scholarly journals On kernel bundles over reducible curves with a node

2020 ◽  
Vol 31 (07) ◽  
pp. 2050054 ◽  
Author(s):  
Sonia Brivio ◽  
Filippo F. Favale
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Opposite Direction ◽  
Smooth Curve ◽  
Ambient Space ◽  
Smooth Case

Given a vector bundle [Formula: see text] on a complex reduced curve [Formula: see text] and a subspace [Formula: see text] of [Formula: see text] which generates [Formula: see text], one can consider the kernel of the evaluation map [Formula: see text], i.e. the kernel bundle [Formula: see text] associated to the pair [Formula: see text]. Motivated by a well-known conjecture of Butler about the semistability of [Formula: see text] and by the results obtained by several authors when the ambient space is a smooth curve, we investigate the case of a reducible curve with one node. Unexpectedly, we are able to prove results which goes in the opposite direction with respect to what is known in the smooth case. For example, [Formula: see text] is actually quite never [Formula: see text]-semistable. Conditions which gives the [Formula: see text]-semistability of [Formula: see text] when [Formula: see text] or when [Formula: see text] is a line bundle are then given.

Double Covers and Metastable Immersions of Spheres

1974 ◽  
Vol 26 (1) ◽  
pp. 145-176 ◽  
Author(s):  
Robert Wells
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Projective Space ◽  
Unit Sphere ◽  
Real Line ◽  
Double Cover ◽  
Sphere Bundle ◽  
Canonical Line Bundle ◽  
Projective Space Bundle ◽  
Double Covers

The real line will be R, Euclidean n-space will be Rn, the unit ball in Rn will be En, the unit sphere in Rn+1 will be Sn, and real projective n-space will be Pn. The canonical line bundle associated with the double cover Sn → Pn will be ηn. If γ is a vector bundle, E(γ) will be its associated cell bundle, S(γ) its associated sphere bundle, P(γ) its associated projective space bundle (P(γ) = S(γ) / (-1)) and T(γ) = E(γ)/S(γ) its Thorn space.

scholarly journals Motivic and $$\ell $$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

2022 ◽  
Vol 28 (2) ◽  
Author(s):  
Massimo Pippi
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Local Ring ◽  
Global Section ◽  
Regular Local Ring ◽  
Regular Scheme ◽  
Zero Locus

AbstractWe study the motivic and $$\ell $$ ℓ -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

scholarly journals Stable Higgs bundles over positive principal elliptic fibrations

2018 ◽  
Vol 5 (1) ◽  
pp. 195-201
Author(s):  
Indranil Biswas ◽  
Mahan Mj ◽  
Misha Verbitsky
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Complex Manifold ◽  
Higgs Bundle ◽  
Compact Complex Manifold ◽  
Holomorphic Line Bundle ◽  
Higgs Bundles ◽  
Stable Vector Bundle ◽  
Hermitian Metric ◽  
The Third

AbstractLet M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B0, where B0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L ⊕ Π ⃰B0, Π ⃰ ɸX), where (B0, ɸX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

scholarly journals GEOMETRY OF THE MODULI OF HIGHER SPIN CURVES

2000 ◽  
Vol 11 (05) ◽  
pp. 637-663 ◽  
Author(s):  
TYLER J. JARVIS
Keyword(s):  
Line Bundle ◽  
Moduli Spaces ◽  
Disjoint Union ◽  
Algebraic Curves ◽  
Quantum Cohomology ◽  
Smooth Curve ◽  
Natural Generalization ◽  
Intersection Theory ◽  
Universal Curve ◽  
Irreducible Components

This article treats various aspects of the geometry of the moduli [Formula: see text] of r-spin curves and its compactification [Formula: see text]. Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand–Dikii (KdVr) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology. For any line bundle [Formula: see text] on the universal curve over the stack of stable curves, there is a smooth stack [Formula: see text] of triples (X, ℒ, b) of a smooth curve X, a line bundle ℒ on X, and an isomorphism [Formula: see text]. In the special case that [Formula: see text] is the relative dualizing sheaf, then [Formula: see text] is the stack [Formula: see text] of r-spin curves. We construct a smooth compactification [Formula: see text] of the stack [Formula: see text], describe the geometric meaning of its points, and prove that its coarse moduli is projective. We also prove that when r is odd and g>1, the compactified stack of spin curves [Formula: see text] and its coarse moduli space [Formula: see text] are irreducible, and when r is even and [Formula: see text] is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when [Formula: see text] is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of [Formula: see text] [15], and also in the study of the cohomological field theory related to Witten's conjecture [16, 34].

scholarly journals Irreducibility of Lagrangian Quot schemes over an algebraic curve

