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Effective Results for Foliations on Smooth Projective Complete Intersection Surfaces
Abstract
We study holomorphic foliations on projective spaces that leave smooth projective complete intersection surfaces M invariant. We determine precisely for which degrees such foliations on M exist. As a consequence, we obtain new bounds for the classical Poincaré problem for smooth projective complete intersection surfaces and prove that previously known bounds for smooth hypersurfaces in ℙ3 are optimal. Furthermore, for a foliation [s] on M with isolated singularities and for degrees beyond an explicit bound that we provide, we show that a section s' has singular scheme containing that of s if and only if s'=ϕ(s) for some global endomorphism ϕ of the tangent bundle of M.
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BibTeX
@misc{olivares2026effective,
title = {{Effective Results for Foliations on Smooth Projective Complete Intersection Surfaces}},
author = {Jorge Olivares and Daniel Posada-Buriticá},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.09877},
eprint = {2608.09877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09877},
}
RIS
TY - UNPB
TI - Effective Results for Foliations on Smooth Projective Complete Intersection Surfaces
AU - Jorge Olivares
AU - Daniel Posada-Buriticá
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2608.09877
UR - https://arxiv.org/abs/2608.09877
AB - We study holomorphic foliations on projective spaces that leave smooth projective complete intersection surfaces M invariant. We determine precisely for which degrees such foliations on M exist. As a consequence, we obtain new bounds for the classical Poincaré problem for smooth projective complete intersection surfaces and prove that previously known bounds for smooth hypersurfaces in ℙ^3 are optimal. Furthermore, for a foliation [s] on M with isolated singularities and for degrees beyond an explicit bound that we provide, we show that a section s' has singular scheme containing that of s if and only if s'=ϕ(s) for some global endomorphism ϕ of the tangent bundle of M.
ER -