Birational Toric Morphisms

• Birational toric morphisms are equivariant maps between toric varieties defined by explicit combinatorial fan refinements and star subdivisions.
• They factorize into elementary modifications—such as blow-ups, blow-downs, and wall-crossings—that drive the Minimal Model Program and related contractions.
• These morphisms impact mirror symmetry, quantum geometry, derived category correspondences, and enumerative invariants, offering concrete computational models.

A birational toric morphism is an equivariant birational map between toric varieties, stacks, or quantum analogs, arising from combinatorial data on their associated fans. These morphisms encode and organize the birational geometry of toric and related spaces through refinements, wall-crossings, and degeneration constructions, translating complex birational phenomena into explicit polyhedral or stack-theoretic moves. Their theory provides structural insights into the Minimal Model Program (MMP), mirror symmetry, quantum geometry, and moduli of birational models.

1. Foundations: Definition and Combinatorial Criteria

Let $X_{\Sigma}$, $X_{\Sigma'}$ be toric varieties defined by fans $\Sigma, \Sigma' \subset N_\mathbb{R}$ for the same lattice $N$. A toric morphism $\phi: X_{\Sigma} \to X_{\Sigma'}$ is a birational toric morphism if the following hold:

• The associated lattice map is an isomorphism ($\Phi = \mathrm{id}_N$), so $N = N'$ and the supports of the fans coincide: $|\Sigma| = |\Sigma'|$.
• There exists a common refinement $\widehat{\Sigma}$ of $\Sigma$ and $\Sigma'$: each is obtained from $\widehat{\Sigma}$ by gluing cones.
• Every birational toric map arises from refining fans through star subdivisions and their inverses. Explicitly, $X_{\Sigma} \dashrightarrow X_{\Sigma'}$ admits a factorization into a sequence of toric blow-ups and blow-downs at torus-invariant centers, corresponding to combinatorial refinement steps on fans (Nobili, 2012).

For toric stacks and orbifolds, the data extends to stacky fans $(N, \Sigma, \beta)$, where $\beta: \mathbb{Z}^{\Sigma(1)} \to N$ assigns lattice generators to rays. Two proper toric DM stacks are T-equivariantly birational if and only if the sublattice attached to each maximal cone coincides where their interiors intersect. Every T-equivariant birational map between toric orbifolds factors via a sequence of star subdivisions and root constructions, encoding all combinatorial and stack-theoretic moves required for birationality (Schmitt, 2023).

2. Factorization: Toric Minimal Model Program and Contraction Types

The birational geometry of toric varieties is fully governed by:

• The toric cone theorem: the Mori cone of curves is generated by classes of invariant curves corresponding to codimension-one cones (Nobili, 2012).
• Each 1-dimensional face represents an extremal contraction—fibering (Mori fiber space), divisorial (blow-down of an invariant divisor), or flip (small contraction followed by flip)—recalled in Reid’s classification.
• By successive contraction of $K_X$-negative extremal rays, the toric MMP produces a sequence of toric birational maps, always ending in a toric Mori fiber space—a fiber space with relatively ample anticanonical divisor and lower-dimensional base. This sequence is functorial and the steps are explicit in fan combinatorics.

Every birational toric morphism is thus factorizable into a sequence of elementary birational modifications: star subdivisions (toric blow-ups), blow-downs, or, in the stacky context, root constructions and their inverses. The factorization theorem ensures uniqueness up to reordering of commuting steps. For quantum toric stacks, the analog holds with weighted blow-ups (of arbitrary, possibly irrational, weights) and cobordism-induced maps forming elementary birational moves (Boivin, 14 Jan 2026).

3. Wall-Crossing, Chamber Structure, and GIT

The secondary (GKZ) fan in $\mathrm{Cl}(X)_\mathbb{R}$ encodes the birational geometry of a toric variety via its chamber structure:

• Maximal cones (chambers) correspond to combinatorially distinct, projective, simplicial fans; crossing a wall corresponds to a birational transformation (flip, flop, or divisorial contraction).
• Each wall-crossing involves a unique primitive collection of rays and a linear circuit relation; the chamber adjacency encodes a minimal birational modification.
• Any two birational toric models differ by wall-crossings—each associated to an extremal contraction—realizing all birational geometry inside the GKZ fan. Each pair of adjacent chambers gives rise to a wall-crossing map that, both in combinatorics and GIT, realizes the elementary birational transformation. The full birational class is organized as the set of all models obtained by moving between chambers (Ballard et al., 2024).

In GIT terms, toric birational models are constructed as GIT quotients for varying linearizations and linear equivalence classes, with the change of model under wall-crossing corresponding to an explicit categorical and geometric transformation (Acosta et al., 2016, Barban et al., 31 Mar 2025).

