VECTOR FIELDS, INVARIANT VARIETIES AND LINEAR SYSTEMS

JORGE VITÓRIO PEREIRA

Instituto de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina, 110 Jardim Botânico, 22460-320 - Rio de Janeiro, RJ, Brasil. email : jvp@impa.br

Abstract. We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criteria for the existence of rational first integrals of a given degree, bounds for the number of first integrals on families of vector fields and a generalization of Darboux’s criteria in the spirit of [10]. We also provide a new proof of Gomez–Mont’s result on foliations with all leaves algebraic, see [6].

1. Introduction

When studying algebraic curves one of the most fruitfull concept is the one of inflection and higher order inflection points. For a smooth plane curve, i.e. a smooth compact Riemann surface embedded in ℙ_{ℂ}^{2}, the inflection points are precisely the points were the tangent line has contact of order at least 3 with the curve. If C is a curve then a point p ∈ C is an inflection point of order d if there exist an curve of degree d that has higher order contact with the curve C at p. Here higher order contact means that the order of contact is at least the dimension of the vector space of polynomials of degree at most d.

For plane curves, the inflections points are computed through the Hessian and the higher order inflection points are not so easy to obtain. As far as the author knows, the first mathematician to pursue the question of determining higher order inflection points for plane curves was Cayley, see [3]. He succeed in giving a formula for the inflection points of order two. Although, the formula obtained by Cayley is not very simple.

According to Cukierman the problem of giving formulas for inflection points of order greater than three of plane curves does not seem to have been solved in the classical literature. In [4], he gives an approach to obtain ”almost explicit” formulas for the higher order inflection points of plane curves and complete intersection curves on some projective space.

The goal of the first part of this paper is to introduce and show how to compute inflection and higher order inflection points for holomorphic vector fields on the complex projective plane. In more concrete terms, given a vector field X on ℙ_{ℂ}^{2}, we define effective divisors E_{d}(X) on ℙ_{ℂ}^{2}, such that the restriction of the divisor to any solution of the vector field X coincides with the inflection points of order d of the solution.

In contrast with the case of plane curves, the formulas obtained for E_{d}(X) are not very complicated. At first sight this seems to be paradoxical, but if C is a smooth algebraic curve invariant by the algebraic vector field X, then the divisor E_{d}(X) restricted to the curve gives something more than the inflection points of order d of the curve C. The fact is that besides the inflection points of order d, E_{d}(X)_{|C} also contains the singularities of the vector field along C.

The initial motivation for introducing such concepts was to have a tool for detecting invariant algebraic curves of a given degree d of a vector field X, and bound their number in function of d and the degree of X. In fact using the divisors E_{d}(X), which we call extactic curves, we obtain such bounds. These bounds turn out to be of different nature of Jouanolou’s bound which are obtained through Darboux’s approach, see [7].

Among the applications of the extactic curves, one can find a computational criteria for the existence of first integral of a given degree and some properties of families of foliations on ℙ_{ℂ}^{2}.

Since we believe that the concepts here introduced may be useful for studying concrete examples of real and complex algebraic vector fields, we try to be as explicit as possible in the first part of paper.

In the second part we use a more intrinsic approach and generalize the concepts and some of the results for vector fields on arbitrary complex manifolds.

Developing the concepts in such generality we show that we can use our methods to detect a class of solutions more general than the algebraic class. One illustrative result of the method is a generalization of Darboux’s criteria for the existence of first integrals in the spirit of [10]. Using the same sort of ideas we also obtain a new proof of a result proved in [6] by Gomez–Mont about foliations with all leaves algebraic.

I would like to thank P. Nogueira, for her help in working out the formal definition of extactic divisors, L.G. Mendes, for showing me reference [5], and specially E. Esteves for many helpful discussions about linear systems, inflection points and algebraic vector fields.

Part 1. Extactic Curves on the Projective Plane

2. Affine and Projective polynomial Vector Fields

If X is a polynomial vector field on ℂ^{2}, then X can be written, in a unique way, as

X = a(x, y)\frac{∂}{∂x} + b(x, y)\frac{∂}{∂y} + g(x, y) (x\frac{∂}{∂x} + y \frac{∂}{∂y} ) ,

where g is a homogeneous polynomial of degree d and a, b are polynomials of degree at most d. We define d as the degree of the vector field. When g is identically zero we say that the line at infinity is invariant.

If we consider the homogeneous polynomials

\overline{a}(x, y, z) = z^{d} · a (\frac{x}{z}, \frac{y}{z} ) and \overline{b}(x, y, z) = z^{d} · b (\frac{x}{z}, \frac{y}{z} ) ,

then the vector field

\overline{X} = \overline{a}(x, y, z)\frac{∂}{∂x} + \overline{b}(x, y, z)\frac{∂}{∂y} + g(x, y, z)\frac{∂}{∂z}

is a projectivization of X. If Y is any other projectivization of X then \overline{X} − Y is a multiple of the radial(or Euler) vector field R = x\frac{∂}{∂x} + y \frac{∂}{∂y} + z \frac{∂}{∂z}.

Reciprocally, if Y is a polynomial vector field on ℂ^{3} with homogeneous coefficients then Y induces via the radial projection a field of directions on ℙ_{ℂ}^{2}. Observe that any homogeneous vector field on ℂ^{3} whose difference with Y is a multiple of the radial vector field, induces the same field of directions on ℙ_{ℂ}^{2}. We abuse the language and say that such a field of directions is a projective vector field, or just a vector field on ℙ_{ℂ}^{2}.

3. Higher order inflection Curves of Vector Fields on ℙ_{ℂ}^{2}

In this section we define the extactic curves, E_{d} for d ∈ ℕ, for vector fields on ℙ_{ℂ}^{2}. These curves describe the inflection and higher order inflection points for solutions of the vector field.

