scholarly journals On the Rationality of Certain Strata of the Lange Stratification of Stable Vector Bundles on Curves

2001 ◽  
Vol 8 (4) ◽  
pp. 665-668
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundles ◽  
Maximal Degree ◽  
Projective Curve ◽  
Fiber Structure ◽  
Generic Fiber ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Natural Way

Abstract Let 𝑋 be a smooth projective curve of genus 𝑔 ≥ 2 and 𝑆(𝑟, 𝑑) the moduli scheme of all rank 𝑟 stable vector bundles of degree 𝑑 on 𝑋. Fix an integer 𝑘 with 0 < 𝑘 < 𝑟. H. Lange introduced a natural stratification of 𝑆(𝑟, 𝑑) using the degree of a rank 𝑘 subbundle of any 𝐸 ∈ 𝑆(𝑟, 𝑑) with maximal degree. Every non-dense stratum, say 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎), has in a natural way a fiber structure ℎ : 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎) → Pic𝑎(𝑋) × Pic𝑏(𝑋) with ℎ dominant. Here we study the rationality or the unirationality of the generic fiber of ℎ.

Maximal Subbundles of Rank 2 Vector Bundles on Projective Curves

2000 ◽  
Vol 43 (2) ◽  
pp. 129-137 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Stable Rank ◽  
Maximal Degree ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Projective Curves ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

AbstractLet E be a stable rank 2 vector bundle on a smooth projective curve X and V(E) be the set of all rank 1 subbundles of E with maximal degree. Here we study the varieties (non-emptyness, irreducibility and dimension) of all rank 2 stable vector bundles, E, on X with fixed deg(E) and deg(L), L ∈ V(E) and such that .

scholarly journals Stable vector bundles on curves: numerical invariants of their extremal subbundles

2001 ◽  
Vol 89 (1) ◽  
pp. 46
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Maximal Degree ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Numerical Invariants

Let $X$ be a smooth projective curve of genus $g \geq 2$ and $E$ a rank $r$ vector bundle on $X$. If $1 \leq t < r$ set $s_t(E)$:= sup $\lbrace t$(deg($E$)) - $r$(deg($F$)), where $F$ is a rank $t$ subsheaf of $E \rbrace $. Here we construct rank $r$ stable vector bundles $E$ on $X$ such that the sequence $ \lbrace s_t(E) \rbrace _{1 \leq t<r}$ has a prescribed value and the set of all subsheaves of $E$ with maximal degree may be explicitely described.

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.

scholarly journals Stable maps of genus zero in the space of stable vector bundles on a curve

2017 ◽  
Vol 28 (11) ◽  
pp. 1750078
Author(s):  
Kiryong Chung ◽  
Sanghyeon Lee
Keyword(s):  
Vector Bundles ◽  
Projective Curve ◽  
Stable Maps ◽  
Genus Zero ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Irreducible Components ◽  
Stable Map ◽  
Genus Formula ◽  
Closed Subvariety

Let [Formula: see text] be a smooth projective curve with genus [Formula: see text]. Let [Formula: see text] be the moduli space of stable rank two vector bundles on [Formula: see text] with a fixed determinant [Formula: see text] for [Formula: see text]. In this paper, as a generalization of Kiem and Castravet’s works, we study the stable maps in [Formula: see text] with genus [Formula: see text] and degree [Formula: see text]. Let [Formula: see text] be a natural closed subvariety of [Formula: see text] which parametrizes stable vector bundles with a fixed subbundle [Formula: see text] for a line bundle [Formula: see text] on [Formula: see text]. We describe the stable map space [Formula: see text]. It turns out that the space [Formula: see text] consists of two irreducible components. One of them parameterizes smooth rational cubic curves and the other parameterizes the union of line and smooth conics.

Stability of bundles with a given filtration on a smooth curve

1999 ◽  
Vol 129 (2) ◽  
pp. 229-234
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundles ◽  
Smooth Curve ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles

LetXbe a smooth projective curve of genusg ≥2. Here we construct stable vector bundles onXequipped with a filtration with suitable numerical properties and with stable graded subquotients.

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.

scholarly journals Opers of higher types, Quot-schemes and Frobenius instability loci

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Kirti Joshi ◽  
Christian Pauly
Keyword(s):  
Vector Bundles ◽  
Line Bundles ◽  
Closed Field ◽  
Projective Curve ◽  
Characteristic P ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Quot Schemes ◽  
Higher Types

In this paper we continue our study of the Frobenius instability locus in the coarse moduli space of semi-stable vector bundles of rank $r$ and degree $0$ over a smooth projective curve defined over an algebraically closed field of characteristic $p>0$. In a previous paper we identified the "maximal" Frobenius instability strata with opers (more precisely as opers of type $1$ in the terminology of the present paper) and related them to certain Quot-schemes of Frobenius direct images of line bundles. The main aim of this paper is to describe for any integer $q \geq 1$ a conjectural generalization of this correspondence between opers of type $q$ (which we introduce here) and Quot-schemes of Frobenius direct images of vector bundles of rank $q$. We also give a conjectural formula for the dimension of the Frobenius instability locus. Comment: 17 pages; Final version Epijournal de G'eom'etrie Alg'ebrique, Volume 4 (2020), Article Nr. 17

scholarly journals Stability of symmetric powers of vector bundles of rank two with even degree on a curve

2021 ◽  
Author(s):  
Jeong-Seop Kim
Keyword(s):  
Vector Bundles ◽  
Intersection Number ◽  
Symmetric Power ◽  
Projective Curve ◽  
Stable Vector Bundle ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Symmetric Powers ◽  
Genus Formula

This paper treats the strict semi-stability of the symmetric powers [Formula: see text] of a stable vector bundle [Formula: see text] of rank [Formula: see text] with even degree on a smooth projective curve [Formula: see text] of genus [Formula: see text]. The strict semi-stability of [Formula: see text] is equivalent to the orthogonality of [Formula: see text] or the existence of a bisection on the ruled surface [Formula: see text] whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of [Formula: see text] with strictly semi-stable [Formula: see text]. Moreover, it is shown that when [Formula: see text] is stable, every symmetric power [Formula: see text] is stable for all but a finite number of [Formula: see text] in the moduli of stable vector bundles of rank [Formula: see text] with fixed determinant of even degree on [Formula: see text].

FIBRÉS STABLES DE PENTE 2

2001 ◽  
Vol 33 (5) ◽  
pp. 535-542 ◽  
Author(s):  
VINCENT MERCAT
Keyword(s):  
Vector Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles

We give here a detailed description of the Brill–Noether loci of stable vector bundles of slope 2 on a smooth projective curve, which is assumed to be nonhyperelliptic.

scholarly journals Moduli spaces of orthogonal and symplectic bundles over an algebraic curve

2008 ◽  
Vol 144 (3) ◽  
pp. 721-733 ◽  
Author(s):  
Olivier Serman
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curve ◽  
Vector Bundles ◽  
Explicit Description ◽  
Representation Space ◽  
Classical Groups ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Polynomial Invariants

AbstractWe prove that, given a smooth projective curve C of genus g≥2, the forgetful morphism $\mathcal {M}_{\mathbf {O}_r} \longrightarrow \mathcal {M}_{\mathbf {GL}_r}$ (respectively $\mathcal M_{\mathbf {Sp}_{2r}}\longrightarrow \mathcal M_{\mathbf {GL}_{2r}}$) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.

Sign in / Sign up

Export Citation Format

Share Document