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

Tomohiro Okuma, Kei-ichi Watanabe, Ken-ichi Yoshida

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 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 $g\ge 2$, 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.

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

Abstract

Let be an excellent two-dimensional normal local ring containing an algebraically closed field. Then is called an elliptic singularity if , where 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 , we provide examples of singularities that is almost Gorenstein (resp. not almost Gorenstein) with . We also provide several examples of determinantal singularities associated with matrices, which include both almost Gorenstein singularities and non-almost Gorenstein singularities.

Paper Structure

This paper contains 16 sections, 33 theorems, 51 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Almost Gorenstein local rings
  4. Cycles and genera
  5. Fundamental exact sequences
  6. The cohomological cycle and the canonical module
  7. The case when $U = H^0(\mathcal{U})$; Cases when $A$ is rational or elliptic.
  8. Elliptic Singularities
  9. The multiplicity of the module $U$
  10. Examples of Elliptic singularities
  11. Examples of cone singularities
  12. Level Cone Singularities and Standard Graded Sings.
  13. The case $D = K_C + P$
  14. The Point Divisors and Weierstrass Semigroups
  15. The case $g=2$
  16. ...and 1 more sections

Key Result

Lemma 2.2

We have the following.

Theorems & Definitions (70)

  • Definition 2.1: cf. GTT
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6: cf. la.rat, karras
  • ...and 60 more