$\mathbb{P}^n$上曲线全形叶状的泛奇点

Sahil Gehlawat, Viêt-Anh Nguyên
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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。
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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.
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