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.

scholarly journals Seshadri constants on some Quot schemes

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Chandranandan Gangopadhyay ◽  
Krishna Hanumanthu ◽  
Ronnie Sebastian
Keyword(s):  
Vector Bundle ◽  
Positive Integer ◽  
Line Bundles ◽  
Seshadri Constants ◽  
Quot Scheme ◽  
Quot Schemes ◽  
Grassmann Bundle

Abstract Let E be a vector bundle of rank n on ℙ 1 {\mathbb{P}^{1}} . Fix a positive integer d. Let 𝒬 ⁢ ( E , d ) {\mathcal{Q}(E,d)} denote the Quot scheme of torsion quotients of E of degree d and let Gr ⁢ ( E , d ) {\mathrm{Gr}(E,d)} denote the Grassmann bundle that parametrizes the d-dimensional quotients of the fibers of E. We compute Seshadri constants of ample line bundles on 𝒬 ⁢ ( E , d ) {\mathcal{Q}(E,d)} and Gr ⁢ ( E , d ) {\mathrm{Gr}(E,d)} .

CLIFFORD'S THEOREM AND HIGHER RANK VECTOR BUNDLES

2002 ◽  
Vol 13 (07) ◽  
pp. 785-796 ◽  
Author(s):  
VINCENT MERCAT
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Smooth Curve ◽  
Clifford Index ◽  
Precise Bound ◽  
Rank 2 ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle ◽  
Rank 2 Bundles

We give here a refinement of the classical Clifford's theorem for the upper bound of the number of independent global sections of a semistable vector bundle on a smooth curve. We also conjecture a new version of this theorem that takes into account the Clifford index of the curve. In the case of a bi-elliptic curve we obtain a very precise bound. Finally we study the case of rank 2 bundles.

scholarly journals On Fano manifolds, which are PK-bundles over P2

1990 ◽  
Vol 120 ◽  
pp. 89-101 ◽  
Author(s):  
Michał Szurek ◽  
Jarosław A. Wiśniewski
Keyword(s):  
Vector Bundle ◽  
Projective Plane ◽  
General Situation ◽  
Complex Projective Plane ◽  
Fano Manifolds ◽  
Rank 2 ◽  
Complex Projective

In our earlier paper [8] we discussed Fano manifolds X that are of the form X = ℙ() with a rank-2 vector bundle on a surface S. Here we study a more general situation of Fano manifolds, ruled over the complex projective plane P2 as Pr-1-bundles, i.e., being of the form ℙ() with -a bundle of rank r ≥ 3 on P2.

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

THE GENERALIZED GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM`

2021 ◽  
Vol 10 (9) ◽  
pp. 3253-3262
Author(s):  
H. Umair ◽  
H. Zainuddin ◽  
K.T. Chan ◽  
Sh.K. Said Husein
Keyword(s):  
Uncertainty Principle ◽  
Symplectic Form ◽  
Uncertainty Relation ◽  
Riemannian Metric ◽  
Hamiltonian Flow ◽  
Geometric Uncertainty ◽  
Projective Hilbert Space ◽  
Space Dynamics ◽  
Geometric Quantum Mechanics ◽  
Complex Projective

Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.

scholarly journals ADJUNCTION FOR VARIETIES WITH A ℂ* ACTION

2020 ◽  
Author(s):  
ELEONORA A. ROMANO ◽  
JAROSŁAW A. WIŚNIEWSKI
Keyword(s):  
Fixed Point ◽  
Line Bundle ◽  
Normal Bundle ◽  
Ample Line Bundle ◽  
Projective Manifold ◽  
Fano Manifolds ◽  
Adjunction Theory ◽  
Rank 2 ◽  
General Orbit ◽  
Complex Projective

Abstract Let X be a complex projective manifold, L an ample line bundle on X, and assume that we have a ℂ* action on (X;L). We classify such triples (X; L;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

Ample Vector Bundles of Curve Genus One

1999 ◽  
Vol 42 (2) ◽  
pp. 209-213 ◽  
Author(s):  
Antonio Lanteri ◽  
Hidetoshi Maeda
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Ample Vector Bundle ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Curve Genus ◽  
Genus One ◽  
Complex Projective ◽  
Zero Locus

AbstractWe investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.

scholarly journals NON-EMPTINESS OF MODULI SPACES OF COHERENT SYSTEMS

2008 ◽  
Vol 19 (07) ◽  
pp. 777-799 ◽  
Author(s):  
L. BRAMBILA-PAZ
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curve ◽  
Coherent System ◽  
Coherent Systems ◽  
Generic Element

Let X be a general smooth projective algebraic curve of genus g ≥ 2 over ℂ. We prove that the moduli space G(α:n,d,k) of α-stable coherent systems of type (n,d,k) over X is empty if k > n and the Brill–Noether number β := β(n,d,n + 1) = β(1,d,n + 1) = g - (n + 1)(n - d + g) < 0. Moreover, if 0 ≤ β < g or β = g, n ∤g and for some α > 0, G(α : n,d,k) ≠ ∅ then G(α : n,d,k) ≠ ∅ for all α > 0 and G(α : n,d,k) = G(α′ : n,d,k) for all α,α′ > 0 and the generic element is generated. In particular, G(α : n,d,n + 1) ≠ ∅ if 0 ≤ β ≤ g and α > 0. Moreover, if β > 0 G(α : n,d,n + 1) is smooth and irreducible of dimension β(1,d,n + 1). We define a dual span of a generically generated coherent system. We assume d < g + n1≤ g + n2and prove that for all α > 0, G(α : n1,d, n1+ n2) ≠ ∅ if and only if G(α : n2,d, n1+ n2) ≠ ∅. For g = 2, we describe G(α : 2,d,k) for k > n.

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