Local invariants of minimal generic curves on rational surfaces

TL;DR

This paper provides explicit formulas to recover the delta invariant of minimal generic curve germs on rational surface singularities from their topology, and characterizes possible delta values for quotient singularities.

Contribution

It introduces explicit formulas for delta invariants of minimal generic curves on rational surfaces and characterizes their possible values in quotient singularities.

Findings

Explicit formulas for delta invariants on rational surface singularities.

Delta invariant values are r-1 or r for quotient singularities.

Lower bound for delta invariant achieved by ordinary r-tuples.

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 of the abstract curve (C,0) from the topology of the embedding. We give explicit formulae whenever (C,0) is minimal generic and (X,0) is rational (as a continuation of previous works of the authors). Additionally we prove that if (X,0) is a quotient singularity, then the delta invariant of (C,0) only admits the values r-1 or r, where r is the number or irreducible components of (C,0). (r-1 realizes the extremal lower bound, valid only for `ordinary r-tuples'.)