Nearly Gorenstein Rational Singularities

• Nearly Gorenstein rational singularities are defined by the inclusion m ⊆ Tr_A(K_A), bridging algebraic and geometric properties in rational surface contexts.
• The canonical trace ideal is realized through global sections on the minimal resolution, using anti-nef divisors like the fundamental cycle to assess singularity type.
• Intersection conditions and combinatorial criteria provide a tractable method to classify these singularities, extending insights to higher-dimensional cyclic quotients.

Nearly Gorenstein rational singularities arise in the study of normal surface and higher-dimensional singularities over an algebraically closed field of characteristic zero, where the relationship between the canonical module and maximal ideal is controlled via the trace ideal. These singularities refine the classical distinction between Gorenstein and non-Gorenstein rational singularities through a homological and geometric lens, with significant implications for the birational geometry and invariant theory of surface singularities.

1. Foundational Definitions and Trace Ideals

Let $(A,\frak m,k)$ denote a two-dimensional normal local domain admitting a resolution of singularities $\pi:X\to\Spec A$, with exceptional divisor $E=\bigcup_{i=1}^nE_i$. The canonical divisor on $X$ is $K_X$, and the canonical $A$-module is $K_A$.

For any finitely generated $A$-module $M$, the trace ideal is defined by

$\Tr_A(M) = \sum_{f\in\Hom_A(M,A)}\Image(f)\subset A.$

In the special case $M=K_A$, the trace ideal identifies as

$\Tr_A(K_A) = K_A\cdot K_A^{-1},$

where $I\cdot I^{-1}$ is computed in the total fraction ring $Q(A)$. When $K_A$ is a canonical ideal, this is structurally similar to trace computations for module endomorphisms.

A singularity $(A, \frak m, k)$ is called nearly Gorenstein if

$\frak m \subseteq \Tr_A(K_A),$

equivalently, $\Tr_A(K_A)=\mathfrak m$ for the non-Gorenstein, nearly Gorenstein case. In this context, the trace ideal is always $\mathfrak m$-primary and integrally closed for rational surface singularities (Maeda et al., 25 Dec 2025, Caminata et al., 2020).

2. Canonical Trace Ideal and Geometric Representation

A key result states that for a rational surface singularity $A$, $\Tr_A(K_A)$ corresponds to the global sections of an invertible sheaf on the minimal resolution $X$. The construction involves anti-nef divisors, specifically the minimal cycle $F$ supported on $E$ such that $K_X+F$ is anti-nef: $\Tr_A(K_A) = H^0\big(X, \mathcal{O}_X(-F)\big).$ This $F$ is effective, anti-nef, and minimal with respect to supporting $K_X+F$ as anti-nef. The fundamental cycle $Z_f$ is the unique minimal positive anti-nef cycle on $X$, characterized by $Z_f\cdot E_i\le0$ for all $i$; $K_X$ is not generally anti-nef itself, hence $F$ is needed for the correction. For $K_X\not\equiv0$, $F\ge Z_f$.

This geometric realization allows direct comparison with the maximal ideal $\mathfrak m$: $H^0(X,\mathcal{O}_X(-Z_f)) = \mathfrak m$, giving a direct criterion for nearly Gorensteinness in terms of cycles on the resolution (Maeda et al., 25 Dec 2025).

3. Main Criteria for Nearly Gorenstein Rational Singularities

The following are equivalent statements for a rational surface singularity $A$ (not Gorenstein):

1. $A$ is nearly Gorenstein.
2. The minimal anti-nef cycle $F$ equals the fundamental cycle $Z_f$.
3. $K_X + Z_f$ is anti-nef.
4. Intersection conditions on the coefficients $z_i$ of $Z_f = \sum_i z_iE_i$ and the negatives of self-intersections $b_i = -E_i^2$ hold in one of the prescribed patterns, e.g., $E$ irreducible, unique $z_{i_0}=2$ with specified intersection numbers, or exactly two $z_{i}=1$.
5. For all $E_i$ with $b_i\ge3$, $Z_f\cdot E_i \le E_i^2+2$.

These criteria connect the algebraic definition via the trace to strict intersection-theoretic or combinatorial conditions on the resolution graph, providing a combinatorial criterion for nearly Gorensteinness (Maeda et al., 25 Dec 2025).

For rational surface singularities that are Gorenstein, $K_X\equiv0$ and $F=0$; in this case, the trace ideal is $A$.

