scholarly journals A NOTE ON RANK TWO STABLE BUNDLES OVER SURFACES

2021 ◽  
pp. 1-14
Author(s):  
GRACIELA REYES-AHUMADA ◽  
L. ROA-LEGUIZAMÓN ◽  
H. TORRES-LÓPEZ
Keyword(s):  
Vector Bundle ◽  
Moduli Spaces ◽  
Stable Bundle ◽  
Stable Vector Bundle ◽  
Stable Bundles ◽  
Injective Morphism ◽  
Rank 2

Abstract. Let π : X → C be a fibration with integral fibers over a curve C and consider a polarization H on the surface X. Let E be a stable vector bundle of rank 2 on C. We prove that the pullback π*(E) is a H-stable bundle over X. This result allows us to relate the corresponding moduli spaces of stable bundles $${{\mathcal M}_C}(2,d)$$ and $${{\mathcal M}_{X,H}}(2,df,0)$$ through an injective morphism. We study the induced morphism at the level of Brill–Noether loci to construct examples of Brill–Noether loci on fibered surfaces. Results concerning the emptiness of Brill–Noether loci follow as a consequence of a generalization of Clifford’s Theorem for rank two bundles on surfaces.

scholarly journals Ample vector bundles on a rational surface (higher rank)

1977 ◽  
Vol 66 ◽  
pp. 77-88
Author(s):  
Toshio Hosoh
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Rational Surface ◽  
Sufficient Condition ◽  
Chern Classes ◽  
Stable Vector Bundle ◽  
Rank 2 ◽  
Higher Rank

In the previous paper [1], we showed that the set of simple vector bundles of rank 2 on a rational surface with fixed Chern classes is bounded and we gave a sufficient condition for an H-stable vector bundle of rank 2 on a rational surface to be ample. In this paper, we shall extend the results of [1] to the case of higher rank.

scholarly journals Ample vector bundles on a rational surface

1975 ◽  
Vol 59 ◽  
pp. 135-148 ◽  
Author(s):  
Toshio Hosoh
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Rational Surface ◽  
Ruled Surface ◽  
Necessary Condition ◽  
Abelian Surface ◽  
Closed Field ◽  
Stable Vector Bundle ◽  
Singular Curve ◽  
Rank 2

On a complete non-singular curve defined over the complex number field C, a stable vector bundle is ample if and only if its degree is positive [3]. On a surface, the notion of the H-stability was introduced by F. Takemoto [8] (see § 1). We have a simple numerical sufficient condition for an H-stable vector bundle on a surface S defined over C to be ample; let E be an H-stable vector bundle of rank 2 on S with Δ(E) = c1(E)2 - 4c2(E) ≧ 0, then E is ample if and only if c1(E) > 0 and c2(E) > 0, provided S is an abelian surface, a ruled surface or a hyper-elliptic surface [9]. But the assumption above concerning Δ(E) evidently seems too strong. In this paper, we restrict ourselves to the projective plane P2 and a rational ruled surface Σn defined over an algebraically closed field k of arbitrary characteristic. We shall prove a finer assertion than that of [9] for an H-stable vector bundle of rank 2 to be ample (Theorem 1 and Theorem 3). Examples show that our result is best possible though it is not a necessary condition (see Remark (1) §2).

scholarly journals Involutions of Rank 2 Higgs Bundle Moduli Spaces

2018 ◽  
pp. 535-550 ◽  
Author(s):  
Oscar García-Prada ◽  
S. Ramanan
Keyword(s):  
Vector Bundle ◽  
Fundamental Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Higgs Field ◽  
Higgs Bundle ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Genus 2 ◽  
Rank 2

This chapter considers the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over ℂ, and studies involutions defined by tensoring the vector bundle with an element α‎ of order 2 in the Jacobian of the curve, combined with multiplication of the Higgs field by ±1. It describes the fixed points of these involutions in terms of the Prym variety of the covering of X defined by α‎, and gives an interpretation in terms of the moduli space of representations of the fundamental group.

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 Stability of tautological bundles on the Hilbert scheme of two points on a surface

2014 ◽  
Vol 214 ◽  
pp. 79-94 ◽  
Author(s):  
Malte Wandel
Keyword(s):  
Vector Bundle ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Stable Vector Bundle ◽  
Free Sheaf ◽  
Smooth Projective Surface ◽  
Rank 2 ◽  
Torsion Free

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

scholarly journals Stability of tautological bundles on the Hilbert scheme of two points on a surface

2014 ◽  
Vol 214 ◽  
pp. 79-94
Author(s):  
Malte Wandel
Keyword(s):  
Vector Bundle ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Stable Vector Bundle ◽  
Free Sheaf ◽  
Smooth Projective Surface ◽  
Rank 2 ◽  
Torsion Free

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

scholarly journals Some results in the theory of vector bundles

1973 ◽  
Vol 52 ◽  
pp. 97-128 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Vector Bundle ◽  
Fuchsian Group ◽  
Vector Bundles ◽  
Unitary Representations ◽  
Stable Bundle ◽  
Singular Curve ◽  
Stable Bundles ◽  
Genus 2 ◽  
Irreducible Unitary Representations ◽  
Geometric Definition

We have several definitions of the positivity of a vector bundle, differentiate definitions, an algebro-geometric definition, a topological definition etc. In § 1 we review the definitions and the relations between them. For a line bundle all the definitions are equivalent and every one agrees that they are reasonable. For a vector bundle, however, the definitions are not necessarily equivalent. One of the main results of this paper is the equivalence of the definitions over a complete non-singular curve. The proof is given in §2. We proved this over an elliptic curve in Umemura [18]. In this case the proof was based on Atiyah’s classification. To prove the equivalence over a curve of genus ≥ 2, the fundamental lemma is; A stable bundle of positive degree is positive in the sense of Nakano. The tool used to prove this lemma is the theory of stable bundles due to Narasimhan and Seshadri [11] —they establish a correspondence between stable bundles and certain types of irreducible unitary representations of a Fuchsian group.

scholarly journals THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

2020 ◽  
pp. 1-23
Author(s):  
MICHELE BOLOGNESI ◽  
NÉSTOR FERNÁNDEZ VARGAS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hyperelliptic Curve ◽  
Geometric Description ◽  
Birational Model ◽  
Rank 2 ◽  
Semistable Vector Bundles ◽  
Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

scholarly journals A rank 2 vector bundle on P4 with 15,000 symmetries

Topology ◽  
1973 ◽  
Vol 12 (1) ◽  
pp. 63-81 ◽  
Author(s):  
G. Horrocks ◽  
D. Mumford
Keyword(s):  
Vector Bundle ◽  
Rank 2
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