scholarly journals On linear series and a conjecture of D. C. Butler

2015 ◽  
Vol 26 (02) ◽  
pp. 1550007 ◽  
Author(s):  
U. N. Bhosle ◽  
L. Brambila-Paz ◽  
P. E. Newstead
Keyword(s):  
Line Bundle ◽  
Linear Subspace ◽  
Linear Series ◽  
Projective Curve ◽  
Coherent Systems ◽  
Natural Context ◽  
Wall Crossing ◽  
The Stability

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H0(L) of dimension n + 1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V ⊗ 𝒪C → L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

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.

Linear stability and stability of syzygy bundles

2018 ◽  
Vol 29 (11) ◽  
pp. 1850080 ◽  
Author(s):  
Abel Castorena ◽  
H. Torres-López
Keyword(s):  
Exact Sequence ◽  
Linear Stability ◽  
Commutative Diagram ◽  
Linear Series ◽  
Stability Conditions ◽  
Stable Bundle ◽  
Projective Curve ◽  
General Curve ◽  
The Stability ◽  
Natural Way

Let [Formula: see text] be a smooth irreducible projective curve and let [Formula: see text] be a complete and generated linear series on [Formula: see text]. Denote by [Formula: see text] the kernel of the evaluation map [Formula: see text]. The exact sequence [Formula: see text] fits into a commutative diagram that we call the Butler’s diagram. This diagram induces in a natural way a multiplication map on global sections [Formula: see text], where [Formula: see text] is a subspace and [Formula: see text] is the dual of a subbundle [Formula: see text]. When the subbundle [Formula: see text] is a stable bundle, we show that the map [Formula: see text] is surjective. When [Formula: see text] is a Brill–Noether general curve, we use the surjectivity of [Formula: see text] to give another proof of the semistability of [Formula: see text], moreover, we fill up a gap in some incomplete argument by Butler: With the surjectivity of [Formula: see text] we give conditions to determine the stability of [Formula: see text], and such conditions imply the well-known stability conditions for [Formula: see text] stated precisely by Butler. Finally we obtain the equivalence between the (semi)stability of [Formula: see text] and the linear (semi)stability of [Formula: see text] on [Formula: see text]-gonal curves.

scholarly journals WALL-CROSSINGS FOR TWISTED QUIVER BUNDLES

2013 ◽  
Vol 24 (05) ◽  
pp. 1350038 ◽  
Author(s):  
BUMSIG KIM ◽  
HWAYOUNG LEE
Keyword(s):  
Abelian Category ◽  
Moment Map ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Wall Crossing ◽  
Quiver Bundles ◽  
The Stability ◽  
The Moment ◽  
Wall Crossing Formula ◽  
Do So

Given a double quiver, we study homological algebra of twisted quiver sheaves with the moment map relation using the short exact sequence of Crawley-Boevey, Holland, Gothen, and King. Then in a certain one-parameter space of the stability conditions, we obtain a wall-crossing formula for the generalized Donaldson–Thomas invariants of the abelian category of framed twisted quiver sheaves on a smooth projective curve. To do so, we closely follow the approach of Chuang, Diaconescu and Pan in the ADHM quiver case, which makes use of the theory of Joyce and Song. The invariants virtually count framed twisted quiver sheaves with the moment map relation and directly generalize the ADHM invariants of Diaconescu.

scholarly journals A pathological case of the C1 conjecture in mixed characteristic

2018 ◽  
Vol 167 (01) ◽  
pp. 61-64 ◽  
Author(s):  
INDER KAUR
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Projective Curve ◽  
Free Sheaf ◽  
Smooth Projective Curve ◽  
Mixed Characteristic ◽  
Locally Free Sheaf ◽  
Fixed Rank ◽  
Pathological Case

AbstractLet K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

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.

SINGULAR CURVES WITH LINE BUNDLES L DEFINED OVER ${\mathbb F}_q$ AND WITH H0(L) = H1(L) = 0

2013 ◽  
Vol 06 (02) ◽  
pp. 1350023
Author(s):  
Edoardo Ballico
Keyword(s):  
Line Bundle ◽  
Line Bundles ◽  
Projective Curve ◽  
Similar Line

Here we prove the existence of several pairs (X, L), where X is a geometrically integral projective curve defined over 𝔽q and L is a line bundle on X defined over 𝔽q and with H0(X, L) = H1(X, L) = 0. These examples are obtained using the existence of similar line bundles on the normalization of X, i.e. a case studied by C. Ballet, C. Ritzenthaler and R. Roland.

On the Image of Certain Extension Maps. I

2007 ◽  
Vol 50 (3) ◽  
pp. 427-433
Author(s):  
Israel Moreno Mejía
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Vector Bundles ◽  
Projective Curve ◽  
Rational Map ◽  
Stable Vector Bundles ◽  
Linear Subspaces ◽  
Smooth Complex ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be a smooth complex projective curve of genus g ≥ 1. Let ξ ∈ J1(X) be a line bundle on X of degree 1. LetW = Ext1(ξn, ξ–1) be the space of extensions of ξn by ξ–1. There is a rational map Dξ : G(n,W) → SUX(n + 1), where G(n,W) is the Grassmannian variety of n-linear subspaces of W and SUX(n + 1) is the moduli space of rank n + 1 semi-stable vector bundles on X with trivial determinant. We prove that if n = 2, then Dξ is everywhere defined and is injective.

scholarly journals Counting Higgs bundles and type quiver bundles

2020 ◽  
Vol 156 (4) ◽  
pp. 744-769
Author(s):  
Sergey Mozgovoy ◽  
Olivier Schiffmann
Keyword(s):  
Line Bundle ◽  
Finite Field ◽  
Moduli Spaces ◽  
Closed Formula ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Closed Expression ◽  
Smooth Projective Curve ◽  
Quiver Bundles ◽  
Fixed Rank

We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus $g$ defined over a finite field, when the twisting line bundle degree is at least $2g-2$ (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type $A$ (finite or affine), obtaining in particular a Harder–Narasimhan-type formula counting semistable $U(p,q)$-Higgs bundles over a smooth projective curve defined over a finite field.

scholarly journals SMALL RATIONAL CURVES ON THE MODULI SPACE OF STABLE BUNDLES

2012 ◽  
Vol 23 (08) ◽  
pp. 1250085 ◽  
Author(s):  
MIN LIU
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Vector Bundles ◽  
Rational Curves ◽  
Projective Curve ◽  
Stable Bundles ◽  
Smooth Projective Curve ◽  
Determinant Formula ◽  
Stable Vector Bundles

For a smooth projective curve C with genus g ≥ 2 and a degree 1 line bundle [Formula: see text] on C, let [Formula: see text] be the moduli space of stable vector bundles of rank r over C with the fixed determinant [Formula: see text]. In this paper, we study the small rational curves on M and estimate the codimension of the locus of the small rational curves. In particular, we determine all small rational curves when r = 3.

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