Toric Varieties with Prescribed Singularities

• The paper introduces a systematic construction of toric varieties using combinatorial data and explicit blow-up techniques to precisely control singularities.
• Detailed analysis of versal deformations and discrepancy computations provides a concrete framework for understanding both local and global singular behaviors.
• Applications in birational geometry, mirror symmetry, and MMP underscore the practical significance of engineering singular patterns in toric varieties.

Toric varieties with prescribed singularities are algebraic varieties constructed from combinatorial data (fans, polytopes) such that their local and global singularity structures are explicitly controlled. These constructions underpin both the paper of singularities in birational geometry and the development of modular deformation spaces, providing testbeds for broader phenomena in singularity theory, mirror symmetry, and birational classification.

1. Formal Construction and Classification of Toric Singularities

An affine toric variety $X_\sigma = \operatorname{Spec}\,k[S]$ is determined by a strongly convex rational polyhedral cone $\sigma \subset N_\mathbb{R}$ and the semigroup $S = \sigma^\vee \cap M$. Singularities of $X_\sigma$ are encoded combinatorially: simplicial cones correspond to quotient singularities, and non-simplicial cones to more complicated behaviors.

Three-dimensional toric singularities and their birational models admit a detailed classification using fan combinatorics and discrepancy computations. Canonical and purely log terminal (plt) blow-ups are completely described by star-subdivisions of cones along primitive lattice vectors, subject to explicit combinatorial and discrepancy conditions. Each such divisorial extraction corresponds to a sequence:

• Start with $(X \ni P)$ (smooth, quotient, or non-$\mathbb{Q}$-factorial singularity).
• Specify the exceptional divisor $E$ (often a weighted projective space).
• Choose $a = (a_1,a_2,a_3) \in N$ such that $a(E,X) = \frac{a_1+a_2+a_3}{r} - 1$ (for index $r$) meets canonicity or plt criteria.
• Perform the star-subdivision to produce $Y$ with the desired singularity (Kudryavtsev, 2014).

Iterative application leads to scaffolds of higher-dimensional toric varieties with prescribed local and global singularity content, provided $\mathbb{Q}$-Gorenstein and toric conditions are maintained.

2. Versal Deformations and Local Deformation Theory

The versal deformation space of a toric singularity $X_\sigma$ is governed by the functor $\operatorname{Def}_X$, whose tangent space $T^1(X_\sigma)_{-R}$ admits explicit combinatorial descriptions. For a fixed primitive character $R \in M$, consider the polyhedron $P_R = \sigma \cap \{v : \langle R, v \rangle = 1\}$. Vertices and edges of $P_R$ encode first-order deformation parameters: one $s$-coordinate per vertex, one $t$-coordinate per edge (and higher $t(c,d)$ for suitable lattice forms $c$).

The key technical assumption is the generation in degree 1 of the submonoid $T_d$ per edge; this is ensured by a "shortness" property—edges must be $k$-short in the sense that the relative lattice length is minimal. Under this hypothesis, all obstructions to lifting deformations vanish, allowing construction of the homogeneous piece of the versal deformation in degree $-R$. The resulting family is described by explicit binomial relations (loop and local equations) and parameterizes all local deformations with prescribed tangent space $T^1_{-R}$ (Altmann et al., 2020).

Illustrative examples include:

• Cyclic quotient singularities with $P$ an interval, generating versal deformations over $A^1$.
• Non-Gorenstein threefold singularities with higher-dimensional base and multiple loop/local relations.

3. Global-to-Local Synthesis: Prescribing Singularities via Complexity

A projective toric variety $X(\Delta)$ built from a fan $\Delta$ can be globally characterized and synthesized by gluing affine toric charts $U_\sigma$. The singularity structure at each point is formal-toric if and only if a complexity invariant vanishes: $c_z(X/Z, B) = \dim X + \rho(X/Z) - \sum b_i,$ where $B = \sum b_i B_i$ is a toric boundary and $\rho$ the relative Picard number. When $c_z = 0$, the pair $(X, \lfloor B \rfloor) \to Z$ is formally isomorphic (after completion) to a toric morphism, providing a geometric criterion to recognize toric singularities purely from divisorial and Picard data (Moraga et al., 2021).

This enables algorithmic construction of toric varieties with prescribed singularities: for each maximal cone $\sigma$, adjust boundary multiplicities to match the indices of primitive ray generators, ensuring log Calabi–Yau conditions and zero complexity globally.

4. Engineering Singularities via Secant Constructions and Simplicial Complexes

Secant varieties of toric embeddings arising from simplicial complexes provide a mechanism for prescribing singular loci in families of toric varieties. The construction $e_\Delta: (\mathbb{C}^n \to \mathbb{C}^N)$ associated to a simplicial complex $\Delta$ yields a toric embedding whose 2-secant, via cumulant coordinate change, is isomorphic to an affine space times a toric variety associated to a lattice polytope $P$ built from the faces of $\Delta$.

The Gorenstein and $\mathbb{Q}$-Gorenstein properties, as well as a complete description of the singular locus, are controlled by lattice-theoretic properties of $P$. Explicit combinatorial recipes allow one to realize arbitrary patterns of singular strata, facilitating the construction of toric varieties—or secant varieties—matching any given combinatorial singularity profile (Khadam et al., 2019).

5. Rationality, k-Rational Singularities, and Vanishing Results

The notion of $k$-rational (higher rational) singularities for toric varieties is explicitly characterized. For an affine toric $X$, and a strong log resolution $f: \widetilde X \to X$ with reduced exceptional divisor $E$, the vanishing

$R^i f_* \Omega^p_{\widetilde X}(\log E) = 0$

holds for all $i > 0, 0 \leq p \leq k$ if and only if every non-simplicial cone in the fan has dimension $>k$. Thus, for simplicial toric varieties, $k$-rationality is controlled by the codimension of the singular locus: $X$ is $k$-rational if and only if $k < \operatorname{codim}_X \operatorname{Sing}(X)$. Non-simplicial toric varieties never satisfy $1$-rationality or above, but are always rational in the classical sense ($k=0$) (Shen et al., 2023).

Worked examples, such as the $A_r$ singularity, illustrate the computation of these vanishings, stressing the combinatorial transparency for toric models.

6. Applications and Examples

Toric singularities with prescribed properties, constructible by the above frameworks, are foundational for testing conjectures in birational geometry, explicit minimal model program (MMP) constructions, mirror symmetry dualities, and the paper of deformation spaces.

Tables such as the following summarize explicit classification and realization techniques:

Framework	Main Construction	Outcome/Control Parameterization
Versal deformation of affine toric cones	Polyhedron $P_R$	Tangent space $T^1_{-R}$, explicit base, family
Toric blow-ups (plt/canonical)	Star-subdivision	Exact exceptional divisor, discrepancy control
Secant toric varieties (simplicial complexes)	Secant–embedding	Gorenstein/$\mathbb{Q}$-Gorenstein, singular loci
Global toric construction via complexity	Boundary-adjusted fan	Formal toric isomorphisms at prescribed points

By modulating combinatorial data—choices of fans, polytopes, simplicial complexes, and divisor indices—researchers can realize any desired local or global pattern of toric singularities within algebraic varieties, with explicit deformation and vanishing-theoretic control (Altmann et al., 2020, Kudryavtsev, 2014, Moraga et al., 2021, Khadam et al., 2019, Shen et al., 2023).