Nearly Gorenstein cyclic quotient singularities

TL;DR

This paper characterizes nearly Gorenstein cyclic quotient singularities in algebraic geometry, providing a necessary and sufficient condition and identifying new classes of such rings.

Contribution

It establishes a complete criterion for nearly Gorenstein property in cyclic quotient singularities and introduces new classes of these rings.

Findings

Derived a necessary and sufficient condition for nearly Gorenstein property

Identified several new classes of nearly Gorenstein cyclic quotient singularities

Enhanced understanding of the structure of cyclic quotient singularities

Abstract

We investigate the nearly Gorenstein property among $d$-dimensional cyclic quotient singularities $\Bbbk[[x_1,\dots,x_d]]^G$, where $\Bbbk$ is an algebraically closed field and $G\subseteq{\rm GL}(d,\Bbbk)$ is a finite small cyclic group whose order is invertible in $\Bbbk$. We prove a necessary and sufficient condition to be nearly Gorenstein that also allows us to find several new classes of such rings.