Local invariants of minimal generic curves on rational surfaces

Abstract

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