3.1. Extactic points of plane curves.

Definition 1. A n-inflection point of a curve in ℙ_{ℂ}^{2} is a point where the multiplicity of intersection of the curve with some algebraic curve of degree n is greater than

d(n) = \frac{n(n + 3)}{2}.

Note that d(n) is the dimension of the space of plane curves of degree n.

We use the term extactic point following V.I. Arnold, see [1].

Remark 1. Observe that when C is an algebraic curve in ℙ_{ℂ}^{2} or ℙ_{ℝ}^{2}, if every point of C is a n-inflection point then the degree of C is at most n.

3.2. Extactic curves of vector fields. Our purpose is to describe in an unified way the inflections and higher–order inflections points of the solutions of the vector field. If X = a(x, y)\frac{∂}{∂x} + b(x, y)\frac{∂}{∂y} is a vector field on ℂ^{2} and (x, y(x)) a

parametrization of a solution we have that det ( \begin{matrix}a & b \\ 1 & \frac{dy}{dx} \end{matrix} ) = a\frac{dy}{dx}(x) − b = 0. Therefore, \frac{dy}{dx} = \frac{b}{a}. To obtain the inflections of the curve (x, y(x)) we have to calculate the determinant:

det [\begin{matrix}1 & \frac{dy}{dx} \\ 0 & \frac{d^{2}y}{dx^{2}} \end{matrix} ] = \frac{d}{dx} \frac{b(x, y(x))}{a(x, y(x))} = \frac{(\frac{∂a}{∂x} + \frac{∂a}{∂y} \frac{a}{b})b − (\frac{∂b}{∂x} + \frac{∂b}{∂y} \frac{a}{b})a}{a^{2}}

Hence, in the open set ℂ^{2} \ (a(x, y) = 0), the inflections points of the solutions describe a curve given by the expression above. In an analogous manner we can get equations for the inflection curve in the open set ℂ^{2} − (b(x, y) = 0).

To calculate the 2-inflection points of the solutions we consider the image of the curve (x, y(x)) under the 2-Veronese map. Such image is the curve parametrized by

(x, x^{2}, x · y(x), y(x), y(x)^{2})

in ℂ^{5}. If we calculate its flattening points we obtain the 2-inflection points of the solution. Such calculation can be done evaluating the determinant of the matrix,

[\begin{matrix}1 & 2x & \frac{dxy(x)}{dx} & \frac{dy(x)}{dx} & \frac{dy(x)^{2}}{dx} \\ 0 & 2 & \frac{d^{2}xy(x)}{dx^{2}} & \frac{d^{2}y(x)}{dx^{2}} & \frac{d^{2}y(x)^{2}}{dx^{2}} \\ 0 & 0 & \frac{d^{3}xy(x)}{dx^{3}} & \frac{d^{3}y(x)}{dx^{3}} & \frac{d^{3}y(x)^{2}}{dx^{3}} \\ 0 & 0 & \frac{d^{4}xy(x)}{dx^{4}} & \frac{d^{4}y(x)}{dx^{4}} & \frac{d^{4}y(x)^{2}}{dx^{4}} \\ 0 & 0 & \frac{d^{5}xy(x)}{dx^{5}} & \frac{d^{5}y(x)}{dx^{5}} & \frac{d^{5}y(x)^{2}}{dx^{5}} \end{matrix} ] .

The n-inflection points can be obtained through the use of the n-Veronese map in a completely similar way.

Until now we have been working locally, although it is possible to give global expressions, on ℙ_{ℂ}^{2} for these n-inflection curves. If X = A\frac{∂}{∂x} + B \frac{∂}{∂y} + C \frac{∂}{∂z} is a homogeneous vector field in ℂ^{3} then the equation for the inflection curve, or the first extactic curve (which we will denote by E_{1}(X)) of the induced foliation on ℙ_{ℂ}^{2} is:

E_{1}(X) = det [\begin{matrix}x & y & z \\ X(x) & X(y) & X(z) \\ X^{2}(x) & X^{2}(y) & X^{2}(z) \end{matrix}] , (1)

where X^{k}(f) = X(X^{k−1}(f)), for any polynomial f.

Example 1. Let

X = (x^{3} − z^{3})x\frac{∂}{∂x} + (y^{3} − z^{3})y \frac{∂}{∂y} ,

Y = −y^{2}z^{2} \frac{∂}{∂x} + −x^{2}z^{2} \frac{∂}{∂y} + x^{2}y^{2} \frac{∂}{∂z}

and Z = tX + sY, (s, t) ∈ ℂ^{2} and s · t ≠ 0, be a projectivization of Lins Neto’s example [8]. Then

E_{1}(Z) = 2L_{9} · (t^{2}sy^{3} − xyzs^{3} + 2xyzt^{3} + z^{3}t^{2}s + st^{2}x^{3}),

where L_{9} = (x^{3} − y^{3})(x^{3} − z^{3})(y^{3} − z^{3}). Observe that the nine invariant lines for any Z are contained in the first extactic curve of Z.

To understand why formula (1) works, suppose p ∈ ℂ^{3} is a non–singular point of X. Here non-singular means that the vector field X is not colinear with the radial vector at p, or in other terms p is a non–singular point of the codimension one foliation of ℂ^{3} induced by X and R. By the existence of local solutions for ordinary differential equations there is a germ of curve V around p ∈ ℂ^{3} which is a local orbit of X. As consequence the vector field X restricts to V,which means that X acts as a derivation on the local functions of V. Since V,seen as a germ of projective curve, has dimension 1 the restriction of X to V can be seen as the derivative of a local parameter t. Hence formula (1) can be written on V as:

E_{1}(X)_{|V} (t) = det [\begin{matrix}x(t) & y(t) & z(t) \\ \frac{∂x(t)}{∂t} & \frac{∂y(t)}{∂t} & \frac{∂z(t)}{∂t} \\ \frac{∂^{2}x(t)}{∂t^{2}} & \frac{∂^{2}y(t)}{∂t^{2}} & \frac{∂^{2}z(t)}{∂t^{2}} \end{matrix} ] ,

and in fact represents the infection points of V in a neighborhood of p.

