Vector bundles near negative curves: moduli and local Euler characteristic

We study moduli spaces of vector bundles on a two-dimensional neighbourhood $Z_k$ of an irreducible curve $\mathrm{\ell }=C{P}^1$ with ${\mathrm{\ell }}^2=-k$ and give an explicit construction of these moduli as stratified 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/ .