scholarly journals Higher codimensional foliations with Kupka singularities

2017 ◽  
Vol 28 (03) ◽  
pp. 1750019
Author(s):  
O. Calvo-Andrade ◽  
M. Corrêa ◽  
A. Fernández-Pérez
Keyword(s):  
Normal Form ◽  
Positive Integer ◽  
Projective Space ◽  
Complete Intersection ◽  
Holomorphic Foliations ◽  
Connected Component ◽  
Codimension One ◽  
Dimension Formula ◽  
Rational Fibration

We consider holomorphic foliations of dimension [Formula: see text] and codimension [Formula: see text] in the projective space [Formula: see text], with a compact connected component of the Kupka set. We prove that if the transversal type is linear with positive integer eigenvalues, then the foliation consists of the fibers of a rational fibration [Formula: see text]. As a corollary, if [Formula: see text] and has a transversal type diagonal with different eigenvalues, then the Kupka component [Formula: see text] is a complete intersection and the leaves of the foliation define a rational fibration. The same conclusion holds if the Kupka set has a radial transversal type. Finally, as an application, we find a normal form for non-integrable codimension-one distributions on [Formula: see text].

scholarly journals ON THE STABILITY OF SYZYGY BUNDLES

2011 ◽  
Vol 22 (04) ◽  
pp. 515-534 ◽  
Author(s):  
IUSTIN COANDĂ
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Vector Bundles ◽  
Elementary Proof ◽  
Base Point ◽  
Vector Spaces ◽  
Similar Argument ◽  
Vanishing Theorem ◽  
The Stability ◽  
Koszul Cohomology

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

scholarly journals Codimension One Holomorphic Distributions on the Projective Three-space

2018 ◽  
Vol 2020 (23) ◽  
pp. 9011-9074 ◽  
Author(s):  
Omegar Calvo-Andrade ◽  
Maurício Corrêa ◽  
Marcos Jardim
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Tangent Bundle ◽  
Projective Variety ◽  
Arbitrary Degree ◽  
Codimension One ◽  
Quot Scheme ◽  
Space Of Distributions ◽  
Three Space

Abstract We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves.

scholarly journals Flags of holomorphic foliations

2011 ◽  
Vol 83 (3) ◽  
pp. 775-786 ◽  
Author(s):  
Rogério S. Mol
Keyword(s):  
Finite Number ◽  
Complex Manifold ◽  
Necessary Conditions ◽  
Holomorphic Foliations ◽  
Lower Dimension ◽  
Higher Dimension ◽  
Basic Properties ◽  
Codimension One ◽  
Tangent Sheaf ◽  
Polar Classes

A flag of holomorphic foliations on a complex manifold M is an object consisting of a finite number of singular holomorphic foliations on M of growing dimensions such that the tangent sheaf of a fixed foliation is a subsheaf of the tangent sheaf of any of the foliations of higher dimension. We study some basic properties oft hese objects and, in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" />, n > 3, we establish some necessary conditions for a foliation, we find bounds of lower dimension to leave invariant foliations of codimension one. Finally, still in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" /> involving the degrees of polar classes of foliations in a flag.

scholarly journals On holomorphic one-forms transverse to closed hypersurfaces

2003 ◽  
Vol 75 (3) ◽  
pp. 265-269
Author(s):  
Toshikazu Ito ◽  
Bruno Scárdua
Keyword(s):  
Affine Space ◽  
Smooth Boundary ◽  
Singular Points ◽  
Real Hypersurfaces ◽  
Holomorphic Foliations ◽  
Codimension One ◽  
Closed Hypersurfaces ◽  
The One

In this note we announce some achievements in the study of holomorphic distributions admitting transverse closed real hypersurfaces. We consider a domain with smooth boundary in the complex affine space of dimension two or greater. Assume that the domain satisfies some cohomology triviality hypothesis (for instance, if the domain is a ball). We prove that if a holomorphic one form in a neighborhood of the domain is such that the corresponding holomorphic distribution is transverse to the boundary of the domain then the Euler-Poincaré-Hopf characteristic of the domain is equal to the sum of indexes of the one-form at its singular points inside the domain. This result has several consequences and applies, for instance, to the study of codimension one holomorphic foliations transverse to spheres.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

2014 ◽  
Vol 24 (07) ◽  
pp. 1450090 ◽  
Author(s):  
Tiago de Carvalho ◽  
Durval José Tonon
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Piecewise Smooth ◽  
Topological Equivalence ◽  
Smooth Vector ◽  
Codimension One ◽  
Piecewise Smooth Vector Fields ◽  
Piecewise Smooth Vector Field

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

scholarly journals Hyperbolicity related problems for complete intersection varieties

2013 ◽  
Vol 150 (3) ◽  
pp. 369-395 ◽  
Author(s):  
Damian Brotbek
Keyword(s):  
Projective Space ◽  
Existence Theorem ◽  
Complete Intersection ◽  
Cotangent Bundle ◽  
Projective Variety ◽  
Complete Intersections ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Intersection Surface

AbstractIn this paper we examine different problems regarding complete intersection varieties of high multidegree in a smooth complex projective variety. First we prove an existence theorem for jet differential equations that generalizes a theorem of Diverio. Then we show how one can deduce hyperbolicity for generic complete intersections of high multidegree and high codimension from the known results on hypersurfaces. Finally, motivated by a conjecture of Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove that a generic complete intersection surface of high multidegree in a projective space of dimension at least four has an ample cotangent bundle.

All symbolic powers and ordinary powers of the defining ideal of a fat nearly-complete intersection are equal

2019 ◽  
Vol 18 (08) ◽  
pp. 1950142 ◽  
Author(s):  
Hassan Haghighi ◽  
Mohammad Mosakhani
Keyword(s):  
Positive Integer ◽  
Complete Intersection ◽  
Containment Problem ◽  
Ideal Formula ◽  
Symbolic Powers

The purpose of this paper is to study the containment problem for the corresponding ideal [Formula: see text] of a special zero dimensional subscheme [Formula: see text] in [Formula: see text], what we call a fat nearly-complete intersection. We show that for every positive integer [Formula: see text], the equality [Formula: see text] holds.

Quarto-quartic birational maps of ℙ3(ℂ)

2017 ◽  
Vol 28 (05) ◽  
pp. 1750037
Author(s):  
Julie Déserti ◽  
Frédéric Han
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Birational Maps ◽  
Dimension Formula ◽  
Trigonal Curves ◽  
Irreducible Components ◽  
Genus Formula ◽  
Complex Projective

We construct a determinantal family of quarto-quartic transformations of a complex projective space of dimension [Formula: see text] from trigonal curves of degree [Formula: see text] and genus [Formula: see text]. Moreover, we show that the variety of [Formula: see text]-birational maps of [Formula: see text] has at least four irreducible components and describe three of them.

A direct proof that toric rank $2$ bundles on projective space split

2020 ◽  
Vol 126 (3) ◽  
pp. 493-496
Author(s):  
David Stapleton
Keyword(s):  
Vector Bundle ◽  
Projective Space ◽  
Complete Intersection ◽  
Vector Bundles ◽  
Direct Proof ◽  
Rank 2 ◽  
Dimensional Projective Space ◽  
Rank 2 Bundles

The point of this paper is to give a short, direct proof that rank $2$ toric vector bundles on $n$-dimensional projective space split once $n$ is at least $3$. This result is originally due to Bertin and Elencwajg, and there is also related work by Kaneyama, Klyachko, and Ilten-Süss. The idea is that, after possibly twisting the vector bundle, there is a section which is a complete intersection.

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