Minimal Anti-nef Cycle in Surface Singularities

• Minimal Anti-nef Cycle is a unique effective divisor on the minimal resolution that ensures K_X + F is anti-nef, characterizing rational surface singularities and nearly Gorenstein properties.
• It realizes the canonical trace ideal as H⁰(X, O_X(-F)) and provides a combinatorial framework for understanding integrally closed ideals.
• Algorithmic methods like Laufer’s computation sequence leverage the dual graph of the resolution to compute F, aiding in the classification of singularities such as cyclic quotients.

A minimal anti-nef cycle, denoted $F$, is a unique effective divisor on the minimal resolution of a two-dimensional normal singularity that encodes both the canonical trace ideal and the nearly Gorenstein property for rational surface singularities. Formally, for a rational surface singularity $A$ with canonical module $K_A$, $F$ is the minimal effective cycle such that $K_X + F$ is anti-nef on the minimal resolution $X$ (i.e., $(K_X + F)\cdot E_i \le 0$ for every exceptional curve $E_i$). This cycle is central in the geometric and algebraic study of rational surface singularities, underpinning the realization of integrally closed trace ideals and providing a combinatorial criterion for nearly Gorensteinness (Maeda et al., 25 Dec 2025).

1. Formal Definition and Structural Properties

Given $(A, \mathfrak m, k)$ a two-dimensional normal local domain over an algebraically closed field, let $\pi: X \to \operatorname{Spec} A$ be the minimal resolution, with exceptional locus $E = \bigcup_{i=1}^n E_i$, where each $E_i$ is a smooth rational curve intersecting transversely.

A divisor $D$ on $X$ is called anti-nef if $D\cdot E_i \le 0$ for all $i$. The minimal anti-nef cycle $F$ is the unique minimal effective divisor such that $K_X + F$ is anti-nef. Equivalently,

$$F = \min \left\{ D \ge 0 \;\big|\; (K_X + D)\cdot E_i \le 0 \text{ for all } i \right\}.$$

Every anti-nef divisor is effective, a consequence of Artin's contractibility criterion. The concept of anti-nefness is crucial because $-D$ being nef relates to the ampleness and contraction theory on surfaces.

2. Relationship to Fundamental Cycle and Canonical Trace Ideal

The fundamental cycle $Z_f$ is the minimal positive anti-nef divisor, i.e.,

$$Z_f = \min \left\{ Z > 0 \mid Z \cdot E_i \le 0\;\forall i \right\}.$$

For a rational singularity (i.e., $p_g(A) = 0$), the minimal anti-nef cycle $F$ satisfies $F \ge Z_f$ if $A$ is not Gorenstein.

The canonical trace ideal is defined as $\operatorname{Tr}_A(K_A) = K_A \cdot K_A^{-1}$, and a significant theorem asserts: $$\operatorname{Tr}_A(K_A) = H^0(X, \mathcal O_X(-F)),$$ which is an integrally closed $\mathfrak m$-primary ideal of $A$ and is computed explicitly by sections vanishing along $F$ (Maeda et al., 25 Dec 2025).

3. Nearly Gorenstein Criterion

A rational surface singularity $A$ is nearly Gorenstein if $\operatorname{Tr}_A(K_A) \supset \mathfrak m$. The minimal anti-nef cycle $F$ provides a criterion:

• $A$ is nearly Gorenstein if and only if $F = Z_f$.
• Equivalently, $K_X + Z_f$ is anti-nef.

This yields combinatorial inequalities: $(Z_f - E_i)\cdot E_i \le 2 \;\forall i,$ and for curves with $E_i^2 \le -3$,

$Z_f \cdot E_i \le E_i^2 + 2.$

These relationships show that $F$ encapsulates not only the trace ideal but also the geometry underlying the nearly Gorenstein property in terms of the dual graph of the resolution.

4. Algorithmic Determination and Example

The cycle $F$ can be computed using the dual graph of the exceptional divisor:

1. Compute $Z_f$ using Laufer’s computation sequence: starting with some $E_{j_1}$, sequentially add $E_{j_k}$ as long as $C_{k-1} \cdot E_{j_k} > 0$, ending with $Z_f$.
2. Compute $F$: Starting from $Z_f$, add $E_j$ whenever $(K_X + C_{k-1}) \cdot E_j > 0$ until for all $i$, $(K_X + C_m) \cdot E_i \le 0$.

Example: For a cyclic quotient singularity $A = k[[u,v]]^{1/n(1,q)}$ with resolution a chain $E_1 - E_2 - \cdots - E_r$ and $E_i^2 = -b_i$, $Z_f$ is determined via a continued-fraction algorithm. $F$ is found as the minimal integral solution to $(K_X + D) \cdot E_i \le 0$ for $D = \sum d_i E_i \ge 0$.

5. Integral Closure and Realization

The realization of the canonical trace ideal via $F$ shows that $H^0(X, \mathcal O_X(-F))$ is integrally closed because it is the full preimage of the ideal under the resolution. The general principle is that ideals associated to effective cycles on a resolution are integrally closed.

On rational singularities, for any divisor $L$ with no fixed part, $$H^0(X, \mathcal O_X(L))^{-1} = H^0(X, \mathcal O_X(-L))$$, and the multiplication of sections respects addition of divisors. For $L = K_X$, this leads directly to the identification of $\operatorname{Tr}_A(K_A)$ with $H^0(X, \mathcal O_X(-F))$.

6. Classifications Based on the Minimal Anti-nef Cycle

Two major classifications of nearly Gorenstein rational surface singularities arise from properties of $F$:

• Fundamental cycle almost reduced: If $Z_f$ has coefficient $1$ on every $E_i$ with $E_i^2 \le -3$, only star-shaped dual graphs labeled $A$, $D$, $E_6$, $E_7$, $E_8$ with a central coefficient $2$ are nearly Gorenstein.
• Non-cyclic quotient singularities: Nearly Gorenstein (non-Gorenstein) quotient singularities correspond to a finite, explicit list of Pinkham–Demazure and sporadic types, with their dual graphs and cycle data provided in the cited classification. The precise forms are as given in Theorem 6.1 of (Maeda et al., 25 Dec 2025).

7. Centrality in the Geometry of Rational Surface Singularities

The minimal anti-nef cycle $F$ serves as a geometric bridge connecting the resolution's intersection theoretic data, integrally closed ideals, and the algebraic property of nearly Gorensteinness for rational surface singularities. Its explicit algorithmic construction enables both effective computation and structural classification, making it a fundamental object in the algebraic and combinatorial investigation of surface singularities (Maeda et al., 25 Dec 2025).