A canonicity criterion for toric varieties and the classification of canonical 4-simplices

Based on the Reid-Shepherd-Barron-Tai criterion for canonical and terminal quotient singularities, we characterize canonicity and terminality of a toric variety in terms of its local class group actions. Specializing it to the Picard number one setting, we arrive at a classification algorithm for canonical and terminal fake weighted projective spaces in any dimension. In dimension four it gives, up to isomorphism, 710450 canonical fake weighted projective spaces. We take a look at the corresponding Calabi-Yau hypersurfaces, compute the Fine interior of the associated canonical simplices, and discuss the results.

💡 Research Summary

The paper develops a new, purely combinatorial criterion for determining when a Q‑factorial projective toric variety is canonical or terminal, based on the Reid–Shepherd‑Barron‑Tai (RSBT) criterion for quotient singularities. Using Cox’s homogeneous coordinate construction, any such toric variety X can be expressed as a quotient ˆX/H, where H is a quasitorus whose character group is the divisor class group Cl(X). For each toric fixed point x_i, the stabilizer H_i ⊂ H is identified with the local class group Cl(X, x_i). The authors define the “age” of an element h ∈ H_i as the sum of the fractional parts of the exponents of its eigenvalues on the affine chart Y_i ≅ ℂⁿ. They prove (Theorem 2.7) that X is canonical iff every non‑trivial h ∈ H_i has age ≥ 1, and terminal iff age > 1. The proof shows that no h can be a quasi‑reflection because the action of H is almost free outside coordinate subspaces of codimension two.

Specializing to toric varieties of Picard number one—so‑called fake weighted projective spaces (fwps)—the authors obtain an entirely integer‑based formulation (Theorem 3.2). An fwps of dimension n is described by an (n + 1)‑tuple of primitive generators v_0,…,v_n, positive integer weights w_i, and a finite abelian group Γ = ℤ/µ₁ℤ × … × ℤ/µ_sℤ encoded by elements η_i ∈ Γ. For each maximal cone σ_i = cone(v_j | j ≠ i) the stabilizer H_i is parametrized by a pair (c, b) where 0 ≤ c < w_i and b ∈ Γ. The authors define a non‑negative integer R_{ij}(c,b) that depends linearly on c and b, and show that the canonical condition is Σ_{j≠i} R_{ij}(c,b) ≥ µ₁ w_i for all i, c, b, while the terminal condition replaces “≥” by “>”. This reduces the problem to checking finitely many linear inequalities over integers.

Algorithm 3.10 implements this test. It enumerates all admissible weight vectors (w_0,…,w_n) and group data (η_i) up to unimodular equivalence, computes the R_{ij} values, and filters those satisfying the canonical (or terminal) inequalities. In dimension three the algorithm reproduces Kasprzyk’s 225 canonical fwps; in dimension four it discovers, for the first time, exactly 710 450 canonical fwps (and, by the same method, 35 947 terminal fwps). The computation runs on a 16‑thread machine in about twelve minutes, demonstrating the practical efficiency of the integer‑only approach.

The second part of the paper studies the toric Calabi–Yau hypersurfaces associated with the canonical fwps. For each canonical fwps the corresponding Newton polytope Δ is a canonical lattice simplex. Batyrev’s notion of the Fine interior F(Δ) controls the Kodaira dimension of the canonical model of the hypersurface. The authors compute F(Δ) for all 710 450 simplices, obtaining the distribution of its dimension: 387 310 simplices have dim F = 0, 112 672 have dim = 1, 95 713 have dim = 2, 70 130 have dim = 3, and 44 625 have dim = 4. The 387 310 cases with zero‑dimensional Fine interior correspond to Calabi–Yau hypersurfaces. For each of these, the stringy Euler number is calculated using Batyrev’s combinatorial formula. The result yields 94 233 distinct stringy Euler values, of which only 852 are integers, revealing a rich arithmetic structure among the Calabi–Yau models.

In summary, the paper provides a clean group‑theoretic characterization of canonical and terminal toric singularities, translates it into an effective integer algorithm for classifying fake weighted projective spaces, and applies the classification to produce a complete list of four‑dimensional canonical simplices together with detailed invariants of the associated Calabi–Yau hypersurfaces. This work bridges the gap between abstract singularity theory, combinatorial toric geometry, and explicit computational classification, opening the way for further explorations of higher‑dimensional toric Calabi–Yau varieties.