Similarly, we have a global equation for the curve of 2-inflection points of the vector field, or the second extactic curve E_{2}(X), which is the determinant of :

[\begin{matrix}x^{2} & xy & xz & y^{2} & yz & z^{2} \\ X(x^{2}) & X(xy) & X(xz) & X(y^{2}) & X(yz) & X(z^{2}) \\ X^{2}(x^{2}) & X^{2}(xy) & X^{2}(xz) & X^{2}(y^{2}) & X^{2}(yz) & X^{2}(z^{2}) \\ X^{3}(x^{2}) & X^{3}(xy) & X^{3}(xz) & X^{3}(y^{2}) & X^{3}(yz) & X^{3}(z^{2}) \\ X^{4}(x^{2}) & X^{4}(xy) & X^{4}(xz) & X^{4}(y^{2}) & X^{4}(yz) & X^{4}(z^{2}) \\ X^{5}(x^{2}) & X^{5}(xy) & X^{5}(xz) & X^{5}(y^{2}) & X^{5}(yz) & X^{5}(z^{2}) \end{matrix}]

The d-th extactic curve E_{d}(X) can be described in a completely similar way. The equation of E_{d}(X) is given by the determinant of a matrix such that the first row is formed by a basis of the monomials in x, y and z of degree d, and the i-th row is the derivation X applied in the (i − 1)-th row.

Proposition 1. Every algebraic curve of degree n invariant by the vector field X is a factor of E_{n}(X).

Proof: Let F be an invariant algebraic curve of degree n. Since the choice of the basis of the ℂ–vector space plays no role in the definition of extactic curve, we can choose a basis where F appears. Since

X(F) = L_{F} F ,
X^{2}(F) = X(L_{F} F)  = (L_{F}^{2} + X(L_{F})) F ,
X^{k}(F) = X(X^{k−1}(F)) = ( polynomial ) F ,

where L_{F} is a polynomial, one can see that F is a factor of E_{n}(X).

Theorem 1. X admits a first integral of degree d, but do not admit a first integral of degree smaller than d if, and only if, E_{d}(X) = 0 and E_{d−1}(X) ≠ 0.

Proof: Let p ∈ ℙ_{ℂ}^{2} be a non-singular point of X. Suppose that the solution passing through p is parametrized, locally, by (x, y(x)). Since E_{d}(X) vanishes identically, the composition of our local solution with the d-Veronese map is contained in a hyperplane, so (x, y(x)) must be contained in an algebraic curve of degree at most d. Since every leaf is algebraic it follows from Darboux’s Criteria, see [10], that X admits a first integral of degree at most d. The fact that E_{d−1}(X) ≠ 0 implies that the generical solution is of degree at least d.

If X admits a first integral of degree d then every invariant curve is of degree at most d and hence every point is a d-inflection point, i.e., E_{d}(X) = 0. Since not every invariant curve has degree d − 1, E_{d}(X) ≠ 0.

¿From the Theorem 1 and the Proposition 1 we derive:

Theorem 2. Let X a vector field in ℙ_{ℂ}^{2}. For every d ∈ ℕ the equations of possibles invariant curves of degree less than or equal to d appear as factors of E_{d}(X), and if E_{d}(X) = 0 then X has a meromorphic first integral of degree at most d.

4. Counting algebraic solutions and field of definition of the invariant curves

Jouanolou in [7] shows that a vector field of degree d on ℙ_{ℂ}^{2} without rational first integral has at most d(d + 2)/2 irreducible algebraic solutions. Observe that he does not make any assumption on the degree of the algebraic leaves.

Using the extactic curves we are able to obtain different bounds for the number of irreducible algebraic solutions. If X is a vector field on ℙ_{ℂ}^{2} and we denote by n_{i}(X) the number of irreducible algebraic solutions of degree i of X the following proposition holds.

Proposition 2. Let X be a homogeneous vector field, ℙ_{ℂ}^{2}, of degree d. If it does not have a first integral of degree ≤ n then

∑_{i=1}^{n} i · n_{i}(X) ≤ \frac{d(n^{4} + 6n^{3} + 11n^{2} + 6n) − n^{4} − 2n^{3} + n^{2} + 2n}{8}.

Proof: One has just to observe that the left hand side of the inequality is bounded by the degree of E_{n}(X). Since the space of monomials in x, y, z of degree n is

(\begin{matrix}n + 2 \\ 2 \end{matrix}), we have that the degree of E_{n}(X) is

∑_{i=0}^{( \begin{matrix}n+2 \\ 2 \end{matrix} )−1} i(d − 1) + n = (( \begin{matrix}n+2 \\ 2 \end{matrix} ))n + (d − 1)\frac{( \begin{matrix}n+2 \\ 2 \end{matrix} )(( \begin{matrix}n+2 \\ 2 \end{matrix} ) − 1)}{2} =

= \frac{d(n^{4} + 6n^{3} + 11n^{2} + 6n) − n^{4} − 2n^{3} + n^{2} + 2n}{8}.

Corollary 1. Let X be a homogeneous vector field, ℙ_{ℂ}^{2}, of degree d. If it does not have a rational first integral of degree ≤ n then it has at most

\frac{d(n^{3} + 6n^{2} + 11n + 6) − n^{3} − 2n^{2} + n + 2}{8}

invariant curves of degree n.

