A Geometric description of almost Gorensteinness for two-dimensional normal singularities

TL;DR

This paper characterizes almost Gorenstein rings for two-dimensional normal singularities using geometric methods and provides criteria and examples, including elliptic and determinantal singularities.

Contribution

It offers a geometric description and criteria for almost Gorensteinness in two-dimensional normal singularities, connecting it with resolution of singularities.

Findings

Elliptic singularities are almost Gorenstein.

Constructed examples of singularities with various fundamental genus values.

Provided examples of determinantal singularities that are both almost Gorenstein and not.

Abstract

Let $A$ be an excellent two-dimensional normal local ring containing an algebraically closed field. Then $A$ is called an elliptic singularity if $p_f(A)=1$, where $p_f$ denotes the fundamental genus. On the other hand, the concept of almost Gorenstein rings was introduced by Barucci and Fr\"oberg for one-dimensional local rings and generalized by Goto, Takahashi and Taniguchi to higher dimension. In this paper, we describe almost Gorenstein rings in geometric language using resolution of singularities and give criterions to be almost Gorenstein. In particular, we show that elliptic singularities are almost Gorenstein. Also, for every integer $g\ge 2$, we provide examples of singularities that is almost Gorenstein (resp. not almost Gorenstein) with $p_f(A)=g$. We also provide several examples of determinantal singularities associated with $2\times 3$ matrices, which include both almost Gorenstein singularities and non-almost Gorenstein singularities.