On the characterization of algebraically integrable plane foliations

Galindo, C.; Monserrat, F.

doi:10.1090/S0002-9947-10-04808-7

On the characterization of algebraically integrable plane foliations
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Trans. Amer. Math. Soc. 362 (2010), 4557-4568 Request permission

Abstract:

We give a characterization theorem for non-degenerate plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree $r$ of a non-degenerate foliation as above provides the minimum number, $r+1$, of points in the projective plane through which pass infinitely many algebraic leaves of the foliation.

References

L. Autonne, Sur la théorie des équations différentielles du premier ordre et du premier degré, J. École Polytech. 61 (1891), 35-122; 62 (1892), 47-180.
Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314, DOI 10.1017/CBO9780511623936
Marco Brunella, Birational geometry of foliations, Monografías de Matemática. [Mathematical Monographs], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2000. Available electronically at http://www.impa.br/Publicacoes/Monografias/Abstracts/brunella.ps. MR 1948251
César Camacho and Paulo Sad, Pontos singulares de equações diferenciais analíticas, 16$^\textrm {o}$ Colóquio Brasileiro de Matemática. [16th Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1987 (Portuguese). MR 953780
Antonio Campillo and Manuel M. Carnicer, Proximity inequalities and bounds for the degree of invariant curves by foliations of $\mathbf P^2_\textbf {C}$, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2211–2228. MR 1407696, DOI 10.1090/S0002-9947-97-01898-9
Antonio Campillo and Jorge Olivares, Polarity with respect to a foliation and Cayley-Bacharach theorems, J. Reine Angew. Math. 534 (2001), 95–118. MR 1831632, DOI 10.1515/crll.2001.036
Manuel M. Carnicer, The Poincaré problem in the nondicritical case, Ann. of Math. (2) 140 (1994), no. 2, 289–294. MR 1298714, DOI 10.2307/2118601
Eduardo Casas-Alvero, Singularities of plane curves, London Mathematical Society Lecture Note Series, vol. 276, Cambridge University Press, Cambridge, 2000. MR 1782072, DOI 10.1017/CBO9780511569326
G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 32 (1878), 60-96; 123-144; 151-200.
A. A. du Plessis and C. T. C. Wall, Application of the theory of the discriminant to highly singular plane curves, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 2, 259–266. MR 1670229, DOI 10.1017/S0305004198003302
E. Esteves and S. Kleiman, Bounds on leaves of one-dimensional foliations, Bull. Braz. Math. Soc. (N.S.) 34 (2003), no. 1, 145–169. Dedicated to the 50th anniversary of IMPA. MR 1993042, DOI 10.1007/s00574-003-0006-3
C. Galindo and F. Monserrat, Algebraic integrability of foliations of the plane, J. Differential Equations 231 (2006), no. 2, 611–632. MR 2287899, DOI 10.1016/j.jde.2006.05.011
J. García de la Fuente, Geometría de los sistemas lineales de series de potencias en dos variables, Ph.D. thesis, Valladolid University, 1989.
G.M. Greuel, G. Pfister and H. Schöenemann, Singular 3.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern, 2005. http://www.singular.uni-kl.de.
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Alcides Lins Neto, Some examples for the Poincaré and Painlevé problems, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 2, 231–266 (English, with English and French summaries). MR 1914932, DOI 10.1016/S0012-9593(02)01089-3
Joseph Lipman, Proximity inequalities for complete ideals in two-dimensional regular local rings, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 293–306. MR 1266187, DOI 10.1090/conm/159/01512
P. Painlevé, Sur les intégrales algébriques des équations différentielles du premier ordre and Mémoire sur les équations différentielles du premier ordre in Oeuvres de Paul Painlevé, Tome II, Éditions du Centre National de la Recherche Scientifique 15, quai Anatole-France, Paris 1974.
H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math. Pures Appl. 3 (7) (1881), 375-442; 3 (8) (1882), 251-296; 4 (1) (1885), 167-244; in Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, 3-84, 95-114.
—, Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré (I), Rend. Circ. Mat. Palermo 5 (1891), 161-191.
—, Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré (II), Rend. Circ. Mat. Palermo 11 (1897), 193-239.
Kyoji Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142 (German). MR 294699, DOI 10.1007/BF01405360
A. Seidenberg, Reduction of singularities of the differential equation $A\,dy=B\,dx$, Amer. J. Math. 90 (1968), 248–269. MR 220710, DOI 10.2307/2373435
Etsuo Yoshinaga and Masahiko Suzuki, Topological types of quasihomogeneous singularities in $\textbf {C}^{2}$, Topology 18 (1979), no. 2, 113–116. MR 544152, DOI 10.1016/0040-9383(79)90029-6
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249, DOI 10.1007/978-3-662-29244-0