Computing instanton numbers of curve singularities

We start with any polynomial p(x, y) defining a plane curve with singularity at 0 ∈ C. Let π: C̃ → C denote the blow-up of C at the origin and let j be a positive integer. The data (j, p) determines a holomorphic bundle E(j, p) on C̃ with splitting type j and extension class p. We then algorithmically compute numerical invariants of the bundle E(j, p) and use them as invariants of the curve. A comparison among these invariants of the curve and some classical invariants can be found in [G3], and some samples are given below in Section 1. The holomorphic bundle E(j, p) and its numerical invariants have interpretation in mathematical physics as instantons on C̃ and numerical invariants of the instantons. Here we mention briefly some properties of these invariants. More details are given in [G3]. Instantons are well known to have a topological invariant called the charge, which in the case of compact surfaces corresponds to a second Chern number of the bundle. For the case of bundles on a blown-up surface, the local Chern number around the exceptional divisor decomposes as a sum of two numerical invariants (denoted w and h), which are local analytic invariants of the bundle. In [BG] it is shown that the pair of invariants (w, h) is strictly finer than the Chern numbers in the following sense. The pair (w, h) provides the coarsest stratification of moduli of instantons on the blown-up plane for which the strata are Hausdorff. In contrast, the stratification by topological charge does not provide Hausdorff strata [BG, Theorem 4.1]. Hence, the pair (w, h) gives strong numerical invariants, detecting more than topological information. The idea of using these invariants for curves is natural given that, as a first step to resolve the singularity at the origin, one blows it up, thus arriving at C̃, the base space of the bundles we construct.