Mori Chamber Decompositions

• Mori chamber decompositions are finite subdivisions of the effective cone of a Mori dream space into convex, rational polyhedral chambers that correspond to distinct birational models.
• They connect variations in GIT quotients and Cox ring gradings to specific birational transformations, such as flips and divisorial contractions in the minimal model program.
• These decompositions facilitate explicit computations in contexts like toric surfaces, blow-ups, and Bott–Samelson varieties by clarifying wall-crossings and moving curves.

A Mori chamber decomposition is the finite subdivision of the effective (or movable) cone of a Mori dream space into convex, rational polyhedral chambers, each corresponding to a distinct birational model arising in the minimal model program (MMP). Mori chambers reflect the variation of GIT quotient models—connecting the birational geometry of projective varieties, especially those with finitely generated Cox rings, to the combinatorics of fans and group actions.

1. Foundations: Mori Dream Spaces, Cones, and Cox Rings

A normal projective variety $X$ over an algebraically closed field of characteristic zero is a Mori dream space (MDS) if:

• $\operatorname{Pic}(X)$ is finitely generated,
• the nef cone $\operatorname{Nef}(X)$ is the convex hull of finitely many semi-ample divisors,
• there are finitely many small $\mathbb{Q}$-factorial modifications (SQMs) $f_i\!: X\dashrightarrow X_i$ such that $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$.

The effective cone $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$ is the closure of the cone generated by effective divisors; the movable cone $\operatorname{Mov}(X)$ consists of effective divisors whose stable base locus has codimension at least 2. A Cox ring $\operatorname{Cox}(X)$ is a finitely generated, multi-graded normal domain, whose spectrum admits a torus action grading induced by $\operatorname{Cl}(X)$. The combinatorics of this grading underlies the structure of Mori chambers, whose walls correspond to loci where the GIT (geometric invariant theory) semistable locus changes (Maslovarić, 2015, Arai et al., 2018, Okawa, 2011, Laface et al., 2018).

2. The Mori Chamber Decomposition: Structure and Governing Principles

The Mori chamber decomposition is a finite fan of convex, rational polyhedral subcones in the effective or movable cone. Each maximal chamber $\operatorname{Pic}(X)$0 is identified by:

$\operatorname{Pic}(X)$1

where $\operatorname{Pic}(X)$2 are birational contractions to $\operatorname{Pic}(X)$3-factorial models $\operatorname{Pic}(X)$4 with nef cone $\operatorname{Pic}(X)$5, and $\operatorname{Pic}(X)$6 denotes the cone spanned by $\operatorname{Pic}(X)$7-exceptional divisors. Chambers have disjoint interiors and their union is $\operatorname{Pic}(X)$8 (Okawa, 2011).

Within each chamber, the Zariski decomposition (writing $\operatorname{Pic}(X)$9 with $\operatorname{Nef}(X)$0 nef and $\operatorname{Nef}(X)$1 effective fixed part) varies linearly (Okawa, 2011). The relative interiors of Mori chambers correspond to birational models of $\operatorname{Nef}(X)$2; two effective divisors are Mori equivalent if they define the same birational contraction.

The chamber walls are facets where birational transformations (flips, divisorial contractions) in the MMP occur. Crossing a wall corresponds to a rational map, and the domain and exceptional locus of the map can be read off from the semistable loci in the GIT language (Maslovarić, 2015).

3. GIT, VGIT, and Combinatorial Realizations

Mori chamber decompositions are the birational manifestation of the variation of GIT quotients (VGIT) for suitable group actions. For $\operatorname{Nef}(X)$3 with reductive $\operatorname{Nef}(X)$4 action and $\operatorname{Nef}(X)$5 an almost factorial domain, the image under the descent map $\operatorname{Nef}(X)$6 of GIT chambers in weight space gives all the Mori chambers in $\operatorname{Nef}(X)$7, where $\operatorname{Nef}(X)$8 is the GIT quotient (Maslovarić, 2015).

For the Cox ring $\operatorname{Nef}(X)$9 of a MDS, graded by $\mathbb{Q}$0, the cone

$\mathbb{Q}$1

admits a decomposition into finitely many maximal ray-ideal cones, identified via GIT by orbits of the torus action. These maximal cones are the Mori chambers (Arai et al., 2018). Their boundaries correspond to hyperplanes defined by the vanishing of certain homogeneous sections; the dual description is a system of linear inequalities encoding the semistable loci. Methods for computing these fans in high-rank situations rely heavily on polyhedral and group-theoretic symmetries (Boehm et al., 2016).

4. Birational Contractions and Wall-Crossing

Each Mori chamber determines a unique birational model; crossing a wall results in a birational transformation (flip or divisorial contraction). For two neighboring chambers, the wall corresponds to a locus where the stable base locus of the associated linear systems changes. If $\mathbb{Q}$2 is the birational map, the domain and exceptional loci correspond to

$\mathbb{Q}$3

with $\mathbb{Q}$4 the GIT quotient map (Maslovarić, 2015). Flips are reflected combinatorially as adjacent cones sharing a facet in the fan structure.

