Numerical invariants for bundles on blow-ups (original) (raw)

Moduli of bundles on the blown-up plane

1998

Let M(j) denote the moduli space of bundles on the blown-up plane which restrict to the exceptional divisor as O(j)+O(-j). We show that there is a topological embedding of M(j) into M(j+1).

Holomorphic bundles on the blown-up plane and the bar construction

Algebraic & Geometric Topology, 2020

Let M 0 = ' BU (k), M 1 = ' (BU (k) × BU (k)). We construct a map from Bar`M 0 , M n 0 , M n 1´t o the rank-stable moduli space of holomorphic bundles on the blowup of P 2 at n points, framed on a rational curve. We show that this map is a homotopy equivalence in the degree 1 and 2 components and in the limit when k → ∞.

Moduli Spaces of Sheaves on General Blow-ups of mathbbP2\mathbb{P}^2mathbbP2

Cornell University - arXiv, 2022

Let X be the blow-up of P 2 along m general points, and A = H − εiEi be a generic polarization with 0 < εi ≪ 1. We classify the Chern characters which satisfy the weak Brill-Noether property, i.e. a general sheaf in MA(v), the moduli space of slope stable sheaves with Chern character v, has at most one non-zero cohomology. We further give a necessary and sufficient condition for the existence of stable sheaves. Our strategy is to specialize to the case when the m points are collinear. 2 Preliminaries Convention. By a surface, we mean a connected smooth projective algebraic surface over C. All sheaves are coherent unless specified. For a surface X and coherent sheaves E and F , we write h i (X, E) = dim H i (X, E), hom(E, F) = dim Hom(E, F), and ext i (E, F) = dim Ext i (E, F). 2.1 Chern characters and Riemann-Roch on surfaces Let E be a torsion-free sheaf on a polarized surface (X, A). Let K(X) Q be the Grothendieck group of X with Q-coefficients. The Chern character ch(E) = (ch 0 (E), ch 1 (E), ch 2 (E)

Holomorphic Rank Two Vector Bundles on Blow-ups

1996

In this paper we study holomorphic rank two vector bundles on the blow up of $ {\bf C}^2$ at the origin. A classical theorem of Birchoff and Grothendieck says that any holomorphic vector bundle on the projective plane bfP1{\bf P}^1bfP1 splits into a sum of line bundles. If EEE is a holomorphic vector bundle over the blow up of $

Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

Communications in Algebra, 2009

We study moduli of vector bundles on a two-dimensional neighbourhood Z k of an irreducible curve ℓ ∼ = P 1 with ℓ 2 = −k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/ ∼ s0571100/Instanton/.

Vector bundles on a neighborhood of an exceptional curve and elementary transformations

Forum Mathematicum, 2000

Let W be the germ of a smooth complex surface around an exceptional curve and let E be a rank 2 vector bundle on W. We study the cohomological properties of a finite sequence {E i } 1≤ i ≤ t of rank 2 vector bundles canonically associated to E. We calculate numerical invariants of E in terms of the splitting types of E i , 1 ≤ i ≤ t. If S is a compact complex smooth surface and E is a rank two bundle on the blow-up of S at a point, we show that all values of c 2 ( E ) -c 2 (π * ( E ) ** ) that are numerically possible are actually attained.

New moduli components of rank 2 bundles on projective space

Sbornik: Mathematics, 2021

We present a new family of monads whose cohomology is a stable rank 2 vector bundle on . We also study the irreducibility and smoothness together with a geometrical description of some of these families. These facts are used to construct a new infinite series of rational moduli components of stable rank 2 vector bundles with trivial determinant and growing second Chern class. We also prove that the moduli space of stable rank 2 vector bundles with trivial determinant and second Chern class equal to 5 has exactly three irreducible rational components. Bibliography: 40 titles.