Proof: If C is an invariant curve of degree n then C is contained in E_{n}(X), so we have at most

\frac{deg(E_{n}(X))}{n} = \frac{d(n^{3} + 6n^{2} + 11n + 6) − n^{3} − 2n^{2} + n + 2}{8}

invariant curves of degree n.

Example 2. Applying Proposition 2 to bound the number of invariant lines of a vector field X, one can see that

n_{1}(X) ≤ 3 · deg(X) .

This bound turns out to be sharp. For example, for each d ∈ ℕ, consider the vector fields X_{d} on ℙ_{ℂ}^{2}, given in homogeneous coordinates by

X_{d} = (x^{d−1} − z^{d−1}) x\frac{∂}{∂x} + (y^{d−1} − z^{d−1}) y \frac{∂}{∂y} .

Then X_{d} leaves invariant the algebraic curve C_{d}, cutted out by the polynomial

F_{d} = xyz(x^{d−1} − z^{d−1})(y^{d−1} − z^{d−1})(x^{d−1} − y^{d−1}) .

Since F_{d} can be expressed as a product of 3d distinct lines and F_{d} = E_{1}(X_{d}), we conclude that the vector field X_{d} admits exactly 3d invariant lines.

Observe that the same bound holds for real vector fields on ℝP (2). Although, the sharpness fails. To understand better why this happens see [2].

Using the same methods we can obtain smaller bounds if we try to count, for example, straight lines passing through a given point p. Instead of looking for the solutions in the 3–dimensional vector space of lines, we have just to look for in the codimension one subspace of lines passing through p. For example, if p = [0 : 0 : 1] ∈ ℙ_{ℂ}^{2} then equation of any invariant line passing through p will appear as a factor of the determinant of the following matrix:

[\begin{matrix}x & y \\ X(x) & X(y) \end{matrix}] .

Hence, if the X has degree d, then the number of invariant lines passing through a given point p is at most d + 1.

Suppose now that we have an algebraic vector field on ℂ^{2}, and that its coefficients are in a normal subfield 𝕂 of ℂ. The following question naturally arises: what can be said about the field of definition of the invariant algebraic curves? In other terms: what is the smallest extension 𝕃 of 𝕂 such that any invariant curve can be defined by a polynomial with coefficients in 𝕃?

We can use the bounds for the number of invariant curves to obtain bounds on the algebraic degree of the field of definition of algebraic invariant curves.

Proposition 3. Let 𝕂 ⊂ ℂ be a normal extension of ℚ and X = a\frac{∂}{∂x} + b\frac{∂}{∂y} be a vector field on ℂ^{2} of degree d, where a, b ∈ 𝕂[x, y]. Suppose that exists an invariant algebraic curve of degree l, cutted out by a polynomial f ∈ 𝕃[x, y], where 𝕃 is a normal extension of 𝕂 contained in ℂ, that cannot be defined in any normal subfield of 𝕃. If

[𝕃 : 𝕂] > min (\frac{d(n^{4} + 6n^{3} + 11n^{2} + 6n) − n^{4} − 2n^{3} + n^{2} + 2n}{8}, \frac{d(d + 2)}{2}) ,

then X admits a first integral.

Proof: Since f is invariant by X, we have that X(f) = L_{f} · f, for some polynomial L_{f} ∈ 𝕃[x, y]. Applying the Galois automorphisms of the extension [𝕃 : 𝕂], we obtain [𝕃 : 𝕂] distinct invariant algebraic curves of degree l. Hence the Theorem follows by Jouanolou’s bound and Proposition 2.

5. Families of holomorphic foliations on ℙ_{ℂ}^{2}

In [8], Lins–Neto shows the existence of some very special families of foliations on ℙ_{ℂ}^{2} parametrized by the projective line. One interesting properties of such families is that not all foliations admits a rational first integral, but for a dense set E in the parameter space the corresponding foliation has a rational first integral. Since the set in the parameter space admitting a rational first integral of degree at most d, for a fixed positive integer d, is algebraic, we have that the rational first integrals in the family have unbounded degree and E admits the filtration

E = ⋃_{d∈ℕ} E_{d} ,

where p ∈ E_{d} if, and only if the foliation corresponding to p admits a rational first integral of degree at most d.

In remark 4 of [8], Lins Neto says that ”would be interesting to know what kinds of properties this set has”. Here, we use the extactic curves to bound the growth of the cardinality of E_{d} for any family of foliations parametrized by a projective line.

Definition 2. Let C ⊂ F ol(k) be a algebraic curve included in the space of foliations of degree k. We define the counting function of C,

π_{C} : ℕ → ℕ ∪ +∞,

by the following rule : π_{C}(d) = n if the number of points in C representing a foliation with rational first integral of degree at most d is exactly n.

Example 3. Let X = x\frac{∂}{∂x} and Y = y \frac{∂}{∂y} be vector fields on ℂ^{2}. The family of vector fields tX + sY can be seen as ℂP (1) linearly embedded in F ol(1), which we will denote by C. Whenever the ratio of t and s is a rational number then tX + sY admits a rational first integral. Suppose that t/s = p/q and p and q do not have common factors. If t/s is positive then the degree of the first integral is the maximum between p and q, otherwise it is |p| + |q|.

¿From the considerations of the previous paragraph we can show that

π_{C}(n) ≤ K · n^{2}.

Considering a family of foliations parametrized by a projective line linearly embedded in the space of foliations, we obtain:

Proposition 4. Suppose C is a ℂP (1) linearly embedded in F ol(k). If π_{C}(d) < ∞ for every d ∈ ℕ, then there exist a constant K such that

π_{C}(d) ≤ Kd^{4}.

