{"title":"$\\mathbb{P}^n$上曲线全形叶状的泛奇点","authors":"Sahil Gehlawat, Viêt-Anh Nguyên","doi":"arxiv-2409.06052","DOIUrl":null,"url":null,"abstract":"Let $\\mathcal{F}_d(\\mathbb{P}^n)$ be the space of all singular holomorphic\nfoliations by curves on $\\mathbb{P}^n$ ($n \\geq 2$) with degree $d \\geq 1.$ We\nshow that there is subset $\\mathcal{S}_d(\\mathbb{P}^n)$ of\n$\\mathcal{F}_d(\\mathbb{P}^n)$ with full Lebesgue measure with the following\nproperties: 1. for every $\\mathcal{F} \\in \\mathcal{S}_d(\\mathbb{P}^n),$ all singular\npoints of $\\mathcal{F}$ are linearizable hyperbolic. 2. If, moreover, $d \\geq 2,$ then every $\\mathcal{F}$ does not possess any\ninvariant algebraic curve.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generic singularities of holomorphic foliations by curves on $\\\\mathbb{P}^n$\",\"authors\":\"Sahil Gehlawat, Viêt-Anh Nguyên\",\"doi\":\"arxiv-2409.06052\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $\\\\mathcal{F}_d(\\\\mathbb{P}^n)$ be the space of all singular holomorphic\\nfoliations by curves on $\\\\mathbb{P}^n$ ($n \\\\geq 2$) with degree $d \\\\geq 1.$ We\\nshow that there is subset $\\\\mathcal{S}_d(\\\\mathbb{P}^n)$ of\\n$\\\\mathcal{F}_d(\\\\mathbb{P}^n)$ with full Lebesgue measure with the following\\nproperties: 1. for every $\\\\mathcal{F} \\\\in \\\\mathcal{S}_d(\\\\mathbb{P}^n),$ all singular\\npoints of $\\\\mathcal{F}$ are linearizable hyperbolic. 2. If, moreover, $d \\\\geq 2,$ then every $\\\\mathcal{F}$ does not possess any\\ninvariant algebraic curve.\",\"PeriodicalId\":501113,\"journal\":{\"name\":\"arXiv - MATH - Differential Geometry\",\"volume\":\"28 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Differential Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06052\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Differential Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06052","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
让 $\mathcal{F}_d(\mathbb{P}^n)$ 是 $\mathbb{P}^n$ ($n \geq 2$)上所有度数为 $d \geq 1 的曲线的奇异全形变换空间。$ Weshow that there is subset $\mathcal{S}_d(\mathbb{P}^n)$ of$\mathcal{F}_d(\mathbb{P}^n)$ with full Lebesgue measure with the followingproperties:1. 对于每一个 $\mathcal{F}\在 \mathcal{S}_d(\mathbb{P}^n)中,$ $mathcal{F}$的所有奇点都是可线性化双曲的。2.此外,如果 $d \geq 2, $ 那么每个 $\mathcal{F}$ 都不具有任何不变的代数曲线。
Generic singularities of holomorphic foliations by curves on $\mathbb{P}^n$
Let $\mathcal{F}_d(\mathbb{P}^n)$ be the space of all singular holomorphic
foliations by curves on $\mathbb{P}^n$ ($n \geq 2$) with degree $d \geq 1.$ We
show that there is subset $\mathcal{S}_d(\mathbb{P}^n)$ of
$\mathcal{F}_d(\mathbb{P}^n)$ with full Lebesgue measure with the following
properties: 1. for every $\mathcal{F} \in \mathcal{S}_d(\mathbb{P}^n),$ all singular
points of $\mathcal{F}$ are linearizable hyperbolic. 2. If, moreover, $d \geq 2,$ then every $\mathcal{F}$ does not possess any
invariant algebraic curve.