Geometry of splice-quotient singularities

TL;DR

This paper provides a new combinatorial-geometric approach to understanding splice-quotient singularities, showing that the divisorial filtration levels are generated by monomials, and offers a new proof of the End Curve Theorem.

Contribution

It introduces an elegant geometric method to analyze splice-quotient singularities and proves that the divisorial filtration is generated by monomials, simplifying previous proofs.

Findings

Divisorial filtration levels are generated by monomials of coordinate functions.

New geometric proof of the End Curve Theorem.

Enhanced understanding of the structure of splice-quotient singularities.

Abstract

We obtain a new important basic result on splice-quotient singularities in an elegant combinatorial-geometric way: every level of the divisorial filtration of the ring of functions is generated by monomials of the defining coordinate functions. The elegant way is the language of of line bundles based on Okuma's description of the function ring of the universal abelian cover. As an easy application, we obtain a new proof of the End Curve Theorem of Neumann and Wahl.