On the geometry of Poincaré's problem for one-dimensional projective foliations

Abstracts

Consideramos o problema de relacionar carateres geométricos extrínsecos de uma variedade projetiva lisa e irredutível, que é invariante por uma folheação holomorfa de dimensão um de um espaço projetivo complexo, a objetos geométricos associados à folheação.

folheações holomorfas; variedades invariantes; classes polares; graus

On the geometry of Poincaré's problem for one-dimensional projective foliations

MARCIO G. SOARES^*

Departamento de Matemática, ICEx, UFMG - 31270-901 Belo Horizonte, Brazil

Manuscript received on June 28, 2001; accepted for publication on July 18, 2001.

ABSTRACT

We consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a one-dimensional holomorphic foliation on a complex projective space, to geometric objects associated to the foliation.

Key words: holomorphic foliations, invariant varieties, polar classes, degrees.

1 INTRODUCTION

S

Simple examples show that, when S is a dicritical separatrix of , the search for a positive solution to the problem is meaningless. The obstruction in this case was given by M. Brunella in (1997), and reads: the number c₁(N) - S ^. S may be negative if S is a dicritical separatrix (here, N is the normal bundle of the foliation). More than that, A. Lins Neto constructs, in (2000), some remarkable families of foliations on providing counterexamples for this problem, all involving singular separatrices and dicritical singularities.

However, as was shown in (Brunella 1997), when S is a non-dicritical separatrix, the number c₁(N) - S ^. S is nonnegative and, in , this means d⁰() + 2 d⁰(S), where d⁰() and d⁰(S) are the degrees of the foliation and of the curve, respectively. Another solution to the problem, in the non-dicritical case, was given by M.M. Carnicer in (1994), using resolution of singularities.

Let us now consider one-dimensional holomorphic foliations on , n 2, that is, morphisms : (m) T , m , m £ 1, with singular set of codimension at least 2. We write m = 1 - d⁰() and call d⁰() 0 the degree of . From now on we will consider d⁰() 2. This is the characteristic number associated to the foliation.

On the other hand, if we consider -invariant algebraic varieties

, it is natural to consider other characters associated to , not just its degree. This is the point of view we address. More precisely, we pose the question of relating extrinsic geometric characters of to geometric objects associated to .

This approach produces some interesting results. Let us illustrate the two-dimensional situation. Suppose we have an -invariant irreducible plane curve S. We associate to a tangency divisor (depending on a pencil ), which is a curve of degree d⁰() + 1 and contains the first polar locus of S. Computing degrees we arrive at d⁰(S) £ d⁰() + 2 in case S is smooth, and at d⁰(S)(d⁰(S) - 1) - ( - 1) £ (d⁰() + 1)d⁰(S) in case S is singular, where is the Milnor number of S at p. This allows us to recover a result of D. Cerveau and A. Lins Neto (1991), which states that if S has only nodes as singularities, then d⁰(S) £ d⁰() + 2, regardless of the singularities of being dicritical or non-dicritical.

In the higher dimensional situation, we obtain relations among polar classes of -invariant smooth varieties and the degree of the foliation.

2 THE TANGENCY DIVISOR OF WITH RESPECT TO A PENCIL

d⁰ tangency divisor

Choose affine coordinates (z₁,..., z_n) such that the hyperplane at infinity, with respect to these, is not -invariant, and let be a vector field representing , where 0 is homogeneous of degree d⁰() and Y_i(z₁,..., z_n) is a polynomial of degree £ d⁰(), 1 £ i £ n. Let H be a generic hyperplane in . Then, the set of points in H which are either singular points of or at which the leaves of are not transversal to H is an algebraic set, noted tang(H,), of dimension n - 2 and degree d⁰() (observe that g(z₁,..., z_n) = 0 is precisely tang(H,)).

DEFINITION. Consider a pencil of hyperplanes = {H_t}t , with axis L^{n - 2}. The tangency divisor of with respect to is

LEMMA 2.1.

is a (possibly singular) hypersurface of degree d⁰() + 1.

