x0 ) = e

γ 2iπ ( nγ β −α)

(x1n ∂∂x0 ). But this would imply that ( nγ β −

γ α)

∈ Z which is impossible,

123

40

W. Costa e Silva

because (α, β) = (α, γ ) = (β, γ ) = 1. Similarly, the axes (x1 = 0) and (x2 = 0) are 

ηt invariant. Case (1) is analogous. Now we see that we can push-forward the foliation to P2 . In this way, we get a family of degree d foliations, all of them leaving invariant the lines (X Y Z = 0) if α > 1 or (Y Z = 0) when α = 1. These foliations will be the candidates to be a deformation of G0 . In fact, since M(d) is an open set we can suppose that this family is inside M(d). We fix the exceptional divisor E 1 (t) to work with and we denote by Gt the restriction P2 , t ∈ D  defined of πw∗ Ft to E 1 (t). Consider the family of mappings f t : P3 in Proposition 4.5. We will consider the family ( f t )t∈D as a family of rational maps E 1 (t); we decrease  if necessary. Note that the map f t : P3 f t ◦ πw (t) : Mw (t)\ ∪ j E j (t) → E 1 (t) extends holomorphically, that is, as an orbifold mapping, to fˆt : Mw (t) → E 1 (t). This is due to the fact that each orbit of the action given by the Expression 6 determines an equivalence class in P2w and is a fiber of the map β

γ

(x0 (t), x1 (t), x2 (t)) → (x0α (t), x1 (t), x2 (t)). The mapping f t can be interpreted as follows. Each fiber of f t meets p j (t) once, which implies that each fiber of fˆt cuts E 1 (t) once outside of the three singular curves in [Mw (t) ∩ E 1 (t)]. Since Mw (t)\ ∪ j E j (t) is biholomorphic to P3 \I (t), after identifying E 1 (t) with P2w , we can imagine that if q ∈ Mw (t)\ ∪ j E j (t) then fˆt (q) is the intersection point of the fiber fˆt−1 ( fˆt (q)) with E 1 (t). We obtain a mapping fˆt : Mw (t) → P2 . It can be extended over the singular set of Mw (t) using Riemann’s Extension Theorem. We shall also denote this extension by fˆt to simplify the notation. We remark, that the blowing-up with weights w := (βγ , αγ , αβ) solves totally the indeterminacy set of f t for each t. With all these ingredients we can define the foliation F˜ t = f t∗ (Gt ) ∈ α,β,γ P B(ν,d , n). This foliation is a deformation of F0 . Based on the previous discussion let us denote F1 (t) = πw (t)∗ (Ft ) and Fˆ1 (t) = πw (t)∗ (F˜ t ). We recall that in a neighborhood of p1 (t) ∈ I (t), Ft is represented by the quasihomogeneous 1-form η (see Eq. 4), we shall denote it by η p1 (t) . Lemma 4.10 If F1 (t) and Fˆ1 (t) are the foliations defined previously, we have that F1 (t)| E 1 (t) = Gˆt = Fˆ1 (t)| E 1 (t)

123

Stability of Branched Pull-Back Projective Foliations

41

where Gˆt is the foliation induced on E 1 (t)  P2w by the quasi-homogeneous 1-form η p1 (t) . Proof This 1-form is invariant by the C∗ -action defined by the Expression 6 and therefore naturally defines a foliation on the weighted projective space E 1 (t)  P2w . This proves the first equality. The second equality follows from the geometrical interpreta

tion of the mapping fˆt : Mw (t) → P2w  P2 , since Fˆ1 (t) = f 1 (t)∗ (Gt ). Now we use the fact that P2w  P2 to obtain the equality Gt = F1 (t)| E 1 (t) = Fˆ1 (t)| E 1 (t) . Let τ1 (t) be a singularity of Gt outside the three invariant straight lines. Since the map t → τ1 (t) ∈ P2 is holomorphic, there exists a holomorphic family of automorphisms of P2 , t → H (t) such that τ1 (t) is kept fixed. Observe that such a singularity has no algebraic separatrix at this point. Fix a local analytic coordinate system (xt , yt ) at τ1 (t) such that the local separatrices are (xt = 0) and (yt = 0), respectively. Observe that the local smooth hypersurfaces along Vˆτ1 (t) = fˆt−1 (τ1 (t)) defined by Xˆ t := (xt ◦ fˆt = 0) and Yˆt := (yt ◦ fˆt = 0) are invariant for Fˆ1 (t). Furthermore, they meet transversely along Vˆτ1 (t) . On the other hand, Vˆτ1 (t) is also contained in the Kupka set of F1 (t). Therefore there are two local smooth hypersurfaces X t := (xt ◦ fˆt = 0) and Yt := (yt ◦ fˆt = 0) invariant for F1 (t) such that: (1) X t and Yt meet transversely along Vˆτ1 (t) . (2) X t ∩ πw (t)−1 ( p1 (t)) = (xt = 0) = Xˆ t ∩ πw (t)−1 ( p1 (t)) and Yt ∩ πw (t)−1 ( p1 (t)) = (yt = 0) = Yˆt ∩ πw (t)−1 ( p1 (t)) (because F1 (t) and Fˆ1 (t) coincide on E 1 (t)). (3) X t and Yt are deformations of X 0 = Xˆ 0 and Y0 = Yˆ0 , respectively. Lemma 4.11 X t = Xˆ t for small t. Proof Let us consider the projection fˆt : Mw (t) → E 1 (t) on a neighborhood of the regular fibre Vˆτ1 (t) , and fix local coordinates xt , yt on E 1 (t) such that X t := (xt ◦ fˆt = ˆ  = Xˆ t ∩ H are (vertical) compact 0). For small , let H = (yt ◦ fˆt = ). Thus  ˆ ˆ  s, are ˆ curves, deformations of 0 = Vτ1 (t) . Set  = X t ∩ Hˆ  . The  s, as the  ˆ compact curves (for t and  small), since X t and X t are both deformations of the same X 0 . Thus for small t, X t is close to Xˆ t . It follows that fˆt ( ) is an analytic curve contained in a small neighborhood of τ1 (t), for small . By the maximum principle, we must have that fˆt ( ) is a point, so that fˆt (X t ) = fˆt (∪  ) is a curve C, that is, X t = fˆt−1 (C). But X t and Xˆ t intersect the exceptional divisor E 1 (t)  P2 along the separatrix (xt = 0) of Gt through τ1 (t). This implies that X t = fˆt−1 (C) = fˆt−1 (xt = 

