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].

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.

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.

ON MAXIMALLY FROBENIUS DESTABILISED VECTOR BUNDLES

2019 ◽  
Vol 99 (2) ◽  
pp. 195-202
Author(s):  
LINGGUANG LI
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Closed Field ◽  
Projective Curve ◽  
Stable Vector Bundle ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

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 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.

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 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.

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.

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