The symmetric signature of cyclic quotient singularities

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 $k[\mkern -2.75mu[ u,v]\mkern -2.75mu]^G$ of rings of formal power series under the action of a cyclic group. 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.