Local invariants on quotient singularities and a genus formula for weighted plane curves

TL;DR

This paper generalizes classical invariants of curve singularities to quotient surface singularities, providing explicit genus formulas for weighted projective plane curves based on local singularity types and degrees.

Contribution

It introduces a generalized Milnor fiber, Milnor number, and delta-invariant for singularities in quotient surfaces, enabling explicit genus calculations for weighted plane curves.

Findings

Generalized Milnor fiber and number for quotient singularities

Derived a delta-invariant formula via Q-resolution

Explicit genus formula for weighted projective plane curves

Abstract

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.