PROOF. Let p be a point in L^{n - 2}, the axis of the pencil. If p sing() then p is necessarily in , otherwise p is a regular point of . In this case, if is the leaf of through p, then either T_p

L^{n - 2} or, T_p together with L^{n - 2} determine a hyperplane H

, and hence we have p tang(H,)

, so that L^{n - 2}

. Now, let p L^{n - 2} be a regular point of and choose a generic line , transverse to L^{n - 2}, passing through p and such that L^{n - 2} and determine a hyperplane H_^b, distinct from H. This line meets at p and at d⁰() further points, counting multiplicities, corresponding to the intersections of with tang(H^b,). Hence has degree d⁰() + 1.

EXAMPLE. If we consider the two-dimensional Jouanolou's example

and the pencil = {(at, bt) : t , (a : b) }, a straightforward manipulation shows that is given, in homogeneous coordinates (X : Y : Z) in , by

3. -INVARIANT SMOOTH IRREDUCIBLE VARIETIES

Let us recall some facts about polar varieties and classes (Fulton 1984). If

is a smooth irreducible algebraic subvariety of , of dimension n - k, and L^{k + j - 2} is a linear subspace, then the j-th polar locus of is defined by

for 0 £ j £ n - k. If L^{k + j - 2} is a generic subspace, the codimension of _j() in is precisely j. The j-th class, (), of is the degree of _j() and, since the cycle associated to _j() is

we have

LEMMA 3.1. Let be a smooth irreducible algebraic variety of dimension n - k, -invariant and not contained in sing(). Then

PROOF. Let us first assume is a linear subspace of . In this case _j = , for j 1, so the first assertion of the lemma is meaningless. Assume then is not a linear subspace and choose a pencil of hyperplanes = {H_t}t , with axis L^{n - 2} generic, so that codim(_{n - k}(),) = n - k. If q

_{n - k}(), then T_q meets L^{n - 2} in a subspace W of dimension at least n - k - 1. If T_q

L^{n - 2} then any hyperplane H_t

contains T_q, if not, a line

T_q,

L^{n - 2},

W consisting of a point determines, together with L^{n - 2}, a hyperplane H_t

such that T_q

H_t. Since is -invariant, we have T_q

T_q

H_t, in case q is not a singular point of , where is the leaf of through q. This implies q tang(H_t,)

, so that _{n - k}()

. Also, it follows from the definition of that is not contained in it.

THEOREM I. Let be a one-dimensional holomorphic foliation on of degree d⁰() 2, with singular set of codimension at least 2, and let be an -invariant smooth irreducible algebraic variety, of dimension n - k, which is not a linear subspace of , and not contained in sing(). Suppose _{n - k - j}()

but _{n - k - j - 1}()

, for some 0 £ j £ n - k - 1. Then

PROOF. Observe that we may assume

_{n - k - j}()

_{n - k - j - 1}() and hence

Bézout's Theorem then gives

COROLLARY 1. Let

sing() be a smooth irreducible complete intersection in , which is not a linear subspace, defined by F₁ = 0,..., F_k = 0 where F

[z₀,..., z_n] is homogeneous of degree d, 1 £ £ k and -invariant, where is as in Theorem I. If _{n - k - j}()

but _{n - k - j - 1}()

then

where is the Wronski (or complete symmetric) function of degree in k variables

Observe that if is a smooth irreducible hypersurface, this reads d⁰() + 2 d⁰(). In (Soares 1997) we showed d⁰() + 1 d⁰(), but assumed to be a non-degenerate foliation on .

Also, in (Soares 2000) the following estimate is obtained, provided n - k is odd and is non-degenerate: if 1 £ k £ n - 2 then

We remark that this estimate is sharper than that given in Corollary 1.

4. THE TWO-DIMENSIONAL CASE

As pointed out in Corollary 1, whenever we have a smooth irreducible -invariant plane curve S, the relation d⁰(S) £ d⁰() + 2 holds because (S) = d⁰(S)(d⁰(S) - 1), regardless of the nature of the singularities of , provided sing() has codimension two.

In order to treat the case of arbitrary irreducible -invariant curves, let us recall the definition (see R. Piene 1978) of the class of a (possibly singular) irreducible curve S in . We let S_reg denote the regular part of S and, for a generic point p in , we consider the subset of S_reg consisting of the points q such that p T_qS_reg. The closure ₁ of in S is the first polar locus of S, and the class (S) of S is its degree. ₁ is a subvariety of codimension 1 whose degree is given by Teissier's formula (Teissier 1973):

where the summation is over all singular points q of S, m_q denotes the Milnor number of S at q and m_q denotes the multiplicity of S at q. Because ₁ is a finite set of regular points in S, revisiting Lemma 3.1 we conclude:

₁

S.