0) = Xˆ t . The algebraic foliations Ft and F˜t have a common transcendental leaf and therefore they are equal. This proves Theorem A in the case n = 3. Suppose now that n ≥ 4. The previous argument implies that if ϒ is a generic ˆ |ϒ . In fact, such planes cut transversely every 3−plane in Pn , we have F(t)|ϒ = F(t)

123

42

W. Costa e Silva

ν strata of the singular set, and I (t) consists of αβγ points. This implies that f t is generic for |t| sufficiently small. We can then repeat the previous argument, finishing the proof of Theorem A. 3

4.3.3 Pull-Back Foliations That can be Characterized by Their Singular Locus In Cerveau and Lins-Neto (1994), the authors have proved that a foliation F on Pn , n ≥ 3 whose Kupka set K (F) contains a codimension two smooth irreducible component, say , which is a complete intersection, has a rational first integral. Moreover, in this case, K (F) = . An analogous result for a particular class of pull-back foliations, was stated in [Cerveau et al. (2001), Theorem B, p.709]. Using the Lemma 4.11 a similar result to [Cerveau et al. (2001), Theorem B, p.709] can be proved for foliations F on Pn , n ≥ 3 which are pull-back of foliations on P2 preserving the three lines (X Y Z ) = 0. Recall from Definition 2.2 the concept of a generic map. Let f ∈ B R M (n, ν, α, β, γ ), I ( f ) its indeterminacy locus and F a foliation on Pn , n ≥ 3. Consider the following properties: P1 : In any point p j ∈ I ( f ) the foliation F has the following local description: there exists an analytic coordinate system (U p j , Z p j ) around p j such that Z p j ( p j ) = 0 ∈ (Cn , 0) and F|(U p j ,Z p j ) can be represented by a quasihomogeneous singularity of type (βγ , αγ , αβ; αβγ (d − 1)) where (a) α = 1 and β and γ are relatively prime numbers, or (b) α > 1 and α, β and γ are pairwise relatively prime numbers. described in Sect. 4.1.1. P2 : There exists a fibre f −1 (q) = V (q) such that V (q) = f −1 (q)\I ( f ) is i=2 (Fi = 0). contained in the Kupka-Set of F and V (q) is not contained in i=0 P3 : V (q) has transversal type X , where X is a germ of vector field on (C2 , 0) with a non algebraic invariant curve and such that 0 ∈ C2 is a non-degenerate / R. singularity with eigenvalues λ1 and λ2 , λλ21 ∈ Lemma 4.11 allows us to prove the following result: Theorem B In the conditions above, if properties P1 , P2 and P3 hold then F is a pull back foliation, F = f ∗ (G), where G is of degree d ≥ 2 on P2 leaving invariant the lines (X Y Z = 0) if α > 1 or the lines Y Z = 0 if α = 1.

5 Concluding Remarks Throughout the text we gave emphasis to the study of pull-back foliations that are obtained from foliations on P2 leaving invariant 3 straight lines. Also we focus on the hypothesis 1 ≤ α < β < γ . With minor modifications it is possible to extend the whole proof of Theorems A and B to the situation 1 = α = β < γ . Let us briefly explain in the next lines the changes that must be made in this broader context. We γ must necessarily consider maps f : Pn P2 given by f = (F0 , F1 , F2 ) where F0 , F1 and F2 are homogeneous polynomials without common factors satisfying deg(F0 ) = deg(F1 ) = γ .deg(F2 ) = ν ≥ 2. Consequently we need to work with foliations in P2

