Abstract
In this work, we construct, for any \(d \ge 2\), a new foliation on \(\mathbb {CP}^2\) of degree d with a unique singular point of multiplicity \(d-1\) without invariant algebraic curves that contain all its separatrices. We also prove that if X is a foliation on \(\mathbb {CP}^2\) with a unique nilpotent singular point, then X has no algebraic leaves. Finally, we characterize logarithmic foliations on \(\mathbb {CP}^2\) with a unique singular point. And we give some new examples of this kind of foliations.
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Acknowledgements
I would like to thank Jawad Snoussi for helpful conversations on this work and for the company during the pandemic, as well as the referees for very useful comments and suggestions which helped to improve this paper.
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This work was supported by Conacyt (Grant 284424)
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Alcántara, C.R. Special foliations on \(\mathbb {CP}^2\) with a unique singular point. Res Math Sci 9, 15 (2022). https://doi.org/10.1007/s40687-022-00311-9
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DOI: https://doi.org/10.1007/s40687-022-00311-9