Algebraic separatrices for non-dicritical foliations on projective spaces of dimension at least four

Degenerate singularities of one dimensional Foliations

We give formulas for the degrees of the spaces of foliations in P2 with a dicritical singularity of prescribed order. Blowing up such singularity induces, generically, a foliation with all but finitely many leaves transversal to the exceptional line; we also find the degree of the locus defined by imposing a leaf of total contact with the exceptional line.

闭亚纯 1-形式

Closed meromorphic 1-forms

本文从闭亚纯 1-形式与平亚纯联络视角综述叶状结构经典理论，并应用于研究半整体分界线代数性及紧 Kähler 流形上数值平凡法丛超曲面的几何。

Invariant Surfaces for Toric Type Foliations in Dimension Three

A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on (C3, 0), without saddle-nodes, has invariant surface. We extend the argument of Cano-Cerveau, done for the nondicritical case, to the compact dicritical components of the exceptional divisor. These components are projective toric surfaces and the isolated invariant branches of the induced foliation extend to global curves. We build the invariant surface as a germ along the singular locus and those global invariant curves. The result of Ortiz-Rosales-Voronin, about the…

三维径向叶状结构

Radial foliations in dimension three

本文针对三维全纯叶状结构，定义近径向与径向叶状空间，证明径向叶状对应“开书”情形，并讨论一般近径向情形。

三维簇上余秩一叶状结构的极小模型计划（MMP）

MMP for co-rank one foliations on threefolds

该研究针对Q-阶乘射影三维簇上余秩一的F-dlt叶状配对，证明了翻转存在、特殊终止、基点自由定理及对数一般型下极小模型存在，推导了相关修正与奇点性质，验证了数值平凡情形的丰饶性，推进了叶状结构极小模型计划研究。

Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds

We provide several applications of the minimal model program to the local and global study of co-rank one foliations on threefolds. Locally, we prove a singular variant of Malgrange's theorem, a classification of terminal foliation singularities and the existence of separatrices for log canonical singularities. Globally, we prove termination of flips, a connectedness theorem on lc centres, a non-vanshing theorem and some hyperbolicity properties of foliations.

On the Camacho-Lins Neto regularity

We work with codimension one foliations in the projective space $\mathbb{P}^{n}$, given a differential one form $ω\in H^0(\mathbb{P}^n,Ω^1_{\mathbb{P}^n}(e))$, such differential form verifies the Frobenius integrability condition $ω\wedge dω=0$. In this work we show that the Camacho-Lins Neto regularity, applied for $ω$, is equivalent to the fact that every first order unfolding of $ω$ is trivial up to isomorphism. We do this by computing the Castelnuovo-Mumford regularity of the ideal $I(ω)$ of first order unfoldings. With this result, we are also showing that the only regular projective foli…