Foliations in algebraic surfaces having a rational first integral

Publicacions Matem`atiques, Vol 41 (1997), 357–373. FOLIATIONS IN ALGEBRAIC SURFACES HAVING A RATIONAL FIRST INTEGRAL Alexis Garc´ ıa Zamora Abstract Given a foliation Fin an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case S=P2some new counter-examples to the classic formulation of the Poincar´e problem are presented. If Sis a rational surface and Fhas singularities of type (1,1) or (1,−1) we prove that the general solution is a non-singular curve. Introduction A foliation Fin a complex algebraic surface Sis a non-identically zero morphism of vector bundles α:L−→ TS, where Lis a line bundle and TS is the tangent bundle of S. In the case Sis the projective plane, there are alternative definitions in global terms and, in particular, there is a notion of degree of a foliation (see Definition 1). Explicitly, the degree of Fis the degree of the polynomials Yi, where the direction field Y= 2  i=0 Yi(x0:x1:x2)∂ ∂xi defines the foliation. In [10] Poincar´e studied the following problem: if Fis a foliation with a rational first integral (that is, all the solutions of Fare algebraic curves) is it possible to bound the degree of the generic solution in terms of the degree of F? An answer to this problem gives a “practical” method to determine when a foliation has a rational first integral. If such a 1991 Mathematics subject classifications: 34A20, 34A26,14H99. 358 A. Garc´ ıa Zamora bound exist then the existence of a first integral is “simply” a problem of algebraic calculus. The answer to this problem is well know to be no. In section 2 some counter-examples are presented. Our first task is then to reformulate the Poincar´e problem. A reasonable formulation is: try to bound the degree of the general solution using information depending only of F, for example the eigenvalues of the singularities of F(we assume that all the singularities of Fare of multiplicity one). In section 1 we present preliminary results and definitions. All this material can be found in [6], [9]or[10]. In section 2 we study the version of the problem presented above. The section starts with the following: Proposition 2.1. Let Fbe a foliation on Swith a first integral and singular points of multiplicity one. Denote the pair of relatively prime integers associated to the eigenvalues of the nodes of Fby (ui,v i), then we have: (1) n  i=1 ri(ui+vi)=−C.(KS+L), (2) g=−C.L 2−ri 2+1. Where Cis a general solution of F, g=geometric genus of C, α:L−→ TS defines F, KS=canonical divisor of S, ri=number of branches of Cthrough a node of F, n=number of nodes of F. Then the counter-examples to the old formulation of the Poincar´e problem are presented; Example 2.1 is well know, Examples 2.2 and 2.3 seem new. In particular Example 2.2, the pencil yd=λxd−2(x−1)(y−1), proves that the genus of the pencils of curves giving the counter-examples can be arbitrary. Some more specific versions of the Poincar´e problem are then presented. We finish the section by showing that if m= deg F>4 and for some cases if m≤4 a bound for the genus of the generic solution is sufficient in order to solve the reformulated Poincar´e’s Problem. Then we introduce the study of certain linear series on the general solution and the concept of linearly equisingularity and prove the following: Foliations having a first integral 359 Proposition 2.3. If the conditions of being linearly equisingular to Cimposes independent conditions on curves of degree dthen g(C)≤δ. Where δ= deg(O˜ C(˜ C)), and ˜ Cπ −→ Cis the minimal embedded resolution of the general solution C. In section 3 we study the foliations in P2having as first integral a generic pencil. The main aim is to prove: Theorem 3.4. Let Fbe a irreducible foliation in a rational surface S with singularities of eigenvalues (1,1) or (1,−1).IfFhas a rational first integral f:S−→ P1, then all the fibers of fare reduced curves and the general one is a non-singular curve meeting transversally any other fiber of f. An important corollary is: Corollary 3.5. Let Fbe a foliation in P2of degree mwith a first integral of degree d, if all the singular points of Fhave eigenvalues (1,1) or (1,−1) then m=2d−2. This section contains a generalization of some results of Poincar´e[10]. Propositions 3.1, 3.2 and 3.3 were proved by Poincar´e for the case S=P2. Even when Poincar´e knew that under the hypothesis of Corollary 3.5 the degree of mis bounded, an explicit bound is not given in [10]. I want to thank CIMAT support and to X. G´omez-Mont for many useful comments. In particular, the proof of Proposition 3.1 was suggested to me by X. G´omez-Mont and C. Danthony. 