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⊂PP2 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 Fp admits a nonconstant holomorphic first integral with Cp as its unique separatrix, and the reduction of Fq contains no saddle-nodes, then F has degree two and is projectively equivalent to the standard cuspidal foliation dd(x2+y3)=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 ℚ. Our second main result concerns degree-two deformations. Finally, for n≥3 we introduce the cuspidal linear pull-back locus 𝒞ncusp, prove that it is a proper closed irreducible algebraic subset of the classical linear pull-back component of Fol2(PPn), 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.
Related
- Rigidity and Affine Deformations of Foliations on Nilpotent and Solvable Contact Lie Algebras
- Effective Results for Foliations on Smooth Projective Complete Intersection Surfaces
- Singular Levi-flats and holomorphic foliations on projective spaces
- Limit cycles of singular algebraic foliations on the real projective plane
- Stability and Logarithmic Foliations on Complex Projective Spaces
BibTeX
@misc{scardua2026cuspidal,
title = {{Cuspidal Cubic Projective Foliations: Global Rigidity, Deformations, and Pull-Backs}},
author = {Bruno Scardua},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26152},
eprint = {2609.26152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26152},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cuspidal Cubic Projective Foliations: Global Rigidity, Deformations, and Pull-Backs
AU - Bruno Scardua
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26152
UR - https://arxiv.org/abs/2609.26152
AB - We study rigidity phenomena for holomorphic foliations associated with irreducible cuspidal cubics. If C⊂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 ℚ. Our second main result concerns degree-two deformations. Finally, for n≥3 we introduce the cuspidal linear pull-back locus 𝒞_n^cusp, prove that it is a proper closed irreducible algebraic subset of the classical linear pull-back component of 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.
ER -