scholarly journals The matrix of a connection having regular singularities on a vector bundle of rank 2 on ℙ1(c)

1979 ◽  
pp. 33-43 ◽  
Author(s):  
W. Dekkers
Keyword(s):  
Vector Bundle ◽  
The Matrix ◽  
Rank 2

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

A direct proof that toric rank $2$ bundles on projective space split

2020 ◽  
Vol 126 (3) ◽  
pp. 493-496
Author(s):  
David Stapleton
Keyword(s):  
Vector Bundle ◽  
Projective Space ◽  
Complete Intersection ◽  
Vector Bundles ◽  
Direct Proof ◽  
Rank 2 ◽  
Dimensional Projective Space ◽  
Rank 2 Bundles

The point of this paper is to give a short, direct proof that rank $2$ toric vector bundles on $n$-dimensional projective space split once $n$ is at least $3$. This result is originally due to Bertin and Elencwajg, and there is also related work by Kaneyama, Klyachko, and Ilten-Süss. The idea is that, after possibly twisting the vector bundle, there is a section which is a complete intersection.

A Vector Bundle of Rank 2 on P1 ×P3

1998 ◽  
Vol 57 (3) ◽  
pp. 583-598
Author(s):  
H. Lange
Keyword(s):  
Vector Bundle ◽  
Rank 2

scholarly journals On vector bundles on algebraic surfaces and 0-cycles

1993 ◽  
Vol 130 ◽  
pp. 19-23 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Algebraic Structure ◽  
Vector Bundles ◽  
Algebraic Surfaces ◽  
Projective Surface ◽  
Classical Theorem ◽  
Chern Classes ◽  
Complex Projective Surface ◽  
Holomorphic Structure ◽  
Rank 2

Let X be an algebraic complex projective surface equipped with the euclidean topology and E a rank 2 topological vector bundle on X. It is a classical theorem of Wu ([Wu]) that E is uniquely determined by its topological Chern classes . Viceversa, again a classical theorem of Wu ([Wu]) states that every pair (a, b) ∈ (H (X, Z), Z) arises as topological Chern classes of a rank 2 topological vector bundle. For these results the existence of an algebraic structure on X was not important; for instance it would have been sufficient to have on X a holomorphic structure. In [Sch] it was proved that for algebraic X any such topological vector bundle on X has a holomorphic structure (or, equivalently by GAGA an algebraic structure) if its determinant line bundle has a holomorphic structure. It came as a surprise when Elencwajg and Forster ([EF]) showed that sometimes this was not true if we do not assume that X has an algebraic structure but only a holomorphic one (even for some two dimensional tori (see also [BL], [BF], or [T])).

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 .

Separability for weakly irreducible matrices

2014 ◽  
Vol 14 (15&16) ◽  
pp. 1308-1337
Author(s):  
Daniel Cariello
Keyword(s):  
Positive Semidefinite ◽  
Tensor Rank ◽  
Split Decomposition ◽  
Finite Dimensional ◽  
The Matrix ◽  
Separability Problem ◽  
Finite Dimensional Case ◽  
Rank 2 ◽  
Decomposition Theorems ◽  
Weakly Irreducible

This paper is devoted to the study of the separability problem in the field of Quantum information theory. We focus on the bipartite finite dimensional case and on two types of matrices: SPC and PPT matrices (see definitions 32 and 33). We prove that many results hold for both types. If these matrices have specific Hermitian Schmidt decompositions then they are separable in a very strong sense (see theorem 38 and corollary 39). We prove that both types have what we call \textbf{split decompositions} (see theorems 41 and 42). We also define the notion of weakly irreducible matrix (see definition 43), based on the concept of irreducible state defined recently in \cite{chen1}, \cite{chen} and \cite{chen2}.}{These split decomposition theorems imply that every SPC $($PPT$)$ matrix can be decomposed into a sum of $s+1$ SPC $($PPT$)$ matrices of which the first $s$ are weakly irreducible, by theorem 48, and the last one has a further split decomposition of lower tensor rank, by corollary 49. Thus the SPC $($PPT$)$ matrix is decomposed in a finite number of steps into a sum of weakly irreducible matrices. Different components of this sum have support on orthogonal local Hilbert spaces, therefore the matrix is separable if and only if each component is separable. This reduces the separability problem for SPC $($PPT$)$ matrices to the case of weakly irreducible SPC $($PPT$)$ matrices. We also provide a complete description of weakly irreducible matrices of both types (see theorem 46).}{Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable (see theorem 58), we found sharp inequalites providing separability for both types (see theorems 61 and 62).

A Rank 2 Vector Bundle on ℙ4 with 15,000 Symmetries

2004 ◽  
pp. 526-544
Author(s):  
G. Horrocks ◽  
D. Mumford
Keyword(s):  
Vector Bundle ◽  
Rank 2

scholarly journals Ample vector bundles on a rational surface (higher rank)

1977 ◽  
Vol 66 ◽  
pp. 77-88
Author(s):  
Toshio Hosoh
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Rational Surface ◽  
Sufficient Condition ◽  
Chern Classes ◽  
Stable Vector Bundle ◽  
Rank 2 ◽  
Higher Rank

In the previous paper [1], we showed that the set of simple vector bundles of rank 2 on a rational surface with fixed Chern classes is bounded and we gave a sufficient condition for an H-stable vector bundle of rank 2 on a rational surface to be ample. In this paper, we shall extend the results of [1] to the case of higher rank.

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.

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