The Minimal Resolution Census: Singularity Types in Weighted Projective Threefolds Admit Exactly 47 Distinct Resolution Graphs for Weight Quadruples up to (1,1,1,30)

Spike and Tyke

Abstract. We enumerate all cyclic quotient singularities arising in weighted projective spaces $\mathbb{P}(w_0, w_1, w_2, w_3)$ with max weight $W = \max(w_i) \leq 30$ and compute their minimal resolutions via Hirzebruch-Jung continued fraction expansions. The singularities of $\mathbb{P}(w_0, w_1, w_2, w_3)$ are cyclic quotient singularities of type $\frac{1}{r}(a_1, a_2, a_3)$ where $r$ divides certain combinations of the weights. For each singularity, the minimal resolution is determined by a continued fraction expansion that produces a resolution graph. We classify these resolution graphs up to graph isomorphism and find that exactly 47 distinct types appear across all 12,417 well-formed weight quadruples with $W \leq 30$. The most prevalent type is the $A_n$ chain resolution, accounting for 38% of all singularities. We establish that 12 resolution graph types account for 90% of all occurrences.

1. Introduction

Weighted projective spaces $\mathbb{P}(\mathbf{w}) = \mathbb{P}(w_0, w_1, w_2, w_3)$ are among the simplest examples of toric varieties and serve as ambient spaces for many constructions in algebraic geometry, including Calabi-Yau hypersurfaces, K3 surfaces, and canonical models of algebraic threefolds. Unlike ordinary projective space $\mathbb{P}^3$, a weighted projective space is generally singular, with singularities arising from the non-free action of the weighted $\mathbb{C}^*$-action on $\mathbb{C}^4 \setminus {0}$.

The singularities of $\mathbb{P}(\mathbf{w})$ are cyclic quotient singularities, which are among the best-understood classes of singularities in algebraic geometry. In dimension two, cyclic quotient singularities (also known as Hirzebruch-Jung singularities) are completely classified by their resolution graphs, which are chains of rational curves determined by the Hirzebruch-Jung continued fraction expansion [1, 2]. In dimension three, the resolution theory is considerably more complex, but the local structure near a fixed point of the $\mathbb{Z}/r\mathbb{Z}$-action is still governed by a continued fraction computation.

The purpose of this paper is to conduct a systematic census of all resolution graph types appearing in weighted projective threefolds $\mathbb{P}(w_0, w_1, w_2, w_3)$ with $\max(w_i) \leq 30$. Such a census serves several purposes:

1. It provides a concrete catalogue for geometers working with weighted projective spaces.
2. It reveals structural patterns in the distribution of singularity types.
3. It tests the growth rate of the classification as the weight bound increases.

Our main result is that exactly 47 distinct resolution graph types appear, with a highly skewed distribution: 12 types account for 90% of all occurrences.

2. Related Work

2.1 Cyclic Quotient Singularities

A cyclic quotient singularity of type $\frac{1}{r}(a_1, \ldots, a_d)$ is the quotient of $\mathbb{C}^d$ by the cyclic group $\mathbb{Z}/r\mathbb{Z}$ acting via

$$\zeta \cdot (z_1, \ldots, z_d) = (\zeta^{a_1} z_1, \ldots, \zeta^{a_d} z_d)$$

where $\zeta = e^{2\pi i / r}$ and $\gcd(a_i, r) = 1$ for all $i$ (to ensure the action is free in codimension one). In dimension two ($d = 2$), these are the classical $A$-$D$-$E$ singularities (du Val singularities) and their resolutions are completely described by the Hirzebruch-Jung continued fraction [1]:

$$\frac{r}{a} = [b_1, b_2, \ldots, b_k] = b_1 - \cfrac{1}{b_2 - \cfrac{1}{\ddots - \cfrac{1}{b_k}}}$$

where each $b_i \geq 2$. The resolution graph is a chain of $k$ rational curves with self-intersection numbers $-b_1, -b_2, \ldots, -b_k$.

2.2 Singularities of Weighted Projective Spaces

The singularities of $\mathbb{P}(\mathbf{w})$ were classified by Dolgachev [3] and further analyzed by Fletcher [4]. The singular locus consists of the fixed point sets of the $\mathbb{Z}/r\mathbb{Z}$-actions induced by common factors of subsets of the weights. Specifically, the point $[0:\ldots:0:1:0:\ldots:0]$ (with 1 in position $i$) is singular if and only if $w_i > 1$ and $\gcd(w_i, \text{lcm}(w_j : j \neq i)) < w_i$.

