  In this paper we extend the concept of Milnor fiber and Milnor number of a
curve singularity allowing the ambient space to be a quotient surface
singularity. A generalization of the local {\delta}-invariant is defined and
described in terms of a Q-resolution of the curve singularity. In particular,
when applied to the classical case (the ambient space is a smooth surface) one
obtains a formula for the classical {\delta}-invariant in terms of a
Q-resolution, which simplifies considerably effective computations. All these
tools will finally allow for an explicit description of the genus formula of a
curve defined on a weighted projective plane in terms of its degree and the
local type of its singularities.
