Cuspidal Cubic Projective Foliations: Global Rigidity, Deformations, and Pull-Backs

Abstract

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.