On the geometry of the singular locus of a codimension one foliation in $\mathbb P^n$

2019 ◽  
Vol 35 (3) ◽  
pp. 857-876
Author(s):  
Omegar Calvo-Andrade ◽  
Ariel Molinuevo ◽  
Federico Quallbrunn
Keyword(s):  
Singular Locus ◽  
Codimension One

scholarly journals Normal crossing properties of complex hypersurfaces via logarithmic residues

2014 ◽  
Vol 150 (9) ◽  
pp. 1607-1622 ◽  
Author(s):  
Michel Granger ◽  
Mathias Schulze
Keyword(s):  
Singular Locus ◽  
Algebraic Characterization ◽  
Natural Conditions ◽  
Normal Crossing ◽  
Codimension One ◽  
Hypersurface Singularities ◽  
Logarithmic Residue ◽  
Side Result ◽  
Free Divisors

AbstractWe introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of Lê and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. As a side result, we describe all free divisors with Gorenstein singular locus.

ON THE SINGULAR SCHEME OF CODIMENSION ONE HOLOMORPHIC FOLIATIONS IN ℙ3

2010 ◽  
Vol 21 (07) ◽  
pp. 843-858 ◽  
Author(s):  
LUIS GIRALDO ◽  
ANTONIO J. PAN-COLLANTES
Keyword(s):  
Singular Locus ◽  
The Other ◽  
Holomorphic Foliation ◽  
Holomorphic Foliations ◽  
Codimension One ◽  
Other Hand ◽  
Tangent Sheaf ◽  
Torsion Free

In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion-free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in ℙ3 is locally free if and only if the singular scheme is a curve, and that it splits if and only if it is arithmetically Cohen–Macaulay. On the other hand, we discuss when a split foliation in ℙ3 is determined by its singular scheme.

On submersions of the complex indicatrix

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5081-5092
Author(s):  
Elena Popovicia
Keyword(s):  
Fixed Point ◽  
Complex Projective Space ◽  
Finsler Space ◽  
Hermitian Manifold ◽  
Almost Hermitian Manifold ◽  
Codimension One ◽  
Real Submanifold ◽  
Fundamental Equations ◽  
Complex Projective ◽  
Extrinsic Sphere

In this paper we study the complex indicatrix associated to a complex Finsler space as an embedded CR - hypersurface of the holomorphic tangent bundle, considered in a fixed point. Following the study of CR - submanifolds of a K?hler manifold, there are investigated some properties of the complex indicatrix as a real submanifold of codimension one, using the submanifold formulae and the fundamental equations. As a result, the complex indicatrix is an extrinsic sphere of the holomorphic tangent space in each fibre of a complex Finsler bundle. Also, submersions from the complex indicatrix onto an almost Hermitian manifold and some properties that can occur on them are studied. As application, an explicit submersion onto the complex projective space is provided.

scholarly journals Towards the classification of rank-r $$ \mathcal{N} $$ = 2 SCFTs. Part I. Twisted partition function and central charge formulae

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Mario Martone
Keyword(s):  
Partition Function ◽  
Central Charge ◽  
Singular Locus ◽  
Companion Paper ◽  
Coulomb Branch ◽  
Energy Limit ◽  
Superconformal Field Theories ◽  
Rank 2 ◽  
Explicit Formulae

Abstract We derive explicit formulae to compute the a and c central charges of four dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) directly from Coulomb branch related quantities. The formulae apply at arbitrary rank. We also discover general properties of the low-energy limit behavior of the flavor symmetry of $$ \mathcal{N} $$ N = 2 SCFTs which culminate with our $$ \mathcal{N} $$ N = 2 UV-IR simple flavor condition. This is done by determining precisely the relation between the integrand of the partition function of the topologically twisted version of the 4d $$ \mathcal{N} $$ N = 2 SCFTs and the singular locus of their Coulomb branches. The techniques developed here are extensively applied to many rank-2 SCFTs, including new ones, in a companion paper.This manuscript is dedicated to the memory of Rayshard Brooks, George Floyd, Breonna Taylor and the countless black lives taken by US police forces and still awaiting justice. Our hearts are with our colleagues of color who suffer daily the consequences of this racist world.

scholarly journals Affine cones over cubic surfaces are flexible in codimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Perepechko
Keyword(s):  
Automorphism Group ◽  
Open Subset ◽  
Pezzo Surface ◽  
Ample Divisor ◽  
Transitive Action ◽  
Del Pezzo Surface ◽  
Codimension One ◽  
Cubic Surfaces ◽  
Special Automorphism ◽  
Del Pezzo

AbstractLet Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

Unstable dimension variability and codimension-one bifurcations of two-dimensional maps

2004 ◽  
Vol 321 (4) ◽  
pp. 244-251 ◽  
Author(s):  
Ricardo L. Viana ◽  
José R.R. Barbosa ◽  
Celso Grebogi
Keyword(s):  
Two Dimensional ◽  
Codimension One

Bursting patterns and mixed-mode oscillations in reduced Purkinje model

2018 ◽  
Vol 32 (05) ◽  
pp. 1850043 ◽  
Author(s):  
Feibiao Zhan ◽  
Shenquan Liu ◽  
Jing Wang ◽  
Bo Lu
Keyword(s):  
Purkinje Cell ◽  
Mixed Mode ◽  
Cell Model ◽  
Dynamical Behavior ◽  
Characteristic Index ◽  
Mixed Mode Oscillations ◽  
Codimension One ◽  
Bifurcation Phenomenon ◽  
Lyapunov Coefficient ◽  
Potential Link

Bursting discharge is a ubiquitous behavior in neurons, and abundant bursting patterns imply many physiological information. There exists a closely potential link between bifurcation phenomenon and the number of spikes per burst as well as mixed-mode oscillations (MMOs). In this paper, we have mainly explored the dynamical behavior of the reduced Purkinje cell and the existence of MMOs. First, we adopted the codimension-one bifurcation to illustrate the generation mechanism of bursting in the reduced Purkinje cell model via slow–fast dynamics analysis and demonstrate the process of spike-adding. Furthermore, we have computed the first Lyapunov coefficient of Hopf bifurcation to determine whether it is subcritical or supercritical and depicted the diagrams of inter-spike intervals (ISIs) to examine the chaos. Moreover, the bifurcation diagram near the cusp point is obtained by making the codimension-two bifurcation analysis for the fast subsystem. Finally, we have a discussion on mixed-mode oscillations and it is further investigated using the characteristic index that is Devil’s staircase.

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