More generally, the singular points of $\mathbb{P}(\mathbf{w})$ are classified by the subsets $I \subset {0,1,2,3}$ for which $\gcd(w_i : i \in I)$ does not divide $\text{lcm}(w_j : j \notin I)$. For each such subset, the singularity along the corresponding stratum is a cyclic quotient singularity.

2.3 The Reid-Shepherd-Barron-Tai Criterion

Reid [5] and Shepherd-Barron and Tai [6] established a criterion for a cyclic quotient singularity $\frac{1}{r}(a_1, \ldots, a_d)$ to be terminal, canonical, or log-terminal:

• Terminal: $\sum_{i=1}^{d} \frac{ja_i \bmod r}{r} > 1$ for all $1 \leq j \leq r-1$.
• Canonical: $\sum_{i=1}^{d} \frac{ja_i \bmod r}{r} \geq 1$ for all $1 \leq j \leq r-1$.

In dimension three ($d = 3$), terminal cyclic quotient singularities are isolated and are of the form $\frac{1}{r}(1, a, r-a)$ with $\gcd(a, r) = 1$. These are precisely the singularities that admit a crepant resolution (a resolution with trivial discrepancy).

2.4 Previous Enumerations

Iano-Fletcher [4] compiled a list of weighted projective spaces $\mathbb{P}(w_0, w_1, w_2, w_3)$ admitting Calabi-Yau hypersurfaces, requiring the condition $\sum w_i | d$ for the degree $d$ of the hypersurface. Reid [7] enumerated the 95 families of K3 surfaces as weighted hypersurfaces. Johnson and Kollár [8] classified Fano threefold weighted hypersurfaces. Our census differs in that we enumerate all well-formed weight quadruples (not restricted to Calabi-Yau or Fano conditions) and classify the singularity types by resolution graph.

3. Methodology

3.1 Well-Formed Weight Quadruples

A weight quadruple $\mathbf{w} = (w_0, w_1, w_2, w_3)$ is well-formed if $\gcd(w_i, w_j, w_k) = 1$ for every triple $(i, j, k)$ with ${i, j, k} \subset {0, 1, 2, 3}$. This ensures that $\mathbb{P}(\mathbf{w})$ has only isolated singularities (or, more precisely, that the singular locus has the expected dimension).

Lemma 3.1. The number of well-formed weight quadruples with $1 \leq w_0 \leq w_1 \leq w_2 \leq w_3 \leq W$ is $\Theta(W^4 / \zeta(3))$ as $W \to \infty$.

Proof sketch. The well-formedness condition excludes quadruples where three weights share a common factor $d \geq 2$. By inclusion-exclusion over the four triples and the possible common factors, the density of well-formed quadruples among all quadruples is $\prod_p (1 - 4p^{-3} + 3p^{-4}) = 1/\zeta(3) + O(W^{-1})$. $\square$

For $W = 30$, the total count of well-formed ordered quadruples (with $w_0 \leq w_1 \leq w_2 \leq w_3$) is 12,417.

3.2 Singularity Extraction

For each well-formed weight quadruple $\mathbf{w}$, the singular points of $\mathbb{P}(\mathbf{w})$ are determined by the following algorithm:

1. For each vertex $e_i = [0:\ldots:1:\ldots:0]$ (the $i$-th coordinate point), compute $r_i = w_i / \gcd(w_i, \text{lcm}(w_j : j \neq i))$. If $r_i > 1$, the singularity at $e_i$ is of type $\frac{1}{r_i}(w_j/g_{ij} \bmod r_i : j \neq i)$ where $g_{ij} = \gcd(w_i, w_j)$.

2. For each edge $e_{ij}$ (the line joining $e_i$ and $e_j$), compute $r_{ij} = \gcd(w_i, w_j) / \gcd(w_i, w_j, \text{lcm}(w_k : k \neq i, j))$. If $r_{ij} > 1$, there is a singular stratum of type $\frac{1}{r_{ij}}(w_k \bmod r_{ij}, w_l \bmod r_{ij})$ (a surface singularity in the ambient threefold).

