On some indices of foliations and applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912788769669120 |
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| author | Fernández-Pérez, Arturo Barroso, Evelia R. García Saravia-Molina, Nancy |
| author_facet | Fernández-Pérez, Arturo Barroso, Evelia R. García Saravia-Molina, Nancy |
| contents | In this paper we establish a relationship between the Milnor number, the $χ$-number, and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the Gómez-Mont--Seade--Verjovsky index, we prove that the difference between the multiplicity and the Tjurina number of a foliation with respect to a reduced curve is independent of the foliation. We also derive a local formula for the Tjurina number of a foliation with respect to a reduced curve. From a global point of view, these results lead to the following consequences: we provide a new proof of a global result regarding the multiplicity of a foliation due to Cerveau-Lins Neto and a new proof of a Soares's inequality for the sum of the Milnor number of an invariant curve of a foliation. Additionally, we obtain bounds for the global Tjurina number of a foliation on the complex projective plane. Finally, we provide an answer to the conjecture posed by Alcántara and Mozo-Fernández about foliations on the complex projective plane having a unique singularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_01090 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On some indices of foliations and applications Fernández-Pérez, Arturo Barroso, Evelia R. García Saravia-Molina, Nancy Complex Variables Algebraic Geometry In this paper we establish a relationship between the Milnor number, the $χ$-number, and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the Gómez-Mont--Seade--Verjovsky index, we prove that the difference between the multiplicity and the Tjurina number of a foliation with respect to a reduced curve is independent of the foliation. We also derive a local formula for the Tjurina number of a foliation with respect to a reduced curve. From a global point of view, these results lead to the following consequences: we provide a new proof of a global result regarding the multiplicity of a foliation due to Cerveau-Lins Neto and a new proof of a Soares's inequality for the sum of the Milnor number of an invariant curve of a foliation. Additionally, we obtain bounds for the global Tjurina number of a foliation on the complex projective plane. Finally, we provide an answer to the conjecture posed by Alcántara and Mozo-Fernández about foliations on the complex projective plane having a unique singularity. |
| title | On some indices of foliations and applications |
| topic | Complex Variables Algebraic Geometry |
| url | https://arxiv.org/abs/2506.01090 |