The space of curvettes of quotient singularities and associated invariants

TL;DR

This paper introduces an explicit formula for an invariant of cyclic quotient surface singularities, utilizing the space of curvettes and other invariants, enhancing understanding of their algebraic and geometric properties.

Contribution

It provides a concrete formula for the invariant $R_X$ based on numerical data and describes the space of curvettes and related invariants explicitly.

Findings

Explicit formula for $R_X$ in terms of $d$ and $q$

Description of the space of curvettes and generic curves

Analysis of invariants like $eta$-numbers and Milnor numbers

Abstract

This paper deals with a complete invariant $R_X$ for cyclic quotient surface singularities. This invariant appears in the Riemann Roch and Numerical Adjunction Formulas for normal surface singularities. Our goal is to give an explicit formula for $R_X$ based on the numerical information of $X$, that is, $d$ and $q$ as in $X=X(d;1,q)$. In the process, the space of curvettes and generic curves is explicitly described. We also define and describe other invariants of curves in $X$ such as the LR-logarithmic eigenmodules, $\delta$-invariants, and their Milnor and Newton numbers.