4. Birational Toric Morphisms in Other Settings: Stacks, Quantum Toric Geometry, and Degenerations

Stack-theoretic generalizations employ stacky fans:

• In the toric orbifold case, birational moves include both star subdivisions and root constructions (taking roots of boundary divisors), with birationality completely characterized by matching sublattices on overlaps.
• In quantum toric geometry, fans may involve irrational rays, and admissible morphisms factor through arbitrary weighted blow-ups and cobordism-induced transformations in higher-dimensional ambient spaces. Here, birational morphisms can interpolate between models with continuous deformations parameterized by real weights (Boivin, 14 Jan 2026).

Degeneration techniques connect birational toric morphisms to:

• Flat toric degenerations (arising from birational sequences and compatible monomial orders in representation theory of algebraic groups) producing birational maps from flag varieties or spherical varieties to toric degenerations. The resulting birational morphism is realized as the central fiber inclusion of a flat family, with associated valuation monoids, Newton–Okounkov bodies, and essential bases providing explicit combinatorial/representation-theoretic models of the degeneration (Fang et al., 2015).
• Mutations of Laurent polynomials, which, under admissible Minkowski decompositions, interpolate between toric varieties as flat projective families and realize birational transformations at the level of Newton polytopes and their associated fans (Ilten, 2012).

5. Categorical and Gromov–Witten Aspects of Birational Toric Morphisms

Birational toric morphisms have profound categorical and enumerative implications:

• The gluing of derived categories along birational models (as in King’s Conjecture) constructs the Cox category as the subcategory generated by pull-backs from all birational models lying in the GKZ fan. The structure of exceptional collections and tilting bundles, as well as the existence of full strong exceptional collections, is controlled by the combinatorics of birational toric maps between models (Ballard et al., 2024).
• Fourier–Mukai transforms associated with birational toric maps relate derived categories, and the full window-categorical structure is explicit in the GKZ decomposition (Ballard et al., 2024).
• On the Gromov–Witten side, toric wall-crossing induces explicit linear transformations between I-functions of different models, matching generating series for Gromov–Witten invariants under birational maps. The relationship is implemented through explicit correspondences in cohomological variables and matches across both varieties and complete intersections in toric models, encoding the deep enumerative correspondence underlying toric birationality (Acosta et al., 2016).

6. Examples and Explicit Constructions

Canonical examples illustrate the abstract principles:

• The blow-up of $\mathbb{P}^2$ at a torus-fixed point is obtained by star subdividing the fan associated to $\mathbb{P}^2$, introducing a new ray and corresponding divisor. The divisorial contraction corresponds to removing a ray whose link is a codimension-1 face (Nobili, 2012).
• Flips and flops, such as between Hirzebruch surfaces $\mathbb{F}_m$, arise from wall-crossings and small contractions flipping a collection of cones with common face (Nobili, 2012, Barban et al., 2021).
• Weighted blow-ups and non-equalized flips generalize Atiyah flips, with the type determined by the C*–action on the local toric model: equalized weights yield classical Atiyah flips; general weights yield nonequalized flips, locally modeled on toric birational maps with arbitrary positive integer weights (Barban et al., 2021).
• Mutations of Laurent polynomials effect birational transformations of Newton polytopes, realizing, for instance, the classical smoothing of $\mathbb{P}(1,1,2)$ to $\mathbb{P}^1 \times \mathbb{P}^1$ (Ilten, 2012).

7. Broader Implications, Moduli, and Computational Aspects

Birational toric morphisms underpin the global geometry and moduli of toric varieties:

• The complete birational class of projective toric varieties of fixed dimension and Picard rank is described by the chamber structure of effective cones and secondary fans. Explicit wall-crossing decompositions provide both conceptual and computational tools for categorizing all birational models.
• In the Mori Dream Space framework, every birational map between toric MDSs is realized by projective toric geometric realizations—projective cobordisms whose extremal GIT quotients correspond to the source and target—explicitly computable via convex-geometric manipulations (moment polytopes, normal fans). Algorithms for enumerating and crossing between birational models trace explicit sequences of wall-crossings and flips (Barban et al., 31 Mar 2025).
• For toric orbifolds and stacks, the classification up to torus-equivariant birational maps yields a finite procedure of stacky star subdivisions and root constructions; for quantum fans and stacks, birationality encompasses more general (possibly irrational) subdivisions and cobordisms, providing a larger moduli space of models (Schmitt, 2023, Boivin, 14 Jan 2026).
• In representation theory, birational toric morphisms appear in the context of toric degenerations of flag varieties and the universal structure of Newton–Okounkov bodies, tightly linking birational geometry, polyhedral combinatorics, and PBW–type filtrations (Fang et al., 2015).

Birational toric morphisms thus serve as the unifying combinatorial and categorical scaffold for understanding and manipulating the birational geometry of toric and closely related algebraic varieties and stacks, with deep consequences for the study of birational models, categorical equivalences, and enumerative invariants.