  We study moduli spaces of vector bundles on a two-dimensional neighbourhood
$Z_k$ of an irreducible curve $\ell = CP^1$ with $\ell^2 = -k$ 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/ .
