Abstract
We present enumerative results for holomorphic foliations by curves on \(\mathbb {P}^{n}\), n ≥ 3, with singularities of positive dimension. Some of the results presented improve previous ones due to Corrêa et al. (Annales de l’institut Fourier, 64(4):1781–1805, 2014) and Costa (Ann Fac Sci Toulouse, Math (6), 15(2):297–321, 2006). We also present an enumerative result bounding the number of isolated singularities in a projective subvariety invariant by a holomorphic foliation by curves on \(\mathbb {P}^{n}\) with a singularity of positive dimension. Moreover, we construct a family of holomorphic foliations by curves on \(\mathbb {P}^{n}\) with a singularity of a positive dimension where its Milnor number is exhibited.
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References
Baum P, Bott R. On the zeros of meromorphic vector-fields. Essays on Topology and Related topics, mémoires dďeés à Georges de Rham. Berlin: Springer; 1970, pp. 29–47.
Corrêa M, JR A, Fernández-Pérez G, Nonato Costa R. Vidal Martins: Foliations by curves with curves as singularities. Annales de l’institut Fourier 2014;64(4): 1781–1805.
Fulton W. Intersection theory. Berlin: Springer; 1984.
Griffiths P, Harris J. 1994. Principles of algebraic geometry. John Wiley & Sons Inc.
Porteous IR. Blowing up Chern class. Proc Cambridge Phil Soc 1960;56:118–124.
Costa GN. Holomorphic foliations by curves on \(\mathbb {P}^{3}\) with non-isolated singularities. Ann Fac Sci Toulouse, Math (6) 2006;15(2):297–321.
Costa GN. Indices Baum-Bott for curves of singularities. Bull Braz Math Soc 2016; 47(3):883–910.
Marcio G. Soares: Projective varieties invariant by one-dimensional foliations. Ann Math 2000;152:369–382.
Soares MG. Bounding Poincaré-Hopf indices and Milnor numbers. Math Nachr 2005;278(6):703–711.
Vainsencher I. Foliations singular along a curve. Trans London Math Soc 2015;2 (1):80–92.
Acknowledgments
The authors wish to express their gratitude to Renato Vidal Martins (UFMG) and John MacQuarrie (UFMG) for several helpful comments concerning this work. The first author was partially supported by CNPq Brazil grant numbers 427388/2016-3 and 301825/2016-5 and PRONEX/FAPERJ.
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Fernández-Pérez, A., Costa, G.N. On Foliations by Curves with Singularities of Positive Dimension. J Dyn Control Syst 26, 581–609 (2020). https://doi.org/10.1007/s10883-019-09466-1
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DOI: https://doi.org/10.1007/s10883-019-09466-1