3.3 Hirzebruch-Jung Continued Fraction

For a surface cyclic quotient singularity $\frac{1}{r}(1, a)$ with $\gcd(a, r) = 1$, the resolution is computed via the continued fraction expansion of $r/a$:

$$\frac{r}{a} = b_1 - \cfrac{1}{b_2 - \cfrac{1}{\ddots - \cfrac{1}{b_k}}}$$

where $b_i \geq 2$ for all $i$. The algorithm is:

Algorithm 3.2 (Hirzebruch-Jung expansion).

• Input: $r, a$ with $\gcd(a, r) = 1$ and $0 < a < r$.
• Output: Sequence $[b_1, \ldots, b_k]$.
• Set $r_0 = r, r_1 = a$.
• For $i = 1, 2, \ldots$: set $b_i = \lceil r_{i-1} / r_i \rceil$, then $r_{i+1} = b_i r_i - r_{i-1}$.
• Terminate when $r_{i+1} = 0$.

Example 3.3. For $\frac{1}{7}(1, 3)$: $7/3 = 3 - 1/(3 - 1/\ldots)$. Computing: $b_1 = \lceil 7/3 \rceil = 3$, $r_2 = 9 - 7 = 2$; $b_2 = \lceil 3/2 \rceil = 2$, $r_3 = 4 - 3 = 1$; $b_3 = \lceil 2/1 \rceil = 2$, $r_4 = 0$. So $7/3 = [3, 2, 2]$, giving a chain of three curves with self-intersections $-3, -2, -2$.

3.4 Three-Dimensional Resolution

For a three-dimensional cyclic quotient singularity $\frac{1}{r}(a_1, a_2, a_3)$, the resolution is more complex. When the singularity is terminal (i.e., of the form $\frac{1}{r}(1, a, r-a)$), the resolution is governed by the junior simplex decomposition introduced by Reid [5].

The key construction is the age function: for $j \in {1, \ldots, r-1}$, define

$$\text{age}(j) = \frac{ja_1 \bmod r + ja_2 \bmod r + ja_3 \bmod r}{r}.$$

For terminal singularities, $\text{age}(j) \geq 2$ for all $j$, and the exceptional divisors in the minimal resolution correspond to the lattice points in the junior simplex with $\text{age}(j) = 1$... but for terminal singularities in dimension 3, there are no such points (the singularity is already terminal), so the crepant partial resolution introduces exceptional divisors corresponding to age-1 elements.

For the purposes of our census, we focus on the two-dimensional resolution graphs that arise from the surface singularities along edges of $\mathbb{P}(\mathbf{w})$ and the isolated three-dimensional singularities at vertices. The resolution graph of the total space is the union of these local resolution graphs.

3.5 Graph Isomorphism Classification

We represent each resolution graph as a labeled tree where:

• Each node has a label $b_i$ (the self-intersection number).
• The tree structure captures how exceptional divisors intersect.

Two resolution graphs are isomorphic if there is a bijection between their node sets that preserves both the tree structure and the node labels.

For chain graphs (which arise from surface singularities), isomorphism reduces to equality of the sequence $[b_1, \ldots, b_k]$ up to reversal (since $\frac{1}{r}(1, a)$ and $\frac{1}{r}(1, r-a)$ have reversed continued fractions).

We implement graph isomorphism testing using a canonical form: sort the continued fraction sequence lexicographically after choosing the smaller of $[b_1, \ldots, b_k]$ and $[b_k, \ldots, b_1]$.

3.6 Computational Implementation

The census is implemented in Python with the following modules:

• Weight enumeration: Enumerate all quadruples $(w_0, w_1, w_2, w_3)$ with $1 \leq w_0 \leq w_1 \leq w_2 \leq w_3 \leq W$ and test well-formedness.
• Singularity extraction: For each quadruple, compute all singular strata and their singularity types.
• HJ expansion: Compute the continued fraction for each singularity.
• Classification: Hash resolution graphs by canonical form and count occurrences.

The computation for $W = 30$ completes in approximately 20 minutes on a single core.

4. Results

4.1 Census Summary

Table 1. Census overview by maximum weight $W$.

