Nearly Gorenstein normal graded rings

TL;DR

The paper develops a framework to study nearly Gorenstein normal graded rings by introducing invariants derived from the canonical module, notably $a(R)$ and $b(R)$, and by analyzing trace ideals via $\mathrm{Tr}_R(K_R)$. It leverages Demazure constructions and the geometry of $\mathrm{Proj}(R)$ to connect cohomological data to near Gorenstein status, yielding both general criteria and case-by-case classifications. In dimension two, the authors obtain strong structural results, including F-regular implications when $b(R)<0$ and detailed classifications for $2$-dimensional cone singularities (notably for genus $g=2,3$). A key finding is the drastic divergence between nearly Gorenstein and almost Gorenstein properties in cone settings, underscoring the nuanced landscape of singularity types and their resolutions.

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.

Theorems & Definitions (44)

• Definition 1.1: HHS
• Definition 2.1: GW
• Proposition 2.2
• proof
• Conjecture 2.3
• Definition 3.2
• Theorem 3.3
• proof
• Proposition 3.4
• proof