Also, sing(S) Í sing(), so that

sing(S)

S

and hence

₁

sing(S)

S.

It follows from Bézout's theorem that

(S) + m_q (d⁰() + 1)d⁰(S)

Therefore we obtain the

THEOREM II. Let S be an irreducible curve, of degree d⁰(S) > 1, invariant by a foliation on , of degree d⁰() 2 with sing() of codimension 2. Then

where the summation extends over all singular points q of S.

This gives at once the following result, first obtained by Cerveau and Lins Neto (1991);

COROLLARY 2. If all the singularities of S are ordinary double points (so that m_q= 1) then

d⁰(S) d⁰() + 2.

Theorem II illustrates one obstruction to solving Poincaré's problem in general, since we cannot estimate the sum å_q(m_q- 1) when dicritical singularities are present. However, if S is an irreducible -invariant algebraic curve, which is a non-dicritical separatrix, then it follows from (Brunella 1997) that

where the sum is over all singular points q of S, B₁^q,..., Br_q^q are the analytic branches of S at q, and GSV denotes the Gomez-Mont/Seade/Verjovsky index.

REMARK. Let S be a non-dicritical separatrix of , so that d⁰(S) £ d⁰() + 2. Assume equality holds in the expression in Theorem II, which amounts to

Hence we conclude d⁰(S) = d⁰() + 2 and S has only ordinary double points as singularities.

5. -INVARIANT SMOOTH IRREDUCIBLE CURVES

Ssing

d₁ + ^... + d_{n - 1}

d⁰() + n

so that

d⁰(S)

provided codim sing() 2. In the general case we have:

COROLLARY 3. Let S sing() be an -invariant smooth irreducible curve of degree d⁰(S) > 1, where is a one-dimensional holomorphic foliation on of degree d⁰() 2, with singular set of codimension at least 2. Then the first class (S) of S satisfies

(S) (d⁰() + 1)d⁰(S),

the geometric genus g of S satisfies

g + 1.

Also, if N(, S) is the number of singularities of along S, then

N(, S) (d⁰() + 1)d⁰(S).

PROOF. Since S is a curve which is not a line, we have to consider only (S) = d⁰(S) and (S). The first inequality follows immediately from Theorem I. To bound the genus we observe that Lefschetz' theorem on hyperplane sections (Lamotke 1981) gives

(S) = 2d⁰(S) + 2g - 2

and the second inequality follows. On the other hand, since S is irreducible and not contained in sing(), Whitney's finiteness theorem for algebraic sets (Milnor 1968) implies that S sing() is connected, and hence N(, S) is necessarily finite. Also,

sing() S Ì

S

and Bézout's theorem implies

N(, S) (d⁰() + 1)d⁰(S).

The first class of a smooth irreducible curve S in was calculated by R. Piene (1976), and is as follows:

(S) = 2(d⁰(S) + g - 1) -

where g is the genus of S and 0 is an integer, called the 0 - th stationary index. It follows from Theorem I that:

COROLLARY 4. With the same hypothesis of Corollary 3

2d⁰(S) - (S) - (d⁰() + 1)d⁰(S).

REMARK ON EXTREMAL CURVES. We can obtain an estimate for d⁰(S) in terms of d⁰() and n 3, provided S is non-degenerate (that is, is not contained in a hyperplane) and extremal (that is, the genus of S attains Castelnuovo's bound). Recall that, for S a smooth non-degenerate curve in of degree d⁰(S) 2n, Castenuovo's bound is (Arbarello et al. 1985):

g(n - 1) + m,

where

d⁰(S) - 1 = m(n - 1) + .

The inequality

g + 1

together with S extremal give, performing a straightforward manipulation:

d⁰(S) 2(d⁰() - 1)(n - 1) + .

ACKNOWLEGMENTS

RESUMO

Palavras-chave: folheações holomorfas, variedades invariantes, classes polares, graus.

LINS NETO A. 2000. Some examples for Poincaré and Painlevé problems. Pre-print IMPA.

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Publication Dates

• Publication in this collection
  12 Dec 2001
• Date of issue
  Dec 2001

History

• Accepted
  18 July 2001
• Received
  28 June 2001