The Symmetric signature of cyclic quotient singularities

TL;DR

This paper computes the symmetric signature for two-dimensional cyclic quotient singularities and shows it coincides with the F-signature, providing insights into invariants of such singularities in characteristic zero.

Contribution

It explicitly calculates the symmetric signature for a class of cyclic quotient singularities and establishes its equality with the F-signature.

Findings

Symmetric signature equals F-signature for these singularities.

Provides explicit computation for two-dimensional cyclic quotient singularities.

Connects symmetric signature with known invariants in characteristic zero.

Abstract

The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the symmetric signature for two-dimensional cyclic quotient singularities, i.e. invariant subrings $k[[u,v]]^G$ of rings of formal power series under the action of a cyclic group $G$. Equivalently, these rings arise as the completions (at the irrelevant ideal) of two-dimensional normal toric rings. We show that for this class of rings the symmetric signature coincides with the $F$-signature.