1. Preliminaries Definition 1. A foliation Fof degree min P2(C) is determined by either of the following objects: a) An algebraic Pfaff equation given by a projective 1-form ω= 2  i=0 ωi(x0:x1:x2)dxi=0 360 A. Garc´ ıa Zamora with ωihomogeneous polynomials of degree m+ 1 and 2  i=0 xiωi≡0. b) A direction field Y= 2  i=0 Yi(x0:x1:x2)∂ ∂xi with Yihomogeneous polynomials of degree m, modulo a sum of G(x0:x1:x2)R, where R= 2  i=0 xi ∂ ∂xi is the radial vector field, and Gis a homogeneous polynomial. c) A map of vector bundles α:L−(m−1) −→ TP2where L−(m−1) = H⊗−(m−1), and His the hyperplane bundle on P2. If the greatest common divisor of (ωi) is equal to one, Fis called an irreducible foliation. Definition 2. The singular set Sof Fis determined by: a) The common zeros of the ωi. b) The points (x0:x1:x2) such that Y0(x0:x1:x2) x0 =Y1(x0:x1:x2) x1 =Y2(x0:x1:x2) x2 . c) The points pwhere α:L−(m−1) −→ TP2is not injective. The condition g.c.d.(ωi) = 1 is equivalent to the singular set of Fis finite. The algebraic multiplicity of the singularities is defined alternatively as: a) In affine coordinates, write a local expression for ωin terms of homogeneous monomials ω= ∞  k=r w1,k(x, y)dx + ∞  k=r w2,k(x, y)dy Foliations having a first integral 361 and the first number rsuch that w1,r or w2,r is not zero is called the algebraic multiplicity of the singularity. b) Analogous way for the local expression of the direction field. Moreover, if Fhas algebraic multiplicity 1 in a singular point pand the eigenvalues corresponding to the linear part of Fin pare non zero the singularity is called of multiplicity one. We assume in the following that Sis a discrete set and that the linear part of the associated local vector field is invertible (i.e., all the singularities have multiplicity one). Definition 3. According with each formulation in Definition 1, a curve Cin P2is called a leaf of Fif: a) It is a solution of the differential equation associated to ω=0. b) The tangent line in each point pof C−S coincides with the line determined by Y(p). c) For each point p,αsend the fibers of L−(m−1) on pinto the tangent line of the curve at p. Lemma 1.1. Let Fbe an irreducible foliation in P2of degree m, then the number of singular points, counted with multiplicity, is m2+m+1. For a proof see [9]. An algebraic solution of Fis an irreducible algebraic curve Cthat is the closure of a leaf of F. Let Cbe defined by F(x0:x1:x2) = 0. Then this can be expressed equivalently as: a) An algebraic irreducible curve Csuch that F|dF ∧ω. b) An algebraic irreducible curve Csuch that F| 2  i=0 Yi ∂F ∂xi . c) An algebraic irreducible curve Csuch that α(L−(m−1) |C)⊂TC. We say that Fhas a rational first integral if there exist a homogeneous rational fraction R(x0:x1:x2) such that dR ∧ω=0. It can be proved that the following are equivalent: i) Fhas an infinite number of algebraic solutions. ii) All the solutions of Fare algebraic. 362 A. Garc´ ıa Zamora iii) All the solutions of Fare the irreducible components of an irreducible pencil of plane curves λF +µG = 0 (i.e., for general (λ, µ) λF +µG = 0 is irreducible). iv) Fhas a rational first integral. (For a proof see [9].) The generic element of λF +µG = 0 is an irreducible curve of degree d. However, it is possible that for some finite number of (λi:µi) the corresponding curve is reducible or non-reduced λiF+µiG= j Unij ij . The values of (λi:µi) which have some nij >1 are called remarkable values. By Bertini’s Theorem the general member of this pencil is irreducible and non-singular away from the base locus, we call such a curve a generic solution of F. We remark that the curves of the pencil λF +µG =0 can be thought as the fibers of a rational map P2f −→ P1. The fact that the pencil is irreducible is equivalent to say that there does not exist a factorization of f: P2g −−−−→P1   h P1 −−−−→ −−− f with hof degree >1. Remarks. 1. If Fhas a rational first integral, then integrating the associated local field in each singular point we see that its linear part must be diagonal and the quotients of its eigenvalues u,vmust be rational (u/v ∈Q). In the following by (u, v) we denote relatively primes integer numbers. Moreover we suppose that u≤vand uv = 0, that is, the singularity is of multiplicity one. Following Poincar´e[10] we adopt the following notation: i) If u/v < 0 the singularity is called a saddle. ii) If 0 < u/v < 1 the singularity is called a monocritical node. iii) If u/v = 1 the singularity is called a dicritical node. 2. Note that if Fhas a first integral then the singularities of the algebraic curves in the associated pencil are determinated by the Foliations having a first integral 363 pair (u, v). More explicitly, if Cis a general solution of Fand F hasanodeoftype(u, v)atp, then the singularity of Cat pis analytically equivalent to r j=1(yu−ajxv), aj∈C(see [10]). A basic fact is: dF.G −F.dG = i,j Unij −1 ij ω and from this it follows at once that: (1.1) 2d−2=dij(nij −1) + m, where deg Uij =dij . We give here another, more geometric proof of (1.1). Lemma 1.2. Let Fbe a foliation in P2of degree m. Suppose that F admits a first integral, and the generic solutions are curves of degree d, then 2d−2=dij(nij −1) + m. Proof: Consider a line Lin P2such that it does not pass through any singular point of Fand is not tangent to any non-generic solution. To each point of Lcorrespond a unique value of (λ:µ), thus we have a rational map i:P1−→ P1, then the Riemann Hurwitz formula says −2=−2d+dij(nij −1)+m, where the ramifications are given by the remarkable values and the m points of tangency of Lwith the solutions of F. Notation. 