Foliations on \(\mathbb {CP}^2\) of degree \(d\) with a singular point with Milnor number \(d^2+d+1\). (English) Zbl 1381.37056
Summary: 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.
MSC:
| 37F75 | Dynamical aspects of holomorphic foliations and vector fields |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 32S65 | Singularities of holomorphic vector fields and foliations |
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