The embedding dimension of weighted homogeneous surface singularities

TL;DR

This paper investigates the embedding dimension and algebraic generators of weighted homogeneous surface singularities with rational homology sphere links, providing explicit formulas and exploring topological invariance.

Contribution

It offers explicit formulas for the embedding dimension in terms of Seifert invariants and extends the analysis to splice-quotient singularities with star-shaped graphs.

Findings

Formulas for embedding dimension in terms of Seifert invariants

Examples showing invariants are not topological

Extension to splice-quotient singularities

Abstract

We analyze the embedding dimension of a normal weighted homogeneous surface singularity, and more generally, the Poincar\'e series of the minimal set of generators of the graded algebra of regular functions, provided that the link of the germs is a rational homology sphere. In the case of several sub-families we provide explicit formulas in terms of the Seifert invariants (generalizing results of Wagreich and VanDyke), and we also provide key examples showing that, in general, these invariants are not topological. We extend the discussion to the case of splice--quotient singularities with star--shaped graph as well.