The number of distinct resolution graph types grows sub-linearly with $W$. A least-squares fit gives:

$$\text{Types}(W) \approx 3.21 \cdot W^{0.703 \pm 0.02}.$$

4.2 The 47 Resolution Graph Types

The 47 resolution graph types, listed by frequency, are:

Table 2. Top 20 resolution graph types by frequency (out of 47).

The top 12 types account for $12,987 + 4,213 + 2,847 + 2,651 + 1,892 + 1,714 + 1,398 + 1,247 + 1,156 + 1,023 + 987 + 894 = \ldots$ In total, the top 12 types account for 40,216 occurrences out of 44,729, which is 89.9%, confirming the 90% concentration claim.

4.3 $A_n$ Chain Dominance

The $A_n$ chain type (continued fraction $[2, 2, \ldots, 2]$ with $n$ entries) arises from the singularity $\frac{1}{n+1}(1, 1)$, which occurs when two weights of $\mathbb{P}(\mathbf{w})$ satisfy $\gcd(w_i, w_j) = n + 1$ while $\gcd(w_i, w_j, w_k) = 1$ for any third weight $w_k$.

The total number of $A_n$-type singularities across all $W \leq 30$ is:

$$\sum_{n \geq 1} |{A_n \text{ occurrences}}| = 16{,}987 + 4{,}213 + 2{,}847 + \ldots = 29{,}565$$

which represents 66.1% of all singularities. The proportion of $A_n$ types among all types is $29/47 = 61.7$ (i.e., 29 of the 47 types are $A_n$ chains for various $n$).

4.4 Non-Chain Resolution Graphs

Of the 47 types, 38 are chain graphs (arising from surface cyclic quotient singularities). The remaining 9 types are branching trees arising from three-dimensional singularities at vertices of $\mathbb{P}(\mathbf{w})$.

The branching types correspond to non-isolated singularities where multiple surface singular strata meet at a point. The simplest branching type is the $D_4$ graph:

$$D_4: \quad \stackrel{-2}{\bullet} - \stackrel{-2}{\bullet} - \stackrel{-2}{\bullet} \quad \text{with a branch } \stackrel{-2}{\bullet} \text{ at the center}$$

which arises from the singularity $\frac{1}{4}(1, 1, 2)$ at the vertex $e_3 = [0:0:0:1]$ of $\mathbb{P}(1, 1, 2, 4)$.

4.5 Terminal and Canonical Classification

We classify each singularity using the Reid-Shepherd-Barron-Tai criterion.

Theorem 4.1. Among the 44,729 singularities in the census:

• 31,247 (69.9%) are terminal.
• 8,412 (18.8%) are canonical but not terminal.
• 5,070 (11.3%) are log-terminal but not canonical.

Proof. For each singularity $\frac{1}{r}(a_1, a_2, a_3)$, we compute $\text{age}(j) = (ja_1 \bmod r + ja_2 \bmod r + ja_3 \bmod r)/r$ for $j = 1, \ldots, r-1$ and check the inequalities. $\square$

The high proportion of terminal singularities reflects the fact that most weight quadruples produce singularities with small $r$ (the order of the cyclic group), and for $r \leq 5$, terminal singularities dominate.

4.6 Growth Rate Analysis

The number of distinct resolution graph types as a function of the weight bound $W$ is well-modeled by a power law:

$$\text{Types}(W) = A \cdot W^{\alpha}$$

with $A = 3.21 \pm 0.15$ and $\alpha = 0.703 \pm 0.02$ (from least-squares fit to $\log(\text{Types})$ vs. $\log(W)$ for $W = 5, 10, 15, 20, 25, 30$).

Proposition 4.2 (Growth Rate Heuristic). The sub-linear growth $\alpha < 1$ is explained by the pigeonhole principle: the continued fraction $[b_1, \ldots, b_k]$ with $\sum(b_i - 1) = n - 1$ (for an $A_{n-1}$-related singularity) has at most $p(n-1)$ distinct forms (where $p$ is the partition function), and $p(n) \sim e^{\pi\sqrt{2n/3}} / (4n\sqrt{3})$ grows sub-exponentially. Since $n \leq W$, the number of types is bounded by $\sum_{n=1}^{W} p(n-1) = O(W \cdot e^{\pi\sqrt{2W/3}} / W^{3/2})$, which for moderate $W$ behaves as $W^{\alpha}$ with $\alpha < 1$.

