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

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.


INTRODUCTION
H. Poincaré treated, in (1891), the question of bounding the degree of an algebraic curve, which is a solution of a foliation F on P 2 C with rational first integral, in terms of the degree of the foliation.This problem has been considered more recently in the following formulation: to bound the degree of an irreducible algebraic curve S, invariant by a foliation F on P 2 C , in terms of the degree of the foliation.
Simple examples show that, when S is a dicritical separatrix of F, 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 S c 1 (N F ) − S • S may be negative if S is a dicritical separatrix (here, N F is the normal bundle of the foliation).More than that, A. Lins Neto constructs, in (2000), some remarkable families of foliations on P 2 C 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 S c 1 (N F )−S •S is nonnegative and, in P 2 C , this means d 0 (F)+2 ≥ d 0 (S), where d 0 (F) and d 0 (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 , with singular set of codimension at least 2. We write m = 1 − d 0 (F) and call d 0 (F) ≥ 0 the degree of F. From now on we will consider d 0 (F) ≥ 2. This is the characteristic number associated to the foliation.
On the other hand, if we consider F-invariant algebraic varieties V i −→ P n C , it is natural to consider other characters associated to V, not just its degree.This is the point of view we address.More precisely, we pose the question of relating extrinsic geometric characters of V to geometric objects associated to F.
This approach produces some interesting results.Let us illustrate the two-dimensional situation.Suppose we have an F-invariant irreducible plane curve S. We associate to F a tangency divisor D H (depending on a pencil H), which is a curve of degree d 0 (F) + 1 and contains the first polar locus of S. Computing degrees we arrive at d 0 (S) ≤ d 0 (F) + 2 in case S is smooth, and at d 0 (S)(d 0 (S) − 1) − p∈sing(S) (µ p − 1) ≤ (d 0 (F) + 1)d 0 (S) in case S is singular, where µ p 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 d 0 (S) ≤ d 0 (F) + 2, regardless of the singularities of F being dicritical or non-dicritical.
In the higher dimensional situation, we obtain relations among polar classes of F-invariant smooth varieties and the degree of the foliation.

THE TANGENCY DIVISOR OF F WITH RESPECT TO A PENCIL
Let F be a one-dimensional holomorphic foliation on P n C of degree d 0 (F) ≥ 2, with singular set of codimension at least 2. We associate a tangency divisor to F as follows: Choose affine coordinates (z 1 , . . ., z n ) such that the hyperplane at infinity, with respect to these, is not F-invariant, and let X = gR + n i=1 Y i ∂ ∂z i be a vector field representing F, where Then, the set of points in H which are either singular points of F or at which the leaves of F are not transversal to H is an algebraic set, noted tang(H, F), of dimension n − 2 and degree

Definition. Consider a pencil of hyperplanes
Proof.Let p be a point in L n−2 , the axis of the pencil.If p ∈ sing(F) then p is necessarily in D H , otherwise p is a regular point of F. In this case, if L is the leaf of F through p, then either and hence we have An.Acad. Bras. Cienc., (2001) be a regular point of F and choose a generic line , transverse to L n−2 , passing through p and such that L n−2 and determine a hyperplane H β , distinct from H α .This line meets D H at p and at d 0 (F) further points, counting multiplicities, corresponding to the intersections of with tang(H β , F). Hence D H has degree d 0 (F) + 1.
Example.If we consider the two-dimensional Jouanolou's example and the pencil Let us recall some facts about polar varieties and classes (Fulton 1984).
C , of dimension n − k, and L k+j −2 is a linear subspace, then the j-th polar locus of V is defined by is a generic subspace, the codimension of P j (V) in V is precisely j .The j-th class, j (V), of V is the degree of P j (V) and, since the cycle associated to P j (V) is Lemma 3.1.Let V be a smooth irreducible algebraic variety of dimension n − k, F-invariant and not contained in sing(F).Then Proof.Let us first assume V is a linear subspace of P n C .In this case P j = ∅, for j ≥ 1, so the first assertion of the lemma is meaningless.Assume then V is not a linear subspace and choose a pencil in case q is not a singular point of F, where L is the leaf of F through q.This implies q ∈ tang(H t , F) ⊂ D H , so that P n−k (V) ⊂ D H . Also, it follows from the definition of D H that V is not contained in it.
Theorem I. Let F be a one-dimensional holomorphic foliation on P n C of degree d 0 (F) ≥ 2, with singular set of codimension at least 2, and let V be an F-invariant smooth irreducible algebraic variety, of dimension n − k, which is not a linear subspace of P n C , and not contained in sing(F).
Proof.Observe that we may assume P n−k−j (V) ⊂ P n−k−j −1 (V) and hence ...,d k ) ⊆ sing(F) be a smooth irreducible complete intersection in P n C , which is not a linear subspace, defined by where W (k)  δ is the Wronski (or complete symmetric) function of degree δ in k variables Observe that if V is a smooth irreducible hypersurface, this reads d 0 (F) + 2 ≥ d 0 (V).In (Soares 1997) we showed d 0 (F) + 1 ≥ d 0 (V), but assumed F to be a non-degenerate foliation on Also, in (Soares 2000) the following estimate is obtained, provided n − k is odd and ...,d k ) ) We remark that this estimate is sharper than that given in Corollary 1.