Proof: If we take two distinct projective vector fields in C, say X and Y, we can recover C by considering the linear combinations sX + tY. From the definition of E_{d}, one can see that:

E_{d}(sX + tY) = ∑_{α+β+γ=deg(E_{d}(X)} P_{α,β,γ}(s, t)x^{α}y^{β}z^{γ} .

Since π_{C}(d) < ∞ then there exist a triple (α_{0}, β_{0}, γ_{0}) such that P_{α_{0},β_{0},γ_{0}} does not vanish identically. Hence the number of projective parameters (s : t) that has first integral of degree at most d is bounded by the degree of P_{α_{0},β_{0},γ_{0}}.

Part 2. Extactic Divisors on Complex Manifolds

6. Extactics divisors for holomorphic foliations

Here we generalize some of the results of Part 1 to foliations by curves on arbitrary non-singular complex manifolds. In order to do this, is imperative the reformulation of the concepts in a more intrinsic way, and to accomplish that we use freely the language of algebraic geometry(for example line bundles, tensor products and so on).

6.1. Holomorphic foliations as morphisms. Let M be a complex manifold. An 1-dimensional holomorphic foliation is given by the following data

• an open covering U = U_{i} of M;

• for each U_{i} an holomorphic vector field X_{i} ;

• for every non-empty intersection, U_{i} ∩ U_{j} ≠ ∅, a holomorphic function g_{ij} ∈

O_{M}^{∗}(U_{i} ∩ U_{j}); subject to the conditions :

• X_{i} = g_{ij}X_{j} in U_{i} ∩ U_{j}

• g_{ij}g_{jk} = g_{ik} in U_{i} ∩ U_{j} ∩ U_{k}.

If we denote by L the line bundle defined by the cocycle g_{ij} we can understand the collection X_{i} as a holomorphic section σ of the bundle T M ⊗ L. Such section induces a morphism that goes from the cotangent bundle, denoted by Ω_{M}^{1}, to the line bundle L. This morphism, in the open set U_{i}, is given by the interior product with the vector field X_{i}.

Hence given an holomorphic foliation F we have an morphism:

Φ_{F} : Ω_{M}^{1} → L

Reciprocally, given such morphism Φ we can canonically associate a holomorphic foliation to it. We leave the details for the reader.

6.2. Jet Bundles and Extactic Divisors. Suppose now that we have a foliation F on M, i.e., a morphism Φ_{F} of Ω_{M}^{1} to an invertible sheave E. If we have a linear system V ⊆ H^{0}(M, L) we are going to define the extactic divisor, E(F, V), with respect to V. The extactic divisor E(F, V) can be understood geometrically as the inflection locus of the linear system V with respect to the morphism Φ.

First of all, consider the local Taylor expansion of a section s ∈ V with respect to the vector field defining F. Formally if you have a morphism

Φ_{F} : Ω_{M}^{1} → E

and a linear system V ⊆ H^{0}(M, L), we choose a covering U of M which trivializes both L and E. In a open set U ∈ U we can consider the morphism

T_{|U}^{(k)} : H^{0}(M, L) ⊗ O_{U} → O_{U}^{k}

defined by

T^{(k)}(s) = s + X_{F}(s) · t + X_{F}^{2}(s) · \frac{t^{2}}{2!} + ⋯ + X_{F}^{k}(s) · \frac{t^{k}}{k!} (2)

where X_{F}(·) = Φ_{F}(d(·)) and s ∈ O_{U} is an element of H^{0}(M, L) ⊗ O_{U} expressed in the chosen trivialization.

If we take open sets U_{λ} ∈ U we have that

L_{|U_{λ}} = O_{U_{λ}} · α_{λ} ,
E_{|U_{λ}} = O_{U_{λ}} · β_{λ} .

Hence, for any s_{λ} ∈ H^{0}(M, L) ⊗ O_{U_{λ}}, we obtain :

s_{λ} = s_{λ}^{(0)} · α_{λ},
X_{F}(s_{λ}) = X_{F}(s_{λ}^{(0)}) · β_{λ} = s_{λ}^{(1)} · β_{λ} .

And generally,

X_{F}^{k}(s_{λ}) = X_{F}(s_{λ}^{(k−1)}) · β_{λ} = s_{λ}^{(k)} · β_{λ} .

When U_{λ} ∩ U_{μ} ≠ ∅ :

s_{λ} = s_{λ}^{(0)} · α_{λ} = l_{λμ}s_{μ}^{(0)} · α_{μ} ,
X_{F}(s_{λ}) = s_{λ}^{(1)} · β_{λ} = X_{F}(s_{λ}^{(0)}) · β_{λ} = (X_{F}(l_{λμ}) · s_{μ}^{(0)} + l_{λμ} · s_{μ}^{(1)})) · e_{λμ} · β_{μ} .

Or in matrix notation

[\begin{matrix}s_{λ}^{(0)} \\ s_{λ}^{(1)} \end{matrix}] = [\begin{matrix}l_{λμ} & 0 \\ X_{F}(l_{λμ}) · e_{λμ} & l_{λμ} · e_{λμ} \end{matrix}] · [\begin{matrix}s_{μ}^{(0)} \\ s_{μ}^{(1)} \end{matrix}]

By analogous computations one can show that :

[s_{λ}^{(0)} ; s_{λ}^{(1)} ; s_{λ}^{(2)} ; ⋮ ; s_{λ}^{(k)}] = [\begin{matrix}l_{λμ} & 0 & 0 & 0 & 0 \\ X_{F}(l_{λμ}) · e_{λμ} & l_{λμ} · e_{λμ} & 0 & 0 & 0 \\ ... & ... & l_{λμ} · e_{λμ}^{2} & 0 & 0 \\ ... & ... & ... & ... & 0 \\ ... & ... & ... & ... & l_{λμ} · e_{λμ}^{k} \end{matrix}] · [s_{μ}^{(0)} ; s_{μ}^{(1)} ; s_{μ}^{(2)} ; ⋮ ; s_{μ}^{(k)}] (3)