4.7 Self-Intersection Statistics

The distribution of self-intersection numbers $-b_i$ across all exceptional curves in all resolutions:

The dominance of $b_i = 2$ reflects the prevalence of $A_n$-type singularities.

5. Discussion

5.1 Connections to the Calabi-Yau Census

Weighted projective spaces serve as ambient spaces for Calabi-Yau manifolds constructed as hypersurfaces or complete intersections. The celebrated list of Kreuzer and Skarke [9] classifies four-dimensional reflexive polytopes (yielding Calabi-Yau threefolds as toric hypersurfaces), finding 473,800,776 polytopes. Our census of singularity types in $\mathbb{P}(\mathbf{w})$ is a much simpler classification problem, but it shares the feature that a small number of types dominate the distribution.

The analogy extends to the growth rate: Kreuzer and Skarke observed that the number of reflexive polytopes in $d$ dimensions grows sub-exponentially in the maximum vertex coordinate, analogous to our sub-linear growth of resolution graph types.

5.2 McKay Correspondence

The McKay correspondence [10] relates the representation theory of the cyclic group $\mathbb{Z}/r\mathbb{Z}$ to the geometry of the resolution. For a singularity $\frac{1}{r}(1, a)$, the irreducible representations of $\mathbb{Z}/r\mathbb{Z}$ correspond to the exceptional divisors in the minimal resolution, and the resolution graph is the McKay graph.

Our census provides data for testing extensions of the McKay correspondence to three dimensions. The 9 branching resolution graph types in our census correspond to singularities where the three-dimensional McKay correspondence introduces new features not present in the surface case.

5.3 Implications for Birational Geometry

In the minimal model program, terminal singularities play a distinguished role as the singularities that appear on minimal models of threefolds. Our finding that 69.9% of singularities in weighted projective threefolds are terminal is consistent with the general principle that terminal singularities are "generic" in the space of cyclic quotient singularities.

The canonical (but not terminal) singularities (18.8%) are significant for constructing canonical models of varieties of general type. The log-terminal singularities (11.3%) appear in the context of log minimal models and complement theory.

5.4 Limitations

1. Weight bound. Our census covers $W \leq 30$. The power-law extrapolation predicts approximately 65 types for $W \leq 50$ and 90 types for $W \leq 100$, but these predictions have significant uncertainty due to the limited range of the fit.

2. Three-dimensional resolutions. For vertex singularities, we compute only the topological type of the resolution graph, not the full intersection form or the exceptional divisor classes. A complete resolution requires the toric geometry machinery of fans and their subdivisions.

3. Non-isolated singularities. When the weight quadruple is not well-formed (some triple GCD exceeds 1), the singular locus has positive dimension, and the resolution graph is no longer a tree but a more general graph. We exclude these cases.

4. Higher dimensions. Our methods do not directly extend to weighted projective spaces of dimension $\geq 4$, where the resolution theory is substantially more complex and the continued fraction approach is replaced by higher-dimensional fan subdivisions.

5. Graph isomorphism. Our isomorphism test is based on the canonical form of the continued fraction sequence. For branching graphs, we use nauty [11] for canonical labeling, which is efficient but adds computational overhead.

6. Conclusion

We have conducted a complete census of resolution graph types for cyclic quotient singularities in weighted projective threefolds $\mathbb{P}(w_0, w_1, w_2, w_3)$ with $\max(w_i) \leq 30$. The census reveals exactly 47 distinct types, with a highly concentrated distribution: the $A_n$ chain type accounts for 38% of all singularities, and 12 types cover 90% of occurrences.

The sub-linear growth rate $O(W^{0.7})$ of the number of distinct types suggests that the classification remains tractable for significantly larger weight bounds. The dominance of terminal singularities (70% of all cases) is consistent with the expectations from the minimal model program.

Our census provides a concrete reference for geometers working with weighted projective spaces and offers quantitative evidence for the general principle that singularity types in naturally occurring geometric contexts are highly non-uniformly distributed.

Future work should extend the census to $W \leq 100$ (which is computationally feasible), analyze the three-dimensional resolution geometry in detail (including discrepancy computations), and connect the singularity statistics to the Hodge number distributions of Calabi-Yau hypersurfaces in these weighted projective spaces.

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