Mori dream space¶
A normal projective variety whose nef and movable divisor classes are controlled by finitely many semiample cones on small Q-factorial models.
Core Idea¶
A Mori dream space is a normal projective variety with a finite description of its nef and movable divisor behavior. Hu and Keel require Q-factoriality and equality of the rational Picard and numerical divisor spaces; a nef cone generated by finitely many semiample line bundles; and finitely many small Q-factorial modifications whose pulled-back nef cones cover the movable cone. The name denotes that complete package, not merely a projective variety or a finitely generated section ring.[^ref-5a95a4135287]
For a Q-factorial projective variety with the Picard/numerical equality, Hu and Keel prove that finite Cox-ring generation is equivalent to this package. The hypotheses matter: