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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.18654 |
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| _version_ | 1866915340904038400 |
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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 | Let $\mathcal{F}$ be a holomorphic foliation at $p\in\mathbb{C}^2$, and $B$ be a separatrix of $\mathcal{F}$. We prove the following Dimca-Greuel type inequality $3μ_p(\mathcal{F},B)-4τ_p(\mathcal{F},B)+GSV_p(\mathcal{F},B)\leq 0$, where $μ_p(\mathcal{F},B)$ is the multiplicity of $\mathcal{F}$ along $B$, $τ_p(\mathcal{F},B)$ is the dimension of the quotient of $\mathbb{C}\{x,y\}$ by the ideal generated by the components of any $1$-form defining $\mathcal{F}$ and any equation of $B$, and $GSV_p(\mathcal{F},B)$ is the \textit{Gómez-Mont-Seade-Verjovsky index} of the foliation $\mathcal{F}$ with respect to $B$. As a consequence, we provide a new proof of the $\frac{4}{3}$-Dimca-Greuel conjecture for singularities of irreducible plane curve germs, with foliations ingredients, that differs from those given by Alberich-Carramiñana, Almirón, Blanco, Melle-Hernández and Genzmer-Hernandes, but it is in line with the idea developed by Wang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18654 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An upper bound for the GSV-index of a foliation Fernández-Pérez, Arturo Barroso, Evelia R. García Saravia-Molina, Nancy Complex Variables Differential Geometry 32S65, 32M25 Let $\mathcal{F}$ be a holomorphic foliation at $p\in\mathbb{C}^2$, and $B$ be a separatrix of $\mathcal{F}$. We prove the following Dimca-Greuel type inequality $3μ_p(\mathcal{F},B)-4τ_p(\mathcal{F},B)+GSV_p(\mathcal{F},B)\leq 0$, where $μ_p(\mathcal{F},B)$ is the multiplicity of $\mathcal{F}$ along $B$, $τ_p(\mathcal{F},B)$ is the dimension of the quotient of $\mathbb{C}\{x,y\}$ by the ideal generated by the components of any $1$-form defining $\mathcal{F}$ and any equation of $B$, and $GSV_p(\mathcal{F},B)$ is the \textit{Gómez-Mont-Seade-Verjovsky index} of the foliation $\mathcal{F}$ with respect to $B$. As a consequence, we provide a new proof of the $\frac{4}{3}$-Dimca-Greuel conjecture for singularities of irreducible plane curve germs, with foliations ingredients, that differs from those given by Alberich-Carramiñana, Almirón, Blanco, Melle-Hernández and Genzmer-Hernandes, but it is in line with the idea developed by Wang. |
| title | An upper bound for the GSV-index of a foliation |
| topic | Complex Variables Differential Geometry 32S65, 32M25 |
| url | https://arxiv.org/abs/2403.18654 |