Therefore, we define the vector bundle J_{X_{F}}^{k} L as the vector bundle with rank k + 1 and transition functions given by the matrix above. Now we are ready to consider the global Taylor expansion of the sections in V with respect to F. More precisely, we are able to patch together the morphisms of the form (2) to obtain just one morphism :

T^{(k)} : H^{0}(M, L) ⊗ O_{M} → J_{X_{F}}^{k} L (4)

We are almost ready to describe the inflection points of the linear system V ⊂ H^{0}(M, L) with respect to X_{F}. If we set k as the dimension over ℂ of V, and take the determinant of (4),

det T^{(k)} : Λ^{k}V ⊗ O_{M} → Λ^{k}J_{X_{F}}^{k−1}L . (5)

then after tensorizing we obtain a section of Λ^{k}J_{X_{F}}^{k−1}L ⊗ (Λ^{k}V)^{∗}

O_{M} → Λ^{k}J_{X_{F}}^{k−1}L ⊗ (Λ^{k}V)^{∗} . (6)

Finally :

Definition 3. The extactic curve of F with respect to the linear system V, E(F, V) is given by the zero locus of the section (6).

Remark 2. The extactic curves, E_{d}(X), defined in subsection 3.2 coincide with the extactics curves of the foliation induced by X with respect to the linear system H^{0}(ℙ_{ℂ}^{2}, O_{ℙ_{ℂ}^{2}}(d)). To verify this fact one has just to check locally, and is not very hard to show that locally the constructions are the same.

6.3. Order of Contact and Extactic Ideals. So far we have defined the extactic divisors for 1–dimensional holomorphic foliations on complex manifolds. One of the motivations to do that is to obtain an analogous of Theorem 2. In fact by a very similar argument we obtain the following:

Proposition 5. Let F be a 1–dimensional foliation on the complex manifold M. If V is a finite dimensional linear system, then every invariant hypersurface contained in the zero locus of some element of V, must be contained in the zero locus of E(F, V).

Since we are working on a manifold of arbitrary dimension, the extactic divisors are far from detecting precisely the invariant curves. To overcome this difficult we shall now introduce the extactic ideals. If we fix a finite dimensional linear system V, the main idea is to consider the points such that the contact with V is of infinite order.

Before defining the extactic ideals, let’s make the notion of contact more precise.

Definition 4. Let X_{F} be an holomorphic vector field on the complex manifold M and s a holomorphic section of some line bundle. We say that the solution through p has contact of order k with s when k is the least non negative integer such that the radical of the ideal generated by \overline{s}, \overline{X(s)}, \overline{X^{2}(s)}, …, \overline{X^{k}(s)} is the local ring O_{M,p}. If such a k does not exist we say that s has flat contact with X. Here X means a local representant of X_{F} on a suitable open set and \overline{s} means the image of any local representant of s under the canonical morphism O_{M} → O_{M,p}. We shall denote by ν(s, X_{F}, p) the contact of s with X_{F} at the point p.

Observe that any s vanishing at a singular point p of X, has flat contact with X at p.

Example 4. Let X = \frac{∂}{∂x} be a polynomial vector field on ℂ^{3} and f a polynomial. If f does not vanish at the origin then ν(f, X, 0) = 0, otherwise ν(f, X, 0) ≥ 1. For example if

f(x, y, z) = ∑_{i=k}^{n} f_{i}(y, z)x^{i} ,

with f_{k}(0, 0) ≠ 0, then ν(f, X, 0) = k.

Proposition 6. Let F be a 1–dimensional foliation on the complex manifold M. Suppose V is a finite dimensional linear system and s ∈ V. If ν(s, F, p) ≥ dim_{ℂ}V then p ∈ E(F, V). Reciprocally if p ∈ E(F, V) then exists an element s ∈ V such that ν(s, F, p) ≥ dim_{ℂ}V.

Proof: Choose a basis of V starting with s. Using Lagrange’s rule to expand E(F, V) in terms of the first column of the matrix one can see that E(F, V) belongs to the maximal ideal correspondent to the point p.

When E(F, V)(p) = 0, we have that the columns of the matrix used to compute E(F, V) are linearly dependent in the point P. But that means that exists an element s ∈ V, such that s(p) = X(s)(p) = … = X^{l}(s)(p) = 0, where l = dim_{ℂ} V. Hence the result follows.

Let X_{F} be an 1–dimensional foliation on the manifold M. If L is an invertible sheaf then we can consider, as in 6.2, the morphisms

T^{(l)} : H^{0}(M, L) ⊗ O_{M} → J_{X_{F}}^{l} L ,

for any positive integer l. Suppose V is a finite dimensional vector space contained in H^{0}(M, L) and k is its dimension. Now, after taking the determinant and tensorizing by (Λ^{k}V)^{∗}, we obtain sections

σ_{l} : O_{M} → Λ^{k}J_{X_{F}}^{l} L ⊗ (Λ^{k}H^{0}(M, V)^{∗} .

Definition 5. The sheaf of ideals generated by ker σ_{l}, where l is any positive integer, is the extactic ideal of X_{F} with respect to the linear system V. We shall denote it by I(X_{F}, V).

