2024/10/31 by Tomohiro Okuma, Okuma, Tomohiro, Keiichi Watanabe +3 · 1 citation
Mathematics · #13G05 #13H10 #14B05 #14J17 #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2410.23911
openalex publication_date 2024/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be an excellent two-dimensional normal local ring containing an algebraically closed field. Then A is called an elliptic singularity if pf(A)=1, where pf denotes the fundamental genus. On the other hand, the concept of almost Gorenstein rings was introduced by Barucci and Fröberg 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≥ 2, we provide examples of singularities that is almost Gorenstein (resp. not almost Gorenstein) with pf(A)=g. We also provide several examples of determinantal singularities associated with 2× 3 matrices, which include both almost Gorenstein singularities and non-almost Gorenstein singularities.