Degree of the first integral of a pencil in \(\mathbb{P}^2\) defined by Lins Neto. (English) Zbl 1281.32028

Summary: Let \(\mathcal{P}_4\) be the linear family of foliations of degree \(4\) in \(\mathbb{P}^2\) introduced by A. Lins Neto, whose set of parameters with first integral \(I_p(\mathcal{P}_4)\) is dense and countable. In this work, we compute explicitly the degree of the rational first integral of the foliations in this linear family, as a function of the parameter.


32S65 Singularities of holomorphic vector fields and foliations
37F75 Dynamical aspects of holomorphic foliations and vector fields
Full Text: DOI arXiv Euclid


[1] M. M. Carnicer, The Poincaré problem in the nondicritical case, Ann. of Math. (2) 140(2) (1994), 289\Ndash294. \smallDOI: 10.2307/2118601. · Zbl 0821.32026
[2] D. Cerveau and A. Lins Neto, Holomorphic foliations in \(\mathbb{CP}(2)\) having an invariant algebraic curve, Ann. Inst. Fourier (Grenoble) 41(4) (1991), 883\Ndash903. · Zbl 0734.34007
[3] G. E. Collins and J. R. Johnson, The probability of relative primality of Gaussian integers, in: “Symbolic and algebraic computation” (Rome, 1988), Lecture Notes in Comput. Sci. 358 , Springer, Berlin, 1989, pp. 252\Ndash258. \smallDOI: 10.1007/3-540-51084-2\(_{-}\)23.
[4] G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré, Bull. Sci. Math. Astron. , Sér. 2, 2(1) (1878), 60\Ndash96, 151\Ndash200. · JFM 10.0214.01
[5] A. García Zamora, Foliations in algebraic surfaces having a rational first integral, Publ. Mat. 41(2) (1997), 357\Ndash373. \smallDOI: 10.5565/PUBLMAT\(_{-}\)41297\(_{-}\)03. · Zbl 0910.32039
[6] P. Griffiths and J. Harris, “Principles of algebraic geometry” , Reprint of the 1978 original, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. · Zbl 0836.14001
[7] T. Hayashida and M. Nishi, Existence of curves of genus two on a product of two elliptic curves, J. Math. Soc. Japan 17 (1965), 1\Ndash16. \smallDOI: 10.2969/jmsj/01710001. · Zbl 0132.41701
[8] J. P. Jouanolou, “Équations de Pfaff algébriques” , Lecture Notes in Mathematics 708 , Springer, Berlin, 1979. · Zbl 0477.58002
[9] A. Lins Neto, Some examples for the Poincaré and Painlevé problems, Ann. Sci. École Norm. Sup. (4) 35(2) (2002), 231\Ndash266. \smallDOI: 10.1016/S0012-9593(02)01089-3. · Zbl 1130.34301
[10] A. Lins Neto, Curvature of pencils of foliations. Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. I, Astérisque 296 (2004), 167\Ndash190. · Zbl 1081.32021
[11] A. Lins Neto and B. Azevedo Scárdua, “Folheações algébricas complexas” , 21 Colóquio Brasileiro de Matemática, Conselho Nacional de Desenvolvimento Científico e Technológico, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1997.
[12] M. McQuillan, Non-commutative Mori Theory, rev. version, Technical Report IHES-M-2001-42 (2001). · Zbl 1078.14503
[13] J. V. Pereira, Vector fields, invariant varieties and linear systems, Ann. Inst. Fourier (Grenoble) 51(5) (2001), 1385\Ndash1405. \smallDOI: 10.5802/aif.1858. · Zbl 1107.37038
[14] H. Poincaré, Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré, Rend. Circ. Mat. Palermo 5(1) (1891), 161\Ndash191. · JFM 23.0319.01
[15] L. Puchuri, Famílias lineares de folheações com curvatura zero numa superfície complexa compacta, PhD thesis, Instituto de Matemática Pura e Aplicada (2010).
[16] M. G. Soares, The Poincaré problem for hypersurfaces invariant by one-dimensional foliations, Invent. Math. 128(3) (1997), 495\Ndash500. \smallDOI: 10.1007/s002220050150. · Zbl 0923.32025
[17] I. Stewart and D. Tall, “Algebraic number theory and Fermat’s last theorem” , Third edition, A K Peters, Ltd., Natick, MA, 2002. · Zbl 0994.11001
[18] A. Walfisz, Uber die Wirksamkeit einiger Abschätzungen trigonometrischer Summen, Acta Arith. 4 (1958), 108\Ndash180. · Zbl 0084.27304
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