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Nearly Gorenstein rational surface singularities

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

TL;DR

The paper develops a practical, resolution-based framework to understand when two-dimensional rational surface singularities are nearly Gorenstein by computing the canonical trace ideal Tr_A(K_A) via anti-nef cycles on a minimal resolution. It proves that Tr_A(K_A) is represented by the minimal anti-nef cycle F with F+K_X anti-nef and that F=Z_f characterizes nearly Gorenstein rational singularities, yielding concrete combinatorial criteria. It then classifies nearly Gorenstein rational singularities in two key cases: almost reduced fundamental cycles and quotient singularities, providing explicit graph-theoretic characterizations and families. The results provide both computational tools and structural insights, linking trace ideals to resolution graphs and expanding the landscape of nearly Gorenstein rational and quotient surface singularities.

Abstract

In this paper, we show that for any rational surface singularity $A$, the canonical trace ideal $\mathrm{Tr}_A(K_A)$ is integrally closed ideal which is represented by the minimal anti-nef cycle $F$ on the minimal resolution of singularities so that $K_X+F$ is anti-nef. Then $F \ge \mathbb Z$ if $A$ is not Gorenstein, where $\mathbb Z$ is the fundamental cycle. As a result, we give a criterion for rational surface singularity $A$ to be nearly Gorenstein. Moreover, we classify all nearly Gorenstein rational singularities in terms of resolution of singularities in the following cases: (a) the fundamental cycle $\mathbb Z$ is almost reduced; (b) quotient singularity.

Nearly Gorenstein rational surface singularities

TL;DR

The paper develops a practical, resolution-based framework to understand when two-dimensional rational surface singularities are nearly Gorenstein by computing the canonical trace ideal Tr_A(K_A) via anti-nef cycles on a minimal resolution. It proves that Tr_A(K_A) is represented by the minimal anti-nef cycle F with F+K_X anti-nef and that F=Z_f characterizes nearly Gorenstein rational singularities, yielding concrete combinatorial criteria. It then classifies nearly Gorenstein rational singularities in two key cases: almost reduced fundamental cycles and quotient singularities, providing explicit graph-theoretic characterizations and families. The results provide both computational tools and structural insights, linking trace ideals to resolution graphs and expanding the landscape of nearly Gorenstein rational and quotient surface singularities.

Abstract

In this paper, we show that for any rational surface singularity , the canonical trace ideal is integrally closed ideal which is represented by the minimal anti-nef cycle on the minimal resolution of singularities so that is anti-nef. Then if is not Gorenstein, where is the fundamental cycle. As a result, we give a criterion for rational surface singularity to be nearly Gorenstein. Moreover, we classify all nearly Gorenstein rational singularities in terms of resolution of singularities in the following cases: (a) the fundamental cycle is almost reduced; (b) quotient singularity.
Paper Structure (6 sections, 16 theorems, 38 equations, 7 figures)

This paper contains 6 sections, 16 theorems, 38 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Cycles on a resolution space
  3. The canonical trace ideals of rational singularities
  4. Criteria to be nearly Gorenstein
  5. Rational singularities having almost reduced fundamental cycle
  6. Quotient singularities

Key Result

Theorem 1.2

Assume that $A$ is a two-dimensional rational singularity, and let $K_A$ be a canonical module of $A$. Let $Z_f$ denote the fundamental cycle on the minimal resolution $X \to \mathop{\mathrm{Spec}}\nolimits A$. Then $\mathop{\mathrm{Tr}}\nolimits_A(K_A)$ is an integrally closed $\mathfrak m$-primary where $F$ is the minimal cycle such that $F+K_X$ is anti-nef (cf. notation (3)). Note that $F \ge Z

Figures (7)

  • Figure 2: The resolution graph of a cyclic quotient singularity
  • Figure 3: A rational singularity which satisfies (4.b)
  • Figure 5: Nearly Gorenstein RTP
  • Figure 6: Examples of Nearly Gorenstein rings with $e(A)=4$
  • Figure 7: Rational Triple Point of type $A_{\ell,m,n}$
  • ...and 2 more figures

Theorems & Definitions (41)

  • Definition 1.1: cf. HHS
  • Theorem 1.2: cf. Theorem \ref{['rat-cantr']}
  • Theorem 1.3: see Theorem \ref{['t:nGrat']}
  • Theorem 1.4: cf. Theorem \ref{['ARnG']}
  • Theorem 1.5: see Theorem $\ref{['Quot-nG']}$, Di
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • ...and 31 more