发表机构
Institute of Mathematics, Federal University of Rio de Janeiro（里约热内卢联邦大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究不可约尖点三次曲线相关的全纯叶状结构，证明次数至多七的整体刚性分类，引入尖点线性拉回轨迹并证明其刚性，主要贡献在于分类与拉回识别。
AI 中文摘要
我们研究与不可约尖点三次曲线相关的全纯叶状结构的刚性现象。若 $C\subset\PP^2$ 是此类三次曲线，我们记 $p$ 为其尖点，$q$ 为其唯一的光滑拐点（或挠点），在该点切线与其三次曲线有三阶接触。我们的第一个主要结果是次数不超过七的内在整体分类：若次数至多为七的叶状结构 $\F$ 保持 $C$ 不变，具有奇异集 $\{p,q\}$，芽 $\F_p$ 允许一个非常数的全纯首次积分且以 $C_p$ 为其唯一分界线，且 $\F_q$ 的约化不含鞍结点，则 $\F$ 的次数为二，且射影等价于标准尖点叶状结构 \\[ \dd(x^2+y^3)=0. \\] 特别地，它允许一个有理首次积分。证明结合了适应不变尖点的多项式正规形、光滑拐点处的整体接触约束，以及由全纯首次积分产生的加权局部方程。这些要素消去了直至次数七的所有情形；仅极端的次数七情形需要最后的在 $\mathbb Q$ 上的精确代数计算。我们的第二个主要结果涉及次数二的形变。最后，对于 $n\geq3$，我们引入尖点线性拉回轨迹 $\mathcal C_n^{\mathrm{cusp}}$，证明它是 $\operatorname{Fol}_2(\PP^n)$ 的经典线性拉回分支的一个真闭不可约代数子集，并证明对于保持在该拉回轨迹中的形变，基叶状结构的刚性。这引出了一个内在的拉回识别问题，更广泛地，引出了对射影叶状结构空间不可约分支内几何上显著的不可约代数子轨迹的研究。
英文摘要
We study rigidity phenomena for holomorphic foliations associated with irreducible cuspidal cubics. If $C\subset\PP^2$ is such a cubic, we denote by $p$ its cusp and by $q$ its unique smooth inflection point (or flex), where the tangent line has contact of order three with the cubic. Our first main result is an intrinsic global classification through degree seven: if a foliation $\F$ of degree at most seven leaves $C$ invariant, has singular set $\{p,q\}$, the germ $\F_p$ admits a nonconstant holomorphic first integral with $C_p$ as its unique separatrix, and the reduction of $\F_q$ contains no saddle-nodes, then $\F$ has degree two and is projectively equivalent to the standard cuspidal foliation \[ \dd(x^2+y^3)=0. \] In particular it admits a rational first integral. The proof combines polynomial normal forms adapted to the invariant cusp, a global contact constraint at the smooth flex, and weighted local equations arising from the holomorphic first integral. These ingredients eliminate all cases up to degree seven; only the extremal degree-seven case requires a final exact algebraic computation over $\mathbb Q$. Our second main result concerns degree-two deformations. Finally, for $n\geq3$ we introduce the cuspidal linear pull-back locus $\mathcal C_n^{\mathrm{cusp}}$, prove that it is a proper closed irreducible algebraic subset of the classical linear pull-back component of $\operatorname{Fol}_2(\PP^n)$, and prove rigidity of the base foliation for deformations which remain in that pull-back locus. This leads to an intrinsic pull-back recognition problem and, more broadly, to the study of geometrically distinguished irreducible algebraic subloci inside irreducible components of spaces of projective foliations.
