Nearly Gorenstein normal graded rings

TL;DR

This paper explores the nearly Gorenstein property of normal graded rings, introducing a new invariant and analyzing conditions under which these rings are nearly Gorenstein, especially in low dimensions and specific singularity types.

Contribution

It introduces the invariant b(R), investigates its relation to nearly Gorenstein rings, and characterizes nearly Gorenstein properties for 2-dimensional cone singularities over curves of genus up to 3.

Findings

b(R)<0 implies log-terminal if R is Q-Gorenstein

Conditions for 2D cone singularities over genus g≤3 curves

Nearly Gorenstein and almost Gorenstein properties differ significantly

Abstract

We investigate nearly Gorenstein property for a normal graded ring $R = \bigoplus_{n\ge 0}R_n$ finitely generated over a field. For that purpose, we investigate ${K_R}^{-1}$, the inverse of $K_R$ (the canonical module of $R$) and introduce a new invariant $b(R)$ of $R$. We investigate nearly Gorenstein property of $R$ using $a(R)$ and $b(R)$ and $m(R)$, the initial degree of $R$. If $b(R)<0$, (and if $R$ is $\mathbb Q$-Gorenstein), then we believe that $R$ is log-terminal -- this is proved if $\dim R=2$ or $R$ is F-pure (or $F$-pure type). Then we determine the condition for a $2$-dimensional cone singularity over a smooth curve of genus $g\le 3$ to be nearly Gorenstein. We observe that ``almost Gorenstein" property and nearly Gorenstein property are drastically different for such rings.