scholarly journals COHERENT SYSTEMS ON THE PROJECTIVE LINE

2020 ◽  
Author(s):  
Peter Newstead ◽  
Montserrat Teixidor i Bigas
Keyword(s):  
Projective Line ◽  
Existence Problem ◽  
Coherent Systems ◽  
Stable Bundles

Abstract It is well known that there are no stable bundles of rank greater than 1 on the projective line. In this paper, our main purpose is to study the existence problem for stable coherent systems on the projective line when the number of sections is larger than the rank. We include a review of known results, mostly for a small number of sections.

scholarly journals COHERENT SYSTEMS OF GENUS 0 II: EXISTENCE RESULTS FOR k ≥ 3

2007 ◽  
Vol 18 (04) ◽  
pp. 363-393 ◽  
Author(s):  
H. LANGE ◽  
P. E. NEWSTEAD
Keyword(s):  
Projective Line ◽  
Existence Results ◽  
Critical Values ◽  
Coherent Systems

In this paper, we continue the investigation of coherent systems of type (n,d,k) on the projective line which are stable with respect to some value of a parameter α. We work mainly with k < n and obtain existence results for arbitrary k in certain cases, together with complete results for k = 3. Our methods involve the use of the "flips" which occur at critical values of the parameter.

COHERENT SYSTEMS OF GENUS 0

2004 ◽  
Vol 15 (04) ◽  
pp. 409-424 ◽  
Author(s):  
H. LANGE ◽  
P. E. NEWSTEAD
Keyword(s):  
Moduli Spaces ◽  
Necessary Conditions ◽  
Projective Line ◽  
Coherent Systems

In this paper we begin the classification of coherent systems (E,V) on the projective line which are stable with respect to some value of a parameter α. In particular we show that the moduli spaces, if non-empty, are always smooth and irreducible of the expected dimension. We obtain necessary conditions for non-emptiness and, when dim V=1 or 2, we determine these conditions precisely. We also obtain partial results in some other cases.

scholarly journals STABLE BUNDLES OF RANK 2 WITH FOUR SECTIONS

2011 ◽  
Vol 22 (12) ◽  
pp. 1743-1762 ◽  
Author(s):  
I. GRZEGORCZYK ◽  
V. MERCAT ◽  
P. E. NEWSTEAD
Keyword(s):  
Lower Bound ◽  
Moduli Space ◽  
Algebraic Curve ◽  
Moduli Space Of Curves ◽  
Coherent Systems ◽  
Stable Bundles ◽  
Geometric Criterion ◽  
General Curve ◽  
Rank 2 ◽  
Koszul Cohomology

This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve C. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve C of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems. The results can also be expressed in terms of Koszul cohomology and the methods provide a useful tool for the study of the geometry of the moduli space of curves.

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 COHERENT SYSTEMS OF GENUS 0 III: COMPUTATION OF FLIPS FOR k = 1

2008 ◽  
Vol 19 (09) ◽  
pp. 1103-1119 ◽  
Author(s):  
H. LANGE ◽  
P. E. NEWSTEAD
Keyword(s):  
Moduli Spaces ◽  
Projective Line ◽  
Explicit Description ◽  
Coherent Systems ◽  
Arbitrary Rank ◽  
Hodge Polynomials

In this paper, we continue the investigation of coherent systems of type (n, d, k) on the projective line which are stable with respect to some value of a parameter α. We consider the case k = 1 and study the variation of the moduli spaces with α. We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank.

scholarly journals Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Projective Line ◽  
Tangent Cones ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Complex Curves ◽  
Complex Dimensions ◽  
Irregular Case ◽  
Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

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