Bounds for sectional genera of varieties invariant under Pfaff fields

Corrêa, Maurício; Jardim, Marcos

doi:10.1215/ijm/1385129951

Cited by 6 publications

(17 citation statements)

References 28 publications

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“…Esteves and Kleiman in [55] have provided bounds for the arithmetic genus of a curve invariant by a foliation by curves, which improve and extend some results of Campillo, Carnicer, and de la Fuente, and of du Plessis and Wall [53]. In [40] we establish upper bounds for the sectional genus of Gorenstein varieties which are invariant by Pfaff systems on projective spaces and in [4] Ballico gave an extension of this result. The work of J.P. Jouanolou in [69] gives an improvement and generalization of the Darboux theory of integrability characterizing the existence of rational first integrals for Pfaff equations of codimension one on P n k , where k is an algebraically closed field of characteristic zero.…”

Section: Introductionmentioning

confidence: 62%

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Analytic varieties invariant by foliations and Pfaff systems

Corrêa

2021

Preprint

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In this work we shall present a survey on problems and results on singular holomorphic foliations and Pfaff systems on complex manifolds assuming that these objects possess invariant analytic varieties. We will focus on recent results which have been motivated by classical works of Darboux, Poincaré and Painlevé on the problem of algebraic integration of singular polynomial differential equations. We present results on Poincaré and Painlevé problem of bounding the degree and the genus of analytic varieties invariant by holomorphic foliations and Pfaff systems. We shall discuss the general ideas of the theory of integrability characterizing the existence of meromorphic first integrals for complex analytic Pfaff equations.

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Section: Introductionmentioning

confidence: 62%

“…with σ denoting the Castelnuovo-Mumford regularity of the singular locus of V . In [40] we prove that if V is nonsingular in codimension 1, then one can take ρ = 1, regardless of σ. That is, under this condition we get that ( 13)…”

Section: Poincaré and Painlevé Problems For Foliations And Pfaff Systemsmentioning

confidence: 91%

Analytic varieties invariant by foliations and Pfaff systems

Corrêa

2021

Preprint

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“…The assertion about irreducible factors follows. If condition 2 holds then one also arrives at (10), but now, given distinct…”

Section: Proof Of Lemmamentioning

confidence: 99%

“…Considering dimension n ≥ 3, much work has been done in the last two decades to classify and characterize invariant surfaces; see for instance the survey [6] by Cerveau, and the work [8] by Cerveau et al on local properties. Further notable contributions concerning invariant algebraic varieties are due to Brunella and Gustavo Mendes [3], Cavalier and Lehmann [5], Corrêa and da Silva Machado [11,12], Corrêa and Jardim [10], Corrêa and Soares [13], Esteves [16], and Soares [29][30][31]. The recent survey by Corrêa [9] collects these and more results, and provides an overview.…”

Section: Introductionmentioning

confidence: 99%

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

et al. 2021

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We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.

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“…Let  𝑋 (1) denote the ample generator of Pic(𝑋), and given a sheaf 𝐹 on 𝑋, we set 𝐹(𝑟) ∶= 𝐹 ⊗  𝑋 (1) ⊗𝑟 , as usual. Let 𝑇𝑋 denote the tangent bundle of 𝑋 and define 𝜏 𝑋 ∶= min{𝑡 ∈ ℤ | 𝐻 0 (𝑇𝑋(𝑡)) ≠ 0}, 𝜌 𝑋 ∶= min{𝑡 ∈ ℤ | 𝐻 0 (Ω 1 𝑋 (𝑡)) ≠ 0} , 𝜈 𝑋 ∶= ∫ 𝑋 𝐻 3 and 𝑐 𝑋 ∶= ∫ 𝑋 𝑐 1 (Ω 1 𝑋 ) ⋅ 𝐻 2 = − ∫ 𝑋 𝑐 1 (𝑇𝑋) ⋅ 𝐻 2 ,…”

Section: Introductionmentioning

confidence: 99%

Foliations by curves on threefolds

Cavalcante

Jardim

Santiago

2022

Mathematische Nachrichten

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We study the conormal sheaves and singular schemes of one‐dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ‐stable whenever the tangent bundle TX$TX$ is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on double-struckP3$\mathbb {P}^3$ and on a smooth quadric hypersurface Q3⊂P4$Q_3\subset \mathbb {P}^4$. Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q3.

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Bounds for sectional genera of varieties invariant under Pfaff fields

Abstract: We establish an upper bound for the sectional genus of varieties which are invariant under Pfaff fields on projective spaces.

Cited by 6 publications

References 28 publications

Analytic varieties invariant by foliations and Pfaff systems

Analytic varieties invariant by foliations and Pfaff systems

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

Foliations by curves on threefolds

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