123

Stability of Branched Pull-Back Projective Foliations

43

leaving invariant just a straight line. Denote by I l1 (d, 2) the set holomorphic foliations on P2 of degree d ≥ 2 leaving invariant only the line Z = 0. In this case, a foliation G can be represented by a polynomial 1-form of the type  = Z A (X, Y, Z ) d X + Z B (X, Y, Z ) dY +C (X, Y, Z ) d Z ; the condition i R  = 0 implies X A+Y B+C = 0. For the set I l1 (d, 2) a similar property as in Proposition 3.2 holds, according to [Lins Neto (1998), Theorem 3, p. 385]. If f and G are generic (see Definition 4.1), the 1,1,γ pull-back foliation F = f ∗ G associated to the pair ( f, G) has degree ν,d = ν(d +   1,1,γ 1,1,γ 1 + γ1 ) − 1. Denote by P B(ν,d , n) the closure in Fol ν,d , n , n ≥ 3 of the set of foliations F of the form f ∗ G. We have that 1,1,γ

Theorem C P B(ν,d , n) is a unirational   1,1,γ Fol ν,d , n for all n ≥ 3, d ≥ 2 and ν ≥ 2.

irreducible

component

of

Once more this is possible because the space P2w , in this new situation, has weights for w = (γ , γ , 1). According to [Iano-Fletcher (2000), Lemma 5.7, p.106], it is also isomorphic to P2 and therefore we can follow the same ideas used previously to obtain similar results. With respect to comparisons on the geometry of a generic element: unlike all previ1,1,γ ously known examples, a generic point of our set P B(ν,d , n) has only one algebraic invariant hypersurface. Consequently these families of irreducible components are new. To obtain an analogous statement as in Theorem B as above, we only have to change the properties P1 and P2 as follows: P1 : In any point p j ∈ I ( f ) the foliation F has the following local description: there exists an analytic coordinate system (U p j , Z p j ) around p j such that Z p j ( p j ) = 0 ∈ (Cn , 0) and F|(U p j ,Z p j ) can be represented by a quasihomogeneous singularity of type (γ , γ , 1; γ (d − 1)) as described in Sect. 4.1.1. P2 : There exists a fibre f −1 (q) = V (q) such that V (q) = f −1 (q)\I ( f ) is contained in the Kupka set of F and V (q) is not contained in (F2 = 0). The property P3 remains unchanged. We also have the following: Theorem D In the conditions above, if properties P1 , P2 and P3 hold then F is a pull back foliation, F = f ∗ (G), where G is of degree d ≥ 2 on P2 that preserves the line Z = 0.

References Calvo-Andrade, O., Cerveau, D., Giraldo, L., Lins, A.: Neto. Irreducible components of the space of foliations associated to the affine Lie algebra. Ergodic Theor. Dynam. Syst. 24(4), 987–1014 (2004) Camacho, C., Lins, A.: Neto. The topology of integrable differential forms near a singularity. Inst. Hautes Études Sci. Publ. Math. 55, 5–35 (1982) Cerveau, D., Lins-Neto, A.: Holomorphic foliations in P2 having an invariant algebraic curve. Ann. Inst. Fourrier 41(4), 883–903 (1991) Cerveau, D., Lins, N.A.: Codimension one foliations in CPn , n ≥ 3, with Kupka components. Astérisque, 222 (1994), 93–132

123

44

W. Costa e Silva

Cerveau, D., Lins, A.: Neto. Irreducible components of the space of holomorphic foliations of degree two in Pn . Ann. Math. 143(2), 577–612 (1996) Cerveau, D., Lins, A.: Neto and S.J. Edixhoven. Pull-back components of the space of holomorphic foliations on CPn , n ≥ 3. J. Algebraic Geom. 10(4), 695–711 (2001) Gómez-Mont, X., Lins, A.: Neto. Structural stability of singular holomorphic foliations having a meromorphic first integral. Topology 30(3), 315–334 (1991) Iano-Fletcher, A.R.: Working with weighted complete intersections. Explicit birational geometry of 3-folds, Cambridge Univ. Press, Cambridge A. Corti, M. Reid (eds.) (2000), 101–173 Jouanolou, J.P.: Équations de Pffaf algèbriques. Lect. Notes in Math., 708, (1979) Kupka, I.: The singularities of integrable structurally stable Pfaffian forms. Proc. Nat. Acad. Sci. USA 52(6), 1431–1432 (1964) Lins, N.A.: Finite determinacy of germs of integrable 1-forms in dimension 3 (a special case). Geometric Dynamics, Springer Lect. Notes in Math. 1007(6), 480–497 (1981) Lins Neto, A.: Algebraic solutions of polynomial differential equations and foliations in dimension two. Holomorphic Dynam., Springer Lect. Notes in Math. 1345, 192–232 (1988) Lins, N.A.: Componentes Irredutíveis dos Espaços de Folheações. 26◦ Colóquio Brasileiro de Matemática, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, (2007) Lins, A., Neto, P.: Sad and B.A. Scárdua. On topological Rigidity of projective foliations. Bull. Soc. math. France 126, 381–406 (1998) Martín-Morales, J.: Monodromy Zeta function formula for embedded Q-resolutions. Rev. Mat. Iberoam. 29(3), 939–967 (2013) Sernesi. E.: Deformations of Algebraic Schemes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 334. Springer-Verlag, (2006)

123