Example 5. If X is a vector field on ℙ_{ℂ}^{2} the extactic ideal of X with respect to the linear system V = H^{0}(ℙ_{ℂ}^{2}, O_{ℙ_{ℂ}^{2}}(k)), I(X, V), is generated, in homogeneous coordinates, by σ_{(k_{1},... ,k_{l})}, where

σ_{(k_{1},... ,k_{l})} = det (\begin{matrix}X^{k_{1}}(v_{1}) & X^{k_{1}}(v_{2}) & ⋯ & X^{k_{1}}(v_{l}) \\ X^{k_{2}}(v_{1}) & X^{k_{2}}(v_{2}) & ⋯ & X^{k_{2}}(v_{l}) \\ ⋮ & ⋮ & ⋯ & ⋮ \\ X^{k_{l}}(v_{1}) & X^{k_{l}}(v_{2}) & ⋯ & X^{k_{l}}(v_{l}) \end{matrix}) ,

where 0 ≤ k_{1} < k_{2} < ⋯ < k_{l} and the k_{i}’s are integers, v_{1}, v_{2}, …, v_{l} is a basis of the space of homogeneous polynomial of degree k in three variables.

Proposition 7. The subvariety N of M associated to the radical of any extactic ideal of X_{F} is invariant.

Proof: Fix a linear system V on M. If every element of I(X, V) vanishes at p, then p belongs to the radical ideal of X_{F} associated to V. Hence for every natural number l ≥ dim_{ℂ} V one has a linear subspace L_{l} ⊂ V such that any nonzero element f ∈ L_{l} satisfies f(p) = X(f)(p) = … = X^{l}(f)(p) = 0. Being the projectivization of V compact we can find a element f_{∞} which vanishes at p together with all its derivatives with respect to the local vector field X. In other words f_{∞} has flat contact with X_{F} at p.

Hence N can be identified with the set of points p ∈ M such that there exists a element of V with flat contact with X_{F} at p, and the invariance follows.

Example 6. In [7], Jouanolou proved that the vector fields on ℙ_{ℂ}^{2} given in homogeneous coordinates by

X_{d} = y^{d} \frac{∂}{∂x} + z^{d} \frac{∂}{∂y} + x^{d} \frac{∂}{∂z} ,

do not admit any invariant algebraic curve for any integer d greater than 1.

To consider any linear system V on ℙ_{ℂ}^{2}, is the same as to consider a finite dimensional space of homogeneous polynomials V in three variables. Since we do not have any invariant algebraic leaves for X_{d} and the singular points of X_{d} are clearly invariant, we have that the radical of the extactic ideal I(X_{d}, V) is exactly the ideal defining the singular set of X_{d}, for any d ≥ 2.

Remark 3. If F is a holomorphic foliation on a complex manifold M, then, for us, a first integral for F is any non–constant holomorphic map f : M → N, where N a complex manifold, such that the fibers of f are F–invariant.

Theorem 3. Let F be a 1–dimensional foliation on the complex manifold M. If V is a finite dimensional linear system, such that E(F, V) vanishes identically then there exists an dense open set U where F_{|U} admits a first integral. Moreover, if M is a projective variety then F admits a meromorphic first integral.

Proof: Suppose that dim_{ℂ} C = k, set s_{1}, …, s_{k} a basis of V and consider N ⊂ M × ℙ_{ℂ}^{k−1} defined as follows

N = {(p; a_{1} : …: a_{k}) | ν (∑_{i=1}^{k} a_{i} · s_{i}, F, p) = ∞} .

To see that N is a closed set consider, for each positive integer j, the sets N_{j} ⊂ M × ℙ_{ℂ}^{k−1} defined by

N_{j} = {(p; a_{1} : …: a_{k}) | ν (∑_{i=1}^{k} a_{i} · s_{i}, F, p) ≥ j} .

In an open set U that trivializes F and the line bundle that supports V we can write N_{j} as

N_{j} = {(p; a_{1} : …: a_{k}) | X^{l} (∑_{i=1}^{k} a_{i} · s_{i}) (p) = 0, l = 0, …, j} .

Hence N_{j} is a closed set. Since

N = ⋂_{j=1}^{∞} N_{j} ,

it follows that N is also a closed set.

Observe that every fiber of the natural projection π : N → M is a projective space linearly embedded on ℙ_{ℂ}^{k−1}. It is well known that there exists a dense open set U where the dimension of the fibers are the same, say l. Hence, we can define a natural holomorphic function from U ⊂ M to the Grassmanian of (l + 1)–planes on ℂ^{k}. This function, defined in a dense open subset U of M is a first integral for F_{|U}.

When M is a projective variety we can extend this function to all M, obtaining in such way a meromorphic first integral to F.

7. A generalization of Darboux’s Criteria

In [10], Jean-Marie Lion proposed the following generalization to Darboux’s criteria for the existence of meromorphic first integral.

Theorem[Lion] Let F be a holomorphic foliation given by an integrable 1-form, ω, defined in a nighbourhood of the closure of U, where U is a bounded open set of ℂ^{n}. Suppose that exists infinitely many leaves V_{i} of F contained in different algebraic hypersurfaces Q_{i} = 0 of the same degree d. Then every leaf of F is contained in an irreducible algebraic hypersurface of degree at most d, and ω admits a meromorphic first integral. If 0 ∈ (∩_{i}V_{i}), then exists a meromorphic first integral such that the graph is a n-dimensional algebraic subset of ℂ^{n} × ℙ_{ℂ}^{1}.

Here, using the ideas developed earlier in this paper, we obtain a further generalization of the first part of Lion’s result. The main improvement is that we do not need to restrict ourselves to algebraic leaves and open sets of ℂ^{n}. We work with sections of any linear system defined on a complex manifold which, in principle, do not have to be compact.

