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 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 Moduli Spaces of Coherent Systems of Small Slope on Algebraic Curves

2009 ◽  
Vol 37 (8) ◽  
pp. 2649-2678 ◽  
Author(s):  
S. B. Bradlow ◽  
O. García-Prada ◽  
V. Mercat ◽  
V. Muñoz ◽  
P. E. Newstead
Keyword(s):  
Moduli Spaces ◽  
Algebraic Curves ◽  
Coherent Systems

scholarly journals On Moduli Spaces of Positive Scalar Curvature Metrics on Highly Connected Manifolds

2020 ◽  
Author(s):  
Michael Wiemeler
Keyword(s):  
Fundamental Group ◽  
Scalar Curvature ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Positive Scalar Curvature ◽  
Spin Manifold ◽  
Homotopy Groups ◽  
Positive Scalar ◽  
Simply Connected ◽  
Highly Connected Manifolds

Abstract Let $M$ be a simply connected spin manifold of dimension at least six, which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar curvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds that includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.

scholarly journals A modular compactification of ℳ1,n from A∞-structures

2019 ◽  
Vol 2019 (755) ◽  
pp. 151-189
Author(s):  
Yankı Lekili ◽  
Alexander Polishchuk
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Rational Singularities ◽  
Picard Groups

AbstractWe show that a certain moduli space of minimal A_{\infty}-structures coincides with the modular compactification {\overline{\mathcal{M}}}_{1,n}(n-1) of \mathcal{M}_{1,n} constructed by Smyth in [26]. In addition, we describe these moduli spaces and the universal curves over them by explicit equations, prove that they are normal and Gorenstein, show that their Picard groups have no torsion and that they have rational singularities if and only if n\leq 11.

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.

scholarly journals HODGE POLYNOMIALS OF SOME MODULI SPACES OF COHERENT SYSTEMS

2013 ◽  
Vol 24 (03) ◽  
pp. 1350014
Author(s):  
CRISTIAN GONZÁLEZ–MARTÍNEZ
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Coherent Systems ◽  
Determinantal Varieties ◽  
Hodge Polynomials

When k < n, we study the coherent systems that come from a BGN extension in which the quotient bundle is strictly semistable. In this case we describe a stratification of the moduli space of coherent systems. We also describe the strata as complements of determinantal varieties and we prove that these are irreducible and smooth. These descriptions allow us to compute the Hodge polynomials of this moduli space in some cases. In particular, we give explicit computations for the cases in which (n, d, k) = (3, d, 1) and d is even, obtaining from them the usual Poincaré polynomials.

scholarly journals Comparison of Poisson structures on moduli spaces

2021 ◽  
Author(s):  
Indranil Biswas ◽  
Francesco Bottacin ◽  
Tomás L. Gómez
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Poisson Structures ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraic Vector ◽  
Pure Dimension ◽  
Stable Sheaves ◽  
Fixed Rank ◽  
Dimension One

AbstractLet X be a complex irreducible smooth projective curve, and let $${{\mathbb {L}}}$$ L be an algebraic line bundle on X with a nonzero section $$\sigma _0$$ σ 0 . Let $${\mathcal {M}}$$ M denote the moduli space of stable Hitchin pairs $$(E,\, \theta )$$ ( E , θ ) , where E is an algebraic vector bundle on X of fixed rank r and degree $$\delta $$ δ , and $$\theta \, \in \, H^0(X,\, {\mathcal {E}nd}(E)\otimes K_X\otimes {{\mathbb {L}}})$$ θ ∈ H 0 ( X , E n d ( E ) ⊗ K X ⊗ L ) . Associating to every stable Hitchin pair its spectral data, an isomorphism of $${\mathcal {M}}$$ M with a moduli space $${\mathcal {P}}$$ P of stable sheaves of pure dimension one on the total space of $$K_X\otimes {{\mathbb {L}}}$$ K X ⊗ L is obtained. Both the moduli spaces $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M are equipped with algebraic Poisson structures, which are constructed using $$\sigma _0$$ σ 0 . Here we prove that the above isomorphism between $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M preserves the Poisson structures.

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 HODGE STRUCTURES OF THE MODULI SPACES OF PAIRS

2010 ◽  
Vol 21 (11) ◽  
pp. 1505-1529 ◽  
Author(s):  
VICENTE MUÑOZ
Keyword(s):  
Vector Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hodge Structure ◽  
Projective Curve ◽  
Hodge Structures ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Algebraic Vector

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Let [Formula: see text] be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of [Formula: see text] are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H1(X). This implies a similar result for the moduli spaces of stable vector bundles over X.

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