2021 ◽  
Author(s):  
Daewoong Cheong ◽  
Insong Choe ◽  
George H. Hitching
Keyword(s):  
Vector Bundle ◽  
Symplectic Form ◽  
Algebraic Curve ◽  
Smooth Curve ◽  
Generic Element ◽  
Quot Scheme ◽  
Quot Schemes ◽  
Rank 2 ◽  
Complex Projective

AbstractLet C be a complex projective smooth curve and W a symplectic vector bundle of rank 2n over C. The Lagrangian Quot scheme $$LQ_{-e}(W)$$ L Q - e ( W ) parameterizes subsheaves of rank n and degree $$-e$$ - e which are isotropic with respect to the symplectic form. We prove that $$LQ_{-e}(W)$$ L Q - e ( W ) is irreducible and generically smooth of the expected dimension for all large e, and that a generic element is saturated and stable.

Sums of stably trivial vector bundles

1980 ◽  
Vol 87 (1) ◽  
pp. 97-107 ◽  
Author(s):  
Jacques Allard
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Vector Bundles ◽  
Real Vector

We say that a real vector bundle ξ over a finite C.W. complex X is stably trivial of type (n, k) or, simply, of type (n, k) if ξ ⊕ kε ≅ nε, where ε denotes a trivial line bundle. The following theorem is an immediate corollary (see (12)) of a theorem of T. Y. Lam ((9), theorem 2).

Linear Codes Obtained from Projective and Grassmann Bundles on Curves

2020 ◽  
Author(s):  
Edoardo Ballico
Keyword(s):  
Vector Bundle ◽  
Elliptic Curves ◽  
Vector Bundles ◽  
Linear Codes ◽  
Smooth Curve ◽  
Set Up ◽  
Semistable Vector Bundle ◽  
Double Coverings

We use split vector bundles on an arbitrary smooth curve defined over Fq to get linear codes (following the general set-up considered by S. H. Hansen and T. Nakashima), generalizing two quoted results by T. Nakashima. If p ≠ 2 for all integers d, g ≥ 2, r > 0 such that either r is odd or d is even we prove the existence of a smooth curve C of genus g defined over Fq and a p-semistable vector bundle E on C such that rank(E) = r, deg(E) = d and E is defined over Fq. Most results for particular curves are obtained taking double coverings or triple coverings of elliptic curves.

scholarly journals ASYMPTOTIC TORSION AND TOEPLITZ OPERATORS

2015 ◽  
Vol 16 (2) ◽  
pp. 223-349 ◽  
Author(s):  
Jean-Michel Bismut ◽  
Xiaonan Ma ◽  
Weiping Zhang
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Differential Form ◽  
Toeplitz Operators ◽  
Vector Bundles ◽  
Direct Image ◽  
Analytic Torsion ◽  
Leading Term ◽  
Positive Line Bundle ◽  
Positive Line

We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles $F_{p}$. For $p\in \mathbf{N}$, the flat vector bundle $F_{p}$ is the direct image of $L^{p}$, where $L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

Characterizing scrolls via the Hilbert curve

2014 ◽  
Vol 25 (11) ◽  
pp. 1450101 ◽  
Author(s):  
Antonio Lanteri
Keyword(s):  
Line Bundle ◽  
Ample Line Bundle ◽  
Smooth Curve ◽  
Smooth Surfaces ◽  
Projective Bundle ◽  
Hilbert Curve ◽  
Reconstruction Problem ◽  
Additional Assumptions

Let X be a projective bundle over a smooth curve and let L be an ample line bundle on X inducing [Formula: see text] on every fiber. The Hilbert curve Γ of such a polarized manifold (X, L) is explicitly described and the reconstruction problem is addressed. In particular, it is shown that the very special geographic shape of Γ allows to recover the adjunction theoretic type of (X, L) for smooth surfaces, for scrolls over a smooth curve, as well as for Veronese bundles under additional assumptions.

scholarly journals PARABOLIC k-AMPLE BUNDLES

2011 ◽  
Vol 22 (11) ◽  
pp. 1647-1660 ◽  
Author(s):  
INDRANIL BISWAS ◽  
FATIMA LAYTIMI
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Parabolic Bundles ◽  
Tautological Line Bundle

We construct projectivization of a parabolic vector bundle and a tautological line bundle over it. It is shown that a parabolic vector bundle is ample if and only if the tautological line bundle is ample. This allows us to generalize the notion of a k-ample bundle, introduced by Sommese, to the context of parabolic bundles. A parabolic vector bundle E* is defined to be k-ample if the tautological line bundle [Formula: see text] is k-ample. We establish some properties of parabolic k-ample bundles.

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