Proximity inequalities and bounds for the degree of invariant curves by foliations of ℙ_{ℂ}²

Campillo, Antonio; Carnicer, Manuel

doi:10.1090/S0002-9947-97-01898-9

Proximity inequalities and bounds for the degree of invariant curves by foliations of $\mathbb {P}_{\mathbb {C}}^2$
HTML articles powered by AMS MathViewer

by Antonio Campillo and Manuel M. Carnicer PDF

Trans. Amer. Math. Soc. 349 (1997), 2211-2228 Request permission

Abstract:

In this paper we prove that if $C$ is a reduced curve which is invariant by a foliation $\mathcal F$ in the complex projective plane then one has $\partial ^{\underline {\circ }} C\leq \partial ^{\underline {\circ }} \mathcal F+2+a$ where $a$ is an integer obtained from a concrete problem of imposing singularities to projective plane curves. If $\mathcal F$ is nondicritical or if $C$ has only nodes as singularities, then one gets $a=0$ and we recover known bounds. We also prove proximity formulae for foliations and we use these formulae to give relations between local invariants of the curve and the foliation.

References

César Camacho and Paulo Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2) 115 (1982), no. 3, 579–595. MR 657239, DOI 10.2307/2007013
César Camacho, Alcides Lins Neto, and Paulo Sad, Topological invariants and equidesingularization for holomorphic vector fields, J. Differential Geom. 20 (1984), no. 1, 143–174. MR 772129
Antonio Campillo, Gérard Gonzalez-Sprinberg, and Monique Lejeune-Jalabert, Enriques diagrams, resolutions and toric clusters, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 329–334 (English, with English and French summaries). MR 1320380
F. Cano, Dicriticalness of a singular foliation, Holomorphic dynamics (Mexico, 1986) Lecture Notes in Math., vol. 1345, Springer, Berlin, 1988, pp. 73–94. MR 980953, DOI 10.1007/BFb0081396
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
E. Casas-Alvero, Infinitely near imposed singularities and singularities of polar curves, Math. Ann. 287 (1990), no. 3, 429–454. MR 1060685, DOI 10.1007/BF01446904
D. Cerveau and A. Lins Neto, Holomorphic foliations in $\textbf {C}\textrm {P}(2)$ having an invariant algebraic curve, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 4, 883–903 (English, with French summary). MR 1150571, DOI 10.5802/aif.1278
D. Cerveau and J.-F. Mattei, Formes intégrables holomorphes singulières, Astérisque, vol. 97, Société Mathématique de France, Paris, 1982 (French). With an English summary. MR 704017
H. Dulac. Recherches sur les points singuliers des équations différentielles. Journal de l’Ecole Polytechnique, $2^e$ série, 9:1–125, 1904.
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Joseph Lipman, Adjoints and polars of simple complete ideals in two-dimensional regular local rings, Bull. Soc. Math. Belg. Sér. A 45 (1993), no. 1-2, 223–244. Third Week on Algebra and Algebraic Geometry (SAGA III) (Puerto de la Cruz, 1992). MR 1316244
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
H. Poincaré. Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré (I and II). Rendiconti del circolo matematico di Palermo, 5 and 11:161–191 and 193–239, 1891 and 1897.
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
O. Zariski. Studies in equisingularity I. American Journal of Math., 87:507–535, 1965.