1. In the following by a foliation Fin P2we understand an irreducible foliation with all its singular points of multiplicity one and with eingenvalues (ui,v i), where ui,v idenote relatively primes integers, ui≤vi. 2. A foliation as above with all its nodes dicritical is called a dicritical foliation. We recall that more generally we have: Definition 4. Let Sbe a complex algebraic surface (irreducible, reduced and non-singular). A foliation Fin Sis given by a non-identically zero morphism of vector bundles α:L−→ TS, 364 A. Garc´ ıa Zamora where Lis a line bundle in Sand TS is the tangent bundle of S.Fhas a rational first integral if the closure of the leaves are algebraic curves or, equivalently, if there exist a rational map Sβ −→ X, with Xa Riemann surface, such that the fibers of βare the leaves of F. Since all the arguments and definitions about singular points of foliations in P2are local, they can be extended at once to the case of algebraic surfaces. 2. Reformulation of the Poincar´e problem One of the fundamental results relating the degree of Fand the nature of its algebraic solutions is the following proposition (this result was proved by Poincar´e[10] in the case S=P2, our argument is a generalization of one due to Cerveau and Lins-Neto [3]). Proposition 2.1. Let Fbe a foliation on S with a first integral and singular points of multiplicity one. Denote the pair of relatively prime integers associated to the eigenvalues of the nodes of Fby (ui,v i), then we have: (1) n  i=1 ri(ui+vi)=−C.(KS+L), (2) g=−C.L 2−ri 2+1. Where Cis a general solution of F, g=geometric genus of C, α:L−→ TS defines F, KS=canonical divisor of S, ri=number of branches of Cthrough a node of F, n=number of nodes of F. Proof: Let ˜ Cπ −→ Cbe the normalization of C, the pullback of the restriction α(L|C) define a morphism ˜α:L|˜ C−→ T˜ Cwhich is not injective in the ripoints on the nodes Pi. This gives a section s∈H0(˜ C,T ˜ C⊗L−1|˜ C) with n i=1 rizeros, thus (2.1) n  i=1 ri=2−2g−C.L. Foliations having a first integral 365 Now, an easy generalization of the argument used in [8, p. 279–280], to compute the genus of a plane curve having simple singularities prove that if an embedded curve Chas singularities equivalent to ri j=1(yui−ajxvi) then the geometric genus g(C) is given by: (2.2) 2g−2=C.C +C.KS−r2 iuivi+ri(ui+vi−1). Combining (2.1) and (2.2) and C.C =n i=1 r2 iuiviwe get (1). To prove (2), it is sufficient to evaluate in the genus formula the relation obtained in (1). Remark. If S=P2, the formulas in 2.1 are: (1) ri(ui+vi)=(m+2)d. (2) 2g−2=(m−1)d−ri. The Poincar´e problem does not have a solution as formulated in the introduction. Here we present two examples: Example 2.1. Consider the foliation in P2given by the solutions of (2.3) pyz.dx +qxz.dy −(p+q)yx.dz =0 with p,q, positive integers, it is easy to see that any curve of the pencil λxpyq+µz(p+q)=0 is a solution of (2.3). We can choose d=p+qarbitrarily large and the foliation (2.3) is of degree 1. Example 2.2. Consider the pencil of algebraic curves defined in affine coordinates by (2.4) yd=λxd−2(x−1)(y−1). We have: Proposition 2.2. (1) The generic element of (2.4) is irreducible. (2) The foliation associated to (2.4) has degree 2and all its singular points are of multiplicity one. (3) g=     d−1 2,if dis odd d−2 2,if dis even. 372 A. Garc´ ıa Zamora P1    P1 −−−−→ −−− f (f1:f2) −−−−→ S (zn 0:zn 1) (fn 1,fn 2the corresponding multiple fibers of f) , and thus the pencil given by fis not irreducible. We conclude that n1=n2= 1 that is, his unramified and then r=1 and the general curve has only one branch through P. Note that using Lemma 1.2 we have obtained: Corollary 3.5. Let Fbe a foliation in P2of degree mwith a first integral of degree d, if all the singular points of Fhave eigenvalues (1,1) or (1,−1) then m=2d−2. References 1. W. Barth, C. Peters and A. Van de Ven,“Compact complex surfaces,” Springer Verlag, 1984. 2. M. Carnicer, The Poincar´e problem in the non-dicritical case, Ann. of Math. 140(2) (1994), 289–294. 3. D. Cerveau and A. Lins Neto, Holomorphic foliations in CP(2) having an invariant algebraic curve, Ann. Inst. Fourier (Grenoble) 41 (1991), 883–903. 4. R. H. Fox, On Frenchel’s conjecture about F-groups, Mat. Tidsskrift B (1952), 61–65. 5. W. Fulton, On the topology of algebraic varieties, Proc. 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Instituto de Matem´aticas UNAM Unidad Morelia Nicol´as Romero 150 Colonia Centro Morelia Michoac´an MEXICO e-mail: [email protected]h.mx Primera versi´o rebuda el 6 de Novembre de 1995, darrera versi´o rebuda el 23 de Gener de 1997