scholarly journals On the Segre invariant for rank two vector bundles on ℙ2

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
L. Roa-Leguizamón ◽  
H. Torres-López ◽  
A. G. Zamora
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Chern Classes ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

Abstract We extend the concept of the Segre invariant to vector bundles on a surface X. For X = ℙ2 we determine what numbers can appear as the Segre invariant of a rank 2 vector bundle with given Chern classes. The irreducibility of strata with fixed Segre invariant is proved and their dimensions are computed. Finally, we present applications to the Brill–Noether Theory for rank 2 vector bundles on ℙ2.

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 .

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 Generalized null correlation bundles

1988 ◽  
Vol 111 ◽  
pp. 13-24 ◽  
Author(s):  
Lawrence Ein
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Topological Type ◽  
Stable Rank ◽  
Chern Classes ◽  
Smooth Variety ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

It is well known that the moduli space of stable rank 2 vector bundles on ℙ2 of the fixed topological type is an irreducible smooth variety ([1], and [8]). There are also many known results on the classification of stable rank 2 vector bundles on ℙ3 with “small” Chern classes.

On complex vector bundles on rational threefolds

1985 ◽  
Vol 97 (2) ◽  
pp. 279-288 ◽  
Author(s):  
Constantin BẮnicẮ ◽  
Mihai Putinar
Keyword(s):  
Vector Bundle ◽  
Algebraic Structure ◽  
Vector Bundles ◽  
Topological Type ◽  
Rational Surface ◽  
Chern Classes ◽  
Complex Vector Bundle ◽  
Complex Vector ◽  
Algebraic Vector ◽  
Rank 2

It is known [14] that every topological complex vector bundle on a smooth rational surface admits an algebraic structure. In [10] one constructs algebraic vector bundles of rank 2 on with arbitrary Chern classes c1, c2 subject to the necessary topological condition c1 c2 = 0 (mod 2). However, in dimensions greater than 2 the Chern classes don't determine the topological type of a vector bundle. In [2] one classifies the topological complex vector bundles of rank 2 on and one proves that any such bundle admits an algebraic structure.

ON SINGULARITIES OF ℳℙ3 (c1, c2)

1998 ◽  
Vol 09 (04) ◽  
pp. 407-419 ◽  
Author(s):  
VINCENZO ANCONA ◽  
GIORGIO OTTAVIANI
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Stable Rank ◽  
Chern Classes ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

Let ℳℙ3(c1,c2) be the moduli space of stable rank-2 vector bundles on ℙ3 with Chern classes c1, c2. We prove the following results: (1) Let k, β, γ be three integers such that k > 0, 0 ≤ β < γ, γ ≥ 2, kγ - (k + 1)β > 0; then the moduli space ℳℙ3(0, kγ2 - (k + 1)β2) is singular (the case k = 2, β = 0 was previously proved by M. Maggesi). (2) Let k, β, γ be three integers, with β and γ odd, such that k > 0, 0 < β < γ, γ ≥ 5, kγ - (k + 1)β + 1 > 0; then the moduli space ℳℙ3(-1,k(γ/2)2 - (k + 1)(β/2)2) + 1/4) is singular. In particular ℳℙ3(0,5), ℳℙ3(-1,6) are singular.

scholarly journals EXTENSIONS AND RANK-2 VECTOR BUNDLES ON IRREDUCIBLE NODAL CURVES

2005 ◽  
Vol 16 (10) ◽  
pp. 1081-1118
Author(s):  
D. ARCARA
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Rational Map ◽  
Nodal Curves ◽  
Nodal Curve ◽  
Stable Vector Bundles ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use the natural rational map [Formula: see text] defined by [Formula: see text] to study a compactification [Formula: see text] of the moduli space [Formula: see text] of semi-stable vector bundles of rank 2 and determinant L on C. In particular, we resolve the indeterminancy of ϕL in the case deg L = 3,4 via a sequence of three blow-ups with smooth centers.

scholarly journals OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

2019 ◽  
Vol 7 ◽  
Author(s):  
A. ASOK ◽  
J. FASEL ◽  
M. J. HOPKINS
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Algebraic Variety ◽  
Vector Bundles ◽  
Necessary Condition ◽  
Chern Classes ◽  
Smooth Complex ◽  
Steenrod Operations ◽  
Complex Algebraic Variety ◽  
Cycle Class

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

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