6次齐次凸叶状结构
Homogeneous Convex Foliations of degree 6
AI总结：
本文研究复射影平面上的齐次凸叶状结构，利用射影对偶性关联叶状结构与对偶平面$d$-网，分类了具三个径向奇点的齐次凸叶状结构并完成6次凸齐次叶状结构分类。
摘要：
本文研究复射影平面$\mathbb{P}^2$上的齐次凸叶状结构。若叶状结构的所有叶（直线除外）均无拐点，则称为凸叶状结构，这类叶状结构构成$\mathbb{P}^2$上$d$次叶状结构空间中的扎里斯基闭子集。利用射影对偶性，每个叶状结构可通过勒让德变换与对偶平面上的$d$-网关联，已知齐次凸叶状结构的勒让德变换是平坦的。第一个主要结果给出了在无穷远线上恰好具有三个径向奇点的齐次凸叶状结构的分类。第二个结果完成了6次凸齐次叶状结构的分类，扩展了先前对4次和5次的分类。
英文摘要：
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 $5$.
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