A CRITERION FOR GETTING A BIG COMPONENT OF THE MODULI OF VECTOR BUNDLES BY CHANGING A POLARIZATION

2001 ◽  
Vol 12 (08) ◽  
pp. 927-942
Author(s):  
KIMIKO YAMADA
Keyword(s):  
Vector Bundles ◽  
Line Bundles ◽  
Closed Field ◽  
Projective Surface ◽  
Chern Classes ◽  
Algebraically Closed Field ◽  
Stable Vector Bundles ◽  
Normal Surfaces ◽  
Smooth Projective Surface ◽  
Moduli Of Vector Bundles

Let MH(c1, c2) be a coarse moduli scheme parameterizing all rank-two H-μ-stable vector bundles with Chern classes (c1, c2) on a smooth projective surface X over an algebraically closed field. For fixed two ample line bundles H and H′, it is known that if c2 is greater than some constant p(X, H, H′) depending on H and H′ then MH(c1, c2) and MH′(c1, c2) are birationally equivalent. In this paper we show that this constant p(X, H, H′) generally does depends on the choice of H and H′. More precisely, we give some example of surface (and c1) on which, for any number K, there exists an integer c2 with c2≥K such that sup H: ample dim MH(c1, c2) = + ∞. This result is available also for normal surfaces.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

THE BOGOMOLOV–MIYAOKA–YAU INEQUALITY FOR LOG CANONICAL SURFACES

2001 ◽  
Vol 64 (2) ◽  
pp. 327-343 ◽  
Author(s):  
ADRIAN LANGER
Keyword(s):  
Vector Bundles ◽  
Universal Cover ◽  
Kodaira Dimension ◽  
Projective Surface ◽  
Chern Classes ◽  
Singular Surfaces ◽  
Smooth Projective Surface ◽  
Log Canonical ◽  
Surface Singularities ◽  
Negative Kodaira Dimension

Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov [1, Theorem 5] proved that c21 [les ] 4c2. This was improved to c21 [les ] 3c2 by Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality c21 [les ] 3c2 is attained, for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only possibility). Further generalizations of inequalities for Chern classes for some singular surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6], Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12], Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].In [8] we introduced Chern classes of reflexive sheaves, using Wahl's local Chern classes of vector bundles on resolutions of surface singularities. Here we apply them to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.

scholarly journals On the Connectedness of the Real Part of Moduli Spaces of Vector Bundles on Real Algebraic Surfaces

2000 ◽  
Vol 68 (1) ◽  
pp. 41-54
Author(s):  
Edoardo Ballico
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Algebraic Surfaces ◽  
Stable Rank ◽  
Projective Surface ◽  
Chern Classes ◽  
The Real ◽  
Smooth Projective Surface ◽  
Real Algebraic Surfaces

AbstractLet X be a smooth projective surface with q(X) = 0 defined over R and M(X;r;c1;c2;H) the moduli space of H-stable rank r vector bundles on X with Chern classes c1 and c2. Assume either r = 3 and X(R) connected or r = 3 and X(R) =ø or r=2 and X(R) = ø. We prove that quite often M is connected.

scholarly journals Stable vector bundles on an algebraic surface

1975 ◽  
Vol 58 ◽  
pp. 25-68 ◽  
Author(s):  
Masaki Maruyama
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Algebraic Curve ◽  
Algebraic Surface ◽  
Vector Bundles ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Stable Vector Bundles ◽  
First Chern Class

Let X be a non-singular projective algebraic curve over an algebraically closed field k. D. Mumford introduced the notion of stable vector bundles on X as follows;DEFINITION ([7]). A vector bundle E on X is stable if and only if for any non-trivial quotient bundle F of E,where deg ( • ) denotes the degree of the first Chern class of a vector bundles and r( • ) denotes the rank of a vector bundle.

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

Reducible vector bundles on a quadric surface

1964 ◽  
Vol 60 (3) ◽  
pp. 421-424 ◽  
Author(s):  
P. E. Newstead ◽  
R. L. E. Schwarzenberger
Keyword(s):  
Vector Bundles ◽  
Product Variety ◽  
Line Bundles ◽  
Closed Field ◽  
Quadric Surface ◽  
Algebraically Closed Field ◽  
Title One

0. Introduction. In a paper (1) with the same title, one of us attempted to classify the reducible k2-bundles on a quadric surface Q = P1 × P1 defined over an algebraically closed field k. This attempt suffered from two defects: first, the classification was given by a one-one correspondence rather than by an algebraic parameter variety, and, secondly, there is a mistake in the proof of Proposition 6 of (1), as a result of which Proposition 6, Proposition 7 and the exceptional clause of the Theorem in (1) are false. To correct these defects we use the definitions, notations and numbering of (1) without comment. Finally, we show how the classification of reducible k2-bundles on P1 × … × P1 (n factors) is determined by that of reducible k2-bundles on P1 × P1. The net effect of these corrections is to underline the moral of (1): while the classification of reducible k2-bundles (but with a distinguished factor of the product variety P1 × … × P1) is as nice as it could possibly be, the classification of reducible k2-bundles alone is as nasty as it could possibly be. Since every decomposable k2-bundle is just a sum of two line bundles, we consider only indecomposable reducible k2-bundles.

scholarly journals VECTOR BUNDLES, DUALITIES AND CLASSICAL GEOMETRY ON A CURVE OF GENUS TWO

2007 ◽  
Vol 18 (05) ◽  
pp. 535-558 ◽  
Author(s):  
QUANG MINH NGUYEN
Keyword(s):  
Vector Bundles ◽  
Double Cover ◽  
Line Bundles ◽  
Theta Divisor ◽  
Quartic Surface ◽  
Stable Vector Bundles ◽  
Symmetric Configuration ◽  
Classical Geometry ◽  
Genus Two ◽  
Linear Sections

Let C be a curve of genus two. We denote by [Formula: see text] the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by Jd the variety of line bundles of degree d on C. In particular, J1 has a canonical theta divisor Θ. The space [Formula: see text] is a double cover of ℙ8 = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual [Formula: see text], where J1 is embedded, there is a unique cubic hypersurface singular along J1, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.

scholarly journals Torus-equivariant vector bundles on projective spaces

1988 ◽  
Vol 111 ◽  
pp. 25-40 ◽  
Author(s):  
Tamafumi Kaneyama
Keyword(s):  
Vector Bundle ◽  
Toric Variety ◽  
Vector Bundles ◽  
Rational Point ◽  
Projective Spaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Algebraic Torus

For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

scholarly journals On Equivariant Vector Bundles on an Almost Homogeneous Variety

1975 ◽  
Vol 57 ◽  
pp. 65-86 ◽  
Author(s):  
Tamafumi Kaneyama
Keyword(s):  
Vector Bundles ◽  
Multiplicative Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Algebraic Torus ◽  
Homogeneous Variety

Let k be an algebraically closed field of arbitrary characteristic. Let T be an n-dimensional algebraic torus, i.e. T = Gm × · · · × Gm n-times), where Gm = Spec (k[t, t-1]) is the multiplicative group.

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 $r\mid d$.

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