Codimension one foliations of degree three on projective spaces

Costa, Raphael Constant da; Lizarbe, Ruben; Pereira, Jorge Vitorio

doi:10.1016/j.bulsci.2021.103092

Cited by 11 publications

(9 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This would lead to a more stringent concept for a family of foliations, which many authors have already studied. Our definition above aims to study the irreducible components of the space of foliations on projective spaces as considered by Cerveau, Lins Neto, and others, see [10,16] and references therein.…”

Section: Families Of Foliationsmentioning

confidence: 99%

“…Proposition 11.6 could be easily proved without using reduction to positive characteristic. Instead of considering the p-th powers of vector fields tangent to a subfoliation of degree one, one could consider the Zariski closure of the subgroup generated by the flow of these vector fields as is done in [16,Subsection 3.4].…”

mentioning

confidence: 99%

“…Our next result seems to be of different nature. It generalizes to arbitrary degree, an irreducible component of Fol 3 (P n C ) found in [16]. In degree three, the original proof relies on the structure theorem for degree three foliations on projective spaces established in [30] (n = 3), and [16, Theorem A] (n ≥ 3).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Codimension one foliations in positive characteristic

Mendson¹,

Pereira²

2022

Preprint

View full text Add to dashboard Cite

We investigate the geometry of codimension one foliations on smooth projective varieties defined over fields of positive characteristic with an eye toward applications to the structure of codimension one holomorphic foliations on projective manifolds. CONTENTS 1. Introduction 1 2. Distributions and foliations 2 Part 1. Geometry of foliations in positive characteristic 4 3. Differential geometry in positive characteristic 4 4. Particular features of foliations in positive characteristic 7 5. Degeneracy divisor of the p-curvature 13 6. Families of foliations 22 7. Integrability of the Cartier transform 24 8. Lifting the kernel of the p-curvature 26 Part 2. The space of holomorphic foliations on projective spaces 29 9. Foliations on complex projective spaces 29 10. Subdistributions of minimal degree 31 11. Codimension two subdistributions of small degree 34 12. Foliations without subdistributions of small degree 40 References 42

show abstract

Section: Families Of Foliationsmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Codimension one foliations in positive characteristic

Mendson¹,

Pereira²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…They provide a classification of such foliations by curves and degree ≤ 3 proving that the foliations of degree 1 or 2 are contained in a pencil of planes or is Legendrian, and are given by the complete intersection of two codimension one distributions, and that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. Recently, a partial classification in degree 3 and codimension one foliations was obtained in [25].…”

Section: Introductionmentioning

confidence: 99%

Dimension two holomorphic distributions on four-dimensional projective space

Calvo-Andrade¹,

Corrêa²,

Fonseca-Quispe³

2022

Preprint

View full text Add to dashboard Cite

We study two-dimensional holomorphic distributions on P 4 . We classify dimension two distributions, of degree at most 2, with either locally free tangent sheaf or locally free conormal sheaf and whose the singular scheme has pure dimension one. We show that the corresponding sheaves are split. Next, we investigate the geometry of such distributions, studying from maximally non-integrable to integrable distributions. In the maximally non-integrable case, we show that the distribution is either of Lorentzian type or a push-forward by a rational map of the Cartan prolongation of a singular contact struture on a weighted projective 3-fold. We study distributions of dimension two in P 4 whose the conormal sheaf are the Horrocks-Mumford sheaves, describing the numerical invariants of their singular schemes which are smooth and connected. Such distributions are maximally non-integrable, uniquely determined by their singular schemes and invariant by a group H 5 ⋊ SL(2, Z 5 ) ⊂ Sp(4, Q), where H 5 is the Heisenberg group of level 5. We prove that the moduli spaces of Horrocks-Mumford distributions are irreducible quasi-projective varieties and we determine their dimensions. Finally, we observe that the space of codimension one distributions, of degree d ≥ 6, on P 4 have a family of degenerated flat holomorphic Riemannian metrics. Moreover, the degeneracy divisors of such metrics consists of codimension one distributions invariant by H 5 ⋊ SL(2, Z 5 ) and singular along a degenerate abelian surface with (1, 5)-polarization and level-5-structure.

show abstract

“…them. More recently, Da Costa, Lizarbe, and Pereira studied the case 𝑑 = 3, 𝑛 ≥ 3 in[dCLP21]. They classify the 18 irreducible components of Fol(3, 𝑛) whose general member does not have a first integral.…”

mentioning

confidence: 99%

Foliations on Complex Manifolds

Araujo¹,

Figueredo²

2022

Notices Amer. Math. Soc.

View full text Add to dashboard Cite

In these notes we survey some aspects of the theory of holomorphic foliations on complex manifolds. The origins of the theory go back to works of Darboux, Poincaré and Painlevé, where it was developed to study solutions of ordinary differential equations on ℂ 2 . We briefly discuss some of the early works on this theory, mostly concerned with the local behavior of the leaves near the singularities. We then move the focus from local to global properties. Birational geometry has had a great influence on the development of a global theory of holomorphic foliations. After reviewing the Enriques-Kodaira classification of projective surfaces and explaining the general philosophy of the Mininal Model Program, we explore some of their recent counterparts for foliations.

show abstract

Codimension one foliations of degree three on projective spaces

Cited by 11 publications

References 23 publications

Codimension one foliations in positive characteristic

Codimension one foliations in positive characteristic

Dimension two holomorphic distributions on four-dimensional projective space

Foliations on Complex Manifolds

Contact Info

Product

Resources

About