Local invariants of minimal generic curves on rational surfaces

Abstract

Let (C, 0) be a reduced curve germ in a normal surface singularity (X, 0). The main goal is to recover the delta invariant δ(C) of the abstract curve (C, 0) from the topology of the embedding (C, 0) ⊂ (X, 0). We give explicit formulae whenever (C, 0) is minimal generic and (X, 0) is rational (as a continuation of [8,9]).Additionally we prove that if (X, 0) is a quotient singularity, then δ(C) only admits the values r −1 or r, where r is the number or irreducible components of (C, 0). (δ(C) = r −1 realizes the extremal lower bound, valid only for 'ordinary r-tuples'.