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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.16632 |
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| _version_ | 1866908645053169664 |
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| author | Pracias, Carla Luza, Maycol Falla |
| author_facet | Pracias, Carla Luza, Maycol Falla |
| contents | In this paper, we study homogeneous convex foliations on the complex projective plane $\mathbb{P}^2$. A foliation is called convex if all of its leaves, except straight lines, have no inflection points, and such foliations form a Zariski closed subset in the space of degree $d$ foliations on $\mathbb{P}^2$. Using projective duality, every foliation can be associated with a $d$-web on the dual plane via its Legendre transform, and it is known that the Legendre transform of a homogeneous convex foliation is flat. Our first main result provides a classification of homogeneous convex foliations admitting exactly three radial singularities on the line at infinity. As a second result, we complete the classification of convex homogeneous foliations of degree $6$, extending previous classifications in degrees $4$ and $5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16632 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homogeneous Convex Foliations of degree 6 Pracias, Carla Luza, Maycol Falla Algebraic Geometry Complex Variables Dynamical Systems In this paper, we study homogeneous convex foliations on the complex projective plane $\mathbb{P}^2$. A foliation is called convex if all of its leaves, except straight lines, have no inflection points, and such foliations form a Zariski closed subset in the space of degree $d$ foliations on $\mathbb{P}^2$. Using projective duality, every foliation can be associated with a $d$-web on the dual plane via its Legendre transform, and it is known that the Legendre transform of a homogeneous convex foliation is flat. Our first main result provides a classification of homogeneous convex foliations admitting exactly three radial singularities on the line at infinity. As a second result, we complete the classification of convex homogeneous foliations of degree $6$, extending previous classifications in degrees $4$ and $5$. |
| title | Homogeneous Convex Foliations of degree 6 |
| topic | Algebraic Geometry Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/2505.16632 |