Abstract
We study holomorphic foliations on \(\mathbb {CP}^2\) of degree d with a singular point with Milnor number \(d^2+d+1\) and non-zero linear part. Given a foliation, using the singular scheme of the foliation and the lexicographical monomial order, we give necessary and sufficient conditions to have this kind of singularities. We show that if the singularity is nilpotent, the Gröbner basis with respect to this order gives us the normal form around the singular point. Finally, we prove that these foliations have no invariant lines and we exhibit a family of foliations with a nilpotent singularity with Milnor number \(d^2+d+1\), for d odd.
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Acknowledgements
I would like to thank Abraham Martín del Campo for helpful conversations on Gröbner Basis and on the use of Macaulay2. As well as the referees for useful comments which helped to improve this paper.
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Alcántara, C.R. Foliations on \(\mathbb {CP}^2\) of degree d with a singular point with Milnor number \(d^2+d+1\) . Rev Mat Complut 31, 187–199 (2018). https://doi.org/10.1007/s13163-017-0239-0
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DOI: https://doi.org/10.1007/s13163-017-0239-0