EXISTENCE OF COHERENT SYSTEMS II

2008 ◽  
Vol 19 (10) ◽  
pp. 1269-1283 ◽  
Author(s):  
MONTSERRAT TEIXIDOR I BIGAS
Keyword(s):  
Vector Bundle ◽  
Vector Space ◽  
Moduli Spaces ◽  
Stability Condition ◽  
Real Parameter ◽  
Coherent System ◽  
Coherent Systems ◽  
Positive Real ◽  
Mild Conditions ◽  
Positive Real Parameter

A coherent system of type (r, d, k) on a curve consists of a vector bundle of rank r and degree d together with a vector space of dimension k of the sections of this vector bundle. There is a stability condition depending on a positive real parameter α that allows to construct moduli spaces for these objects. This paper shows non-emptiness of these moduli spaces when k > r for any α under some mild conditions on the degree and genus.

scholarly journals EXISTENCE OF COHERENT SYSTEMS

2008 ◽  
Vol 19 (04) ◽  
pp. 449-454 ◽  
Author(s):  
MONTSERRAT TEIXIDOR I. BIGAS
Keyword(s):  
Vector Bundle ◽  
Vector Space ◽  
Moduli Spaces ◽  
Stability Condition ◽  
Real Parameter ◽  
Coherent System ◽  
Coherent Systems ◽  
Positive Real ◽  
Mild Conditions ◽  
Positive Real Parameter

A coherent system of type (r, d, k) on a curve consists of a vector bundle of rank r and degree d together with a vector space of dimension k of the sections of this vector bundle. There is a stability condition depending on a positive real parameter α that allows to construct moduli spaces for these objects. This paper shows non-emptiness of these moduli spaces when k > r for any α under some mild conditions on the degree and genus.

(t, ℓ)-STABILITY AND COHERENT SYSTEMS

2019 ◽  
Vol 62 (3) ◽  
pp. 661-672
Author(s):  
L. BRAMBILA-PAZ ◽  
O. MATA-GUTIÉRREZ
Keyword(s):  
Moduli Spaces ◽  
Finite Family ◽  
Real Parameter ◽  
Projective Curve ◽  
Coherent Systems ◽  
Positive Real ◽  
Positive Real Parameter ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be a non-singular irreducible complex projective curve of genus g ≥ 2. The concept of stability of coherent systems over X depends on a positive real parameter α, given then a (finite) family of moduli spaces of coherent systems. We use (t, ℓ)-stability to prove the existence of coherent systems over X that are α-stable for all allowed α > 0.

scholarly journals Coherent Systems and Brill–Noether Theory

2003 ◽  
Vol 14 (07) ◽  
pp. 683-733 ◽  
Author(s):  
S. B. Bradlow ◽  
O. García-Prada ◽  
V. Muñoz ◽  
P. E. Newstead
Keyword(s):  
Vector Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Coherent System ◽  
Coherent Systems ◽  
Stable Bundles ◽  
Algebraic Vector Bundle ◽  
Picard Groups ◽  
Algebraic Vector ◽  
The Stability

Let X be a curve of genus g. A coherent system on X consists of a pair (E,V), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyze the cases k=1,2,3 and n=2 explicitly. For small values of α, the moduli spaces of coherent systems are related to the Brill–Noether loci, the subschemes of the moduli spaces of stable bundles consisting of those bundles with at least a prescribed number of independent sections. The study of coherent systems is applied to find the dimension, prove the irreducibility, and in some cases calculate the Picard groups of the Brill–Noether loci with k≤3.

scholarly journals NON-EMPTINESS OF MODULI SPACES OF COHERENT SYSTEMS

2008 ◽  
Vol 19 (07) ◽  
pp. 777-799 ◽  
Author(s):  
L. BRAMBILA-PAZ
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curve ◽  
Coherent System ◽  
Coherent Systems ◽  
Generic Element

Let X be a general smooth projective algebraic curve of genus g ≥ 2 over ℂ. We prove that the moduli space G(α:n,d,k) of α-stable coherent systems of type (n,d,k) over X is empty if k > n and the Brill–Noether number β := β(n,d,n + 1) = β(1,d,n + 1) = g - (n + 1)(n - d + g) < 0. Moreover, if 0 ≤ β < g or β = g, n ∤g and for some α > 0, G(α : n,d,k) ≠ ∅ then G(α : n,d,k) ≠ ∅ for all α > 0 and G(α : n,d,k) = G(α′ : n,d,k) for all α,α′ > 0 and the generic element is generated. In particular, G(α : n,d,n + 1) ≠ ∅ if 0 ≤ β ≤ g and α > 0. Moreover, if β > 0 G(α : n,d,n + 1) is smooth and irreducible of dimension β(1,d,n + 1). We define a dual span of a generically generated coherent system. We assume d < g + n1≤ g + n2and prove that for all α > 0, G(α : n1,d, n1+ n2) ≠ ∅ if and only if G(α : n2,d, n1+ n2) ≠ ∅. For g = 2, we describe G(α : 2,d,k) for k > n.