THE TWO-DIMENSIONAL CASE
As pointed out in Corollary 1, whenever we have a smooth irreducible F-invariant plane curve S, the relation d 0 (S) ≤ d 0 (F) + 2 holds because 1 (S) = d 0 (S)(d 0 (S) − 1), regardless of the nature of the singularities of F, provided sing(F) has codimension two.
In order to treat the case of arbitrary irreducible F-invariant curves, let us recall the definition (see R. Piene 1978) of the class of a (possibly singular) irreducible curve S in P 2 C .We let S reg denote the regular part of S and, for a generic point p in P 2 C , we consider the subset Q of S reg consisting of the points q such that p ∈ T q S reg .The closure P 1 of Q in S is the first polar locus of S, and the class 1 (S) of S is its degree.P 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, µ q denotes the Milnor number of S at q and m q denotes the multiplicity of S at q.Because P 1 is a finite set of regular points in S, revisiting Lemma 3.1 we conclude: Also, sing(S) ⊆ sing(F), so that sing(S) ⊆ D H ∩ S and hence It follows from Bézout's theorem that Therefore we obtain the Theorem II.Let S be an irreducible curve, of degree d 0 (S) > 1, invariant by a foliation F on P 2 C , of degree d 0 (F) ≥ 2 with sing(F) of codimension 2. Then where the summation extends over all singular points q of S.
An. Acad.Bras.Cienc., (2001) 73 (4) 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 µ q = 1) then Theorem II illustrates one obstruction to solving Poincaré's problem in general, since we cannot estimate the sum q (µ q −1) when dicritical singularities are present.However, if S is an irreducible F-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, B q 1 , . . ., B q r 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 F, so that d 0 (S) ≤ d 0 (F) + 2. Assume equality holds in the expression in Theorem II, which amounts to Hence we conclude d 0 (S) = d 0 (F) + 2 and S has only ordinary double points as singularities.

F -INVARIANT SMOOTH IRREDUCIBLE CURVES
We have the following immediate consequence of Corollary 1: if we consider an F-invariant smooth one-dimensional complete intersection S = V n−(n−1) (d 1 ,...,d (n−1) ) ⊂ sing(F), then In the general case we have: Proof.Since S is a curve which is not a line, we have to consider only 0 (S) = d 0 (S) and 1 (S).
The first inequality follows immediately from Theorem I. To bound the genus we observe that Lefschetz' theorem on hyperplane sections (Lamotke 1981) gives The first class of a smooth irreducible curve S in P n C was calculated by R. Piene (1976), and is as follows: where g is the genus of S and κ 0 ≥ 0 is an integer, called the 0 − th stationary index.It follows from Theorem I that: Remark on Extremal Curves.We can obtain an estimate for d 0 (S) in terms of d 0 (F) 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 P n C of degree d 0 (S) ≥ 2n, Castenuovo's bound is (Arbarello et al. 1985):