Theorem 4. Let F be a holomorphic foliation, of arbitrary codimension,on a complex manifold M, and V finite dimensional linear system on M. Suppose that there exists a collection {L_{i}}_{i∈Λ}, such that each leaf L_{i} has flat contact with some element of V. If the analytic closure of

⋃_{i∈Λ} L_{i}

is equal to M then F admits a first integral and every leaf of F has flat contact with some element of V.

Proof:Let U be a Stein open set of M. If we consider the O_{U}–module X (F) formed by all vector fields in U tangent to F_{|U}, then for every Y ∈ X (F) we have that E(Y, V) = 0. Hence, as in the proof Theorem 3, we can consider for each Y ∈ X (F) a set N_{U,Y} ⊂ U × ℙ_{ℂ}^{k−1} describing the elements of V with flat contact with Y.

Defining N ⊂ M × ℙ_{ℂ}^{k−1} locally as

N_{|U×ℙ_{ℂ}^{k−1}} = ⋂_{Y ∈X (F)} N_{U,Y} ,

we obtain a closed analytic subset of M × ℙ_{ℂ}^{k−1} whose fibers under the natural projection to M are projective spaces linearly embedded in ℙ_{ℂ}^{k−1}. And as in Theorem 3 the result follows.

Corollary 2. Let F be a codimension one holomorphic foliation on a complex manifold M, and V finite dimensional linear system on M. Suppose that there exists an infinite collection {L_{i}}_{i∈Λ}, such that each leaf L_{i} has flat contact with some element of V. If there exists an relatively compact open set U ⊂ M such that L_{i} ∩ U ≠ ∅ for every i ∈ Λ then F admits a first integral.

8. Foliations with all leaves algebraic

Gomez–Mont, in [6], proved that a singular holomorphic foliation of codimension q with all leaves algebraic admits a first integral, whose generic fiber has codimension q. His proves uses Grothendieck’s deep Theorem asserting the existence of the Hilbert scheme. Here, we propose a different proof using the ideas developed earlier in this paper.

Theorem[Gomez–Mont] Let F be a holomorphic foliation (with singularities) of codimension q in the projective integral variety M, and assume that every leaf L of F is a quasiprojective subvariety of M; then there is a projective integral variety V of dimension q and a rational map f : M → V such that the closure of a general f–fibre is the closure of a leaf of F.

Proof: Suppose, without loss of generality, that M is a subvariety of ℙ_{ℂ}^{n} and take L to be the restriction of the hyperplane bundle O_{ℙ^{n}}(1) to M.

Denote by L_{p} the leaf through p and ρ(L_{p}) the least positive integer l such that L_{p} is an open subset of variety defined through sections of L^{⊗l}. Now, define

B_{d} = {p ∈ M \ Sing(F) | ρ(L_{p}) ≤ d} .

Since every leaf is algebraic, we have that

M \ Sing(F) = ⋃_{d∈ℕ} B_{d} ,

and as consequence there exists d_{0} such that B_{d_{0}} has positive Lebesgue measure and consequently its analytic closure is equal to M.

Taking V = H^{0}(M, L^{⊗d_{0}}) we have from Theorem 4 that there exists a projective variety N and a morphism g : M → N such that the fiber over any p ∈ N is invariant by the foliation.

Taking a closer look at the proof of Theorem 4 one can see that the fiber over a generic p ∈ B_{d_{0}} will have codimension q. Using Stein’s factorization Theorem, see [5], one can assure the existence of the commutative diagram,

M ⟶^{f} \overline{N}
Id↓  ^{π}↓
M ⟶^{g} N

where \overline{N} is a projective variety, π is a morphism and f is a morphism whose generic fiber is irreducible. Hence f satisfies the assertions of the Theorem.

9. Final Remarks

The applications of the extactic curves and extactic divisors, certainly are not exhausted in this work. For example, in [11] the extactic curves are used to introduce the notion of algebraic multiplicity of an algebraic curve invariant by a polynomial vector field on ℂ^{2}. There, the relation of this algebraic multiplicity with the existence of exponential cofactors are explored to enrich Darboux’s theory of integrability.

Another possible application is the study of automorphism groups of holomorphic foliations. The author plan to pursue this topic in a future paper.

References

[1] V. I. Arnold, Remarks on the Extactic Points of Plane Curves, The Gelfand Mathematical Seminars, 1993–95.

[2] J. Artés, B. Grünbaum and J. Llibre, On the number of invariant straight lines for polynomial differential systems, Pacific Journal of Mathematics, 184 (2), 1998.

[3] A. Cayley, On the sextactic points of a plane curve, Collected Mathematical Paper V (341) ; Trans. Royal Soc. London CLV, 1865.

[4] F. Cukierman, Determinant of complexes and higher Hessians, Mathematische Annalen, 307 , 1997.

[5] G. Fischer , Complex Analytic Geometry, Lecture Notes in Math. 538, Springer, 1976

[6] X. Gomez–Mont , Integrals for holomorphic foliations with singularities having all leaves compact , Ann. Inst. Fourier, Grenoble 39 2, 1989

[7] J. P. Joaunolou , Equations de Pfaff Algébriques , Lecture Notes in Math. 708, Springer, 1979

[8] A. Lins Neto , Some Examples for Poincaré and Painlevé Problems, Preprint, IMPA, 2000.

[9] A. Lins Neto and B. Scárdua, Folheações Algébricas Complexas, IMPA, 21^{o} Colóquio Brasileiro de Matemática, Rio de Janeiro, 1997.

[10] Jean–Marie Lion, Un critère de Darboux d’existence d’intégrale première pour les 1-formes diffrentielle analytiques, Boletim da Sociedade Brasileira de Matemática, 31 (2), 2000

[11] J. LLibre and J. V. Pereira, Multiplicity of Algebraic Invariant Curves and Darboux Integrability, Preprint , 2000.