For each chamber, the section ring $\mathbb{Q}$5 for a big divisor $\mathbb{Q}$6 in the interior is isomorphic to the $\mathbb{Q}$7-invariant subring in the corresponding GIT setting. The chamber structure precisely controls which birational modifications are realized via variation of these data (Maslovarić, 2015, Arai et al., 2018).

5. Applications: Cox Rings, Quiver Moduli, and Explicit Constructions

The Mori chamber decomposition has concrete manifestations in several contexts.

• Quiver Moduli: For a quiver $\mathbb{Q}$8 without oriented cycles and a stability parameter $\mathbb{Q}$9 with stability=semistability, the moduli space $f_i\!: X\dashrightarrow X_i$0 is a MDS and its Mori chamber decomposition is governed by a finite arrangement of "King walls" in $f_i\!: X\dashrightarrow X_i$1; each chamber corresponds to a familiar birational model (e.g., $f_i\!: X\dashrightarrow X_i$2, Hirzebruch surface) (Maslovarić, 2015).
• Cox Ring and Demazure Construction: Fixing a chamber $f_i\!: X\dashrightarrow X_i$3 in the graded cone, one obtains a multi-section (Demazure) ring $f_i\!: X\dashrightarrow X_i$4 associated to a birational model $f_i\!: X\dashrightarrow X_i$5 and explicit $f_i\!: X\dashrightarrow X_i$6-divisors $f_i\!: X\dashrightarrow X_i$7. The relation $f_i\!: X\dashrightarrow X_i$8 demonstrates the recovery of any Cox ring of an MDS via this chamber structure (Arai et al., 2018).
• Explicit Examples: In low dimensions (toric surfaces, blow-ups of projective space, Bott--Samelson varieties), all possible birational contractions, the structure of $f_i\!: X\dashrightarrow X_i$9, $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$0, $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$1, and the intersection-theoretic nature of their walls can be computed explicitly (Merz et al., 2017, Massarenti, 2018, Ballard et al., 2013, Brambilla et al., 2024). For Bott--Samelson varieties, for example, every movable divisor is nef, and the Mori chambers correspond to the divisorial contractions along the unique extremal rays (Merz et al., 2017).

6. Stable Base Locus and Comparison of Decompositions

The stable base locus decomposition (SBLD) refines the effective cone into regions where the stable base locus $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$2 is constant. In general, the Mori chamber decomposition (MCD) is a nontrivial refinement of SBLD: every Mori chamber is contained in a unique SBLD chamber, but SBLD chambers may unite several Mori chambers (Laface et al., 2018). The two coincide in certain settings (e.g., Picard rank two under explicit combinatorial criteria), but in general, there are explicit counterexamples. The chambers and their walls reflect both the birational geometry of $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$3 and the combinatorics of its Cox ring grading.

7. Duality, Moving Curves, and Weyl Chamber Decompositions

Recent progress includes the study of duality between cones of divisors and cones of moving curves, using the Mori chamber decomposition. For a MDS $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$4 of dimension $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$5 and $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$6,

$$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$7

forms a rational polyhedral cone with a chamber decomposition induced from the Mori fan. The dual cones $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$8 are generated by $$\operatorname{Mov}(X) = \bigcup_i f_i^*(\operatorname{Nef}(X_i))$$9-moving curve classes. When $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$0 is a MDS, $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$1, the cone generated by all classes of $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$2-moving curves, possibly on a SQM model (Brambilla et al., 2023).

For blowups of projective space at points, Weyl group actions on curve and divisor classes control the wall structure of $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$3 and its chamber decomposition: Weyl $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$4-planes define the walls via intersection numbers $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$5, and each chamber corresponds to a distinct configuration of contained Weyl $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$6-planes. In non-MDS cases, the infinite proliferation of Weyl $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$7-planes leads to infinite (Weyl) chamber decompositions, with conjectures relating these to nef chamber decompositions in borderline cases (Brambilla et al., 2024).

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References:

• (Okawa, 2011) "On images of Mori dream spaces"
• (Maslovarić, 2015) "Quotients of spectra of almost factorial domains and Mori dream spaces"
• (Arai et al., 2018) "Demazure construction for Z^n-graded Krull domains"
• (Laface et al., 2018) "On Mori chamber and stable base locus decompositions"
• (Brambilla et al., 2023) "Duality and polyhedrality of cones for Mori dream spaces"
• (Brambilla et al., 2024) "Birational geometry of blowups via Weyl chamber decompositions and actions on curves"
• (Merz et al., 2017) "On the Mori theory and Newton-Okounkov bodies of Bott-Samelson varieties"
• (Boehm et al., 2016) "Computing GIT-fans with symmetry and the Mori chamber decomposition of $\operatorname{Eff}(X)\subset N^1(X)_\mathbb{R}$8"
• (Massarenti, 2018) "On the birational geometry of spaces of complete forms I: collineations and quadrics"
• (Ballard et al., 2013) "The Mori Program and Non-Fano Toric Homological Mirror Symmetry"