scholarly journals ON THE GEOMETRY OF MODULI SPACES OF COHERENT SYSTEMS ON ALGEBRAIC CURVES

2007 ◽  
Vol 18 (04) ◽  
pp. 411-453 ◽  
Author(s):  
S. B. BRADLOW ◽  
O. GARCÍA-PRADA ◽  
V. MERCAT ◽  
V. MUÑOZ ◽  
P. E. NEWSTEAD
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curves ◽  
Homotopy Groups ◽  
Coherent System ◽  
Coherent Systems ◽  
Picard Groups ◽  
Algebraic Vector ◽  
The Stability ◽  
Almost All

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E,V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for different values of α when k ≤ n and the variation of the moduli spaces when we vary α. As a consequence, for sufficiently large α, we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case k = n - 1 explicitly, and give the Poincaré polynomials for the case k = n - 2. In an appendix, we describe the geometry of the "flips" which take place at critical values of α in the simplest case, and include a proof of the existence of universal families of coherent systems when GCD (n,d,k) = 1.

scholarly journals Existence of Three Solutions for a Nonlinear Fractional Boundary Value Problem via a Critical Points Theorem

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Chuanzhi Bai
Keyword(s):  
Boundary Value Problem ◽  
Critical Points ◽  
Boundary Value ◽  
Real Parameter ◽  
Fractional Boundary Value Problem ◽  
Positive Real ◽  
Three Solutions ◽  
Positive Real Parameter

This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows:(d/dt)((1/2)0Dtα-1(0CDtαu(t))-(1/2)tDTα-1(tCDTαu(t)))+λa(t)f(u(t))=0, a.e.  t∈[0,T],u(0)=u(T)=0,whereα∈(1/2,1], andλis a positive real parameter. The approach is based on a critical-points theorem established by G. Bonanno.

COHERENT SYSTEMS WITH MANY SECTIONS ON PROJECTIVE CURVES

2006 ◽  
Vol 17 (03) ◽  
pp. 263-267 ◽  
Author(s):  
E. BALLICO
Keyword(s):  
Vector Bundle ◽  
Linear Subspace ◽  
Sufficient Conditions ◽  
Coherent System ◽  
Projective Curve ◽  
Coherent Systems ◽  
Smooth Projective Curve ◽  
Type D ◽  
Projective Curves

Here we give sufficient conditions for the existence of an α-stable (for all α > 0) coherent system (i.e. a vector bundle together with a linear subspace of its global sections) of type (d,n,k), k > n, on a smooth projective curve.

scholarly journals The Existence of Invariant Tori and Quasiperiodic Solutions of the Nosé–Hoover Oscillator

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Yanmin Niu ◽  
Xiong Li
Keyword(s):  
Invariant Tori ◽  
Equivalent Form ◽  
Real Parameter ◽  
Dimensional System ◽  
Reversible System ◽  
Positive Real ◽  
3 Dimensional ◽  
Quasiperiodic Solutions ◽  
Positive Real Parameter ◽  
Twist Theorem

In this paper, we consider an equivalent form of the Nosé–Hoover oscillator, x ′ = y , y ′ = − x − y z ,   and   z ′ = y 2 − a , where a is a positive real parameter. Under a series of transformations, it is transformed into a 2-dimensional reversible system about action-angle variables. By applying a version of twist theorem established by Liu and Song in 2004 for reversible mappings, we find infinitely many invariant tori whenever a is sufficiently small, which eventually turns out that the solutions starting on the invariant tori are quasiperiodic. The discussion about quasiperiodic solutions of such 3-dimensional system is relatively new.

scholarly journals On the Periodicity of a Difference Equation with Maximum

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Ali Gelisken ◽  
Cengiz Cinar ◽  
Ibrahim Yalcinkaya
Keyword(s):  
Difference Equation ◽  
Open Problem ◽  
Initial Conditions ◽  
Rational Numbers ◽  
Real Parameter ◽  
Positive Real ◽  
Positive Real Parameter

We investigate the periodic nature of solutions of the max difference equationxn+1=max⁡{xn,A}/(xnxn−1),n=0,1,…, whereAis a positive real parameter, and the initial conditionsx−1=Ar−1andx0=Ar0such thatr−1andr0are positive rational numbers. The results in this paper answer the Open Problem 6.2 posed by Grove and Ladas (2005).

scholarly journals Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammad Sajid
Keyword(s):  
Fixed Points ◽  
Research Work ◽  
Period Doubling ◽  
Periodic Points ◽  
Real Parameter ◽  
Chaotic Behaviour ◽  
Positive Real ◽  
The Real ◽  
Positive Real Parameter ◽  
Two Parameters

The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

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