4. Special Cases: Almost Reduced and Quotient Singularities

Almost Reduced Fundamental Cycle

A rational singularity has an almost reduced fundamental cycle if every component $E_i$ with $b_i\ge3$ has $z_i=1$ in $Z_f$. The classification in this case comprises the extended Dynkin (A–D–E) graphs possibly with one central curve of higher multiplicity. Specifically:

• Type $A_n^{\mathrm{nG}}$: Line with all $z_i=1$.
• Type $D_n^{\mathrm{nG}}$: $D_n$-graph with two ends of multiplicity $1$ and the central arm potentially higher.
• Types $E_6, E_7, E_8$ with one or two central larger multiplicities. The multiplicities on nodes match those in ordinary A–D–E cycles, except for possible central enhancements (Maeda et al., 25 Dec 2025).

Quotient Singularities

Quotient singularities of the form $A = k[[x, y]]^G$ with $G\subset GL(2,k)$ and no cyclic quotient appear as rational, log-terminal surface singularities. The classification of nearly Gorenstein but non-Gorenstein quotient singularities is as follows: $$D = \tfrac12P_1 + \tfrac12P_2 - \frac{ks-(k-1)}{(k+1)s-k}P_3,\quad k\ge0,\, s\ge3,$$ with finitely many exceptional cases for types $(2,3,3)$, $(2,3,4)$, $(2,3,5)$. The resolution graphs are star-shaped as for Du Val singularities, but with a branch of larger multiplicity in the anti-nef cycle (Maeda et al., 25 Dec 2025).

5. Cyclic Quotient Singularities and Higher Dimensions

For $R = k[[x_1,\ldots,x_d]]^G$ with $G$ a small cyclic subgroup of $GL(d,k)$ of type $\frac1r(a_1,\ldots,a_d)$ (i.e., $G$ acts diagonally by a primitive $r$-th root of unity), $R$ is nearly Gorenstein if for each $1\le i<d$, permutation $\sigma\in S_d$, and $i$-tuple $(\alpha_1,\ldots,\alpha_i)$ with $\sum_j \alpha_j<r$ and $\sum_j \alpha_j a_{\sigma(j)}\equiv0\pmod r$, there exist $1\le\beta_j\le\alpha_j+1$ such that

$$\sum_{j=1}^i \beta_j a_{\sigma(j)} \equiv -\sum_{k=i+1}^d a_{\sigma(k)} \pmod r.$$

In dimension 2, all cyclic quotient singularities are nearly Gorenstein (Caminata et al., 2020). Table 1 in (Caminata et al., 2020) precisely records Gorenstein, nearly Gorenstein, and non-nearly Gorenstein cases for small $n,d$.

The canonical module and its trace satisfy

$\omega_R \cong (fR)^G, \qquad \tr(\omega_R) = (fR)^G\cdot(R:_{Q(R)} (fR))^G,$

where $f = x_1\cdots x_d$ is $G$-canonical, and $R:_{Q(R)} (fR)$ is generated by monomials $M$ with $Mf \in R$.

6. Illustrative Examples and Structural Insights

Several explicit structures and examples elucidate the nearly Gorenstein property:

• Du Val (A–D–E) Singularities: $K_X\equiv0$, $F=0$, $\Tr_A(K_A)=A$.
• Almost Reduced Non-Gorenstein: The $D_n^{\mathrm{nG}}$ case with increased central coefficient shows $K_X+Z_f$ remains anti-nef, but $K_X\not\equiv0$.
• Quotient Example: Branch lengths $(1,1,s-2)$ and divisor $D=\tfrac12P_1+\tfrac12P_2-\tfrac{2s-1}{3s-2}P_3$ yield $F=Z_f$.
• Non-Nearly Gorenstein: The rational quotient graph for $(2,3,5)$ with two $(-3)$-curves meeting a $(-2)$-curve fails the anti-nefness condition, hence $\Tr_A(K_A)\subsetneq\mathfrak m$.
• Higher Dimensional Cyclic Quotients: $R=\frac1n(1,m,n-1)$, $n\ge3$, $\gcd(m,n)=1$ satisfies $\res(R)=m$, so nearly Gorenstein iff $m=1$ (Caminata et al., 2020).

In all cases, the criteria permit a direct, combinatorial or monomial-verification approach to determining nearly Gorensteinness.

7. Broader Context and Connections

Nearly Gorenstein rational singularities generalize the class of Gorenstein singularities while preserving essential geometric features, notably rationality, Cohen–Macaulayness, and $\mathbb{Q}$-Gorenstein-ness. Nearly Gorensteinness ensures that the canonical trace ideal defines the same closed point as the maximal ideal, and these rings are always Gorenstein on the punctured spectrum. This underscores their role as a bridge between strictly Gorenstein cases and more generic rational singularities. Invariant-theoretic implications arise in quotient singularity settings, relating the group structure directly to canonical and trace-theoretic properties (Maeda et al., 25 Dec 2025, Caminata et al., 2020).