某些正方形的几何学分类
Géométrie classique de certains feuilletages quadratiques
摘要：
复射影平面上的二次叶理集$\mathscr{F}(2;2)$可以与作用于其上的14维射影空间的\textsc{Zariski}的开集$\mathrm{Aut}(\mathbb{P}^2(\mathbb{C})).$相一致。我们对只有一个奇异性的$\mathbb{P}^2(\mathbb{C}),$二次叶理的自同构进行分类。只有四个这样的叶理直到共轭;而其中三个有一个动态，可以很容易地描述的动态的第四个仍然是神秘的。这种分类也使我们能够描述$\mathrm{Aut}(\mathbb{P}^2(\mathbb{C}))$对$\mathscr{F}(2;2).$的作用。一方面我们证明了轨道的维数大于6，并且恰好有两个$6;$维轨道。另一方面我们得到了$\mathscr{F} (2;2)$中的一般轨道的闭包至少包含7个~7维轨道和恰好一个$6.$维轨道
英文摘要：
The set $\mathscr{F}(2;2)$ of quadratic foliations on the complex projective plane can be identified with a \textsc{Zariski}'s open set of a projective space of dimension 14 on which acts $\mathrm{Aut}(\mathbb{P}^2(\mathbb{C})).$ We classify, up to automorphisms of $\mathbb{P}^2(\mathbb{C}),$ quadratic foliations with only one singularity. There are only four such foliations up to conjugacy; whereas three of them have a dynamic which can be easily described the dynamic of the fourth is still mysterious. This classification also allows us to describe the action of $\mathrm{Aut}(\mathbb{P}^2(\mathbb{C}))$ on $\mathscr{F}(2;2).$ On the one hand we show that the dimension of the orbits is more than 6 and that there are exactly two orbits of dimension $6;$ on the other hand we obtain that the closure of the generic orbit in $\mathscr{F} (2;2)$ contains at least seven orbits of dimension~7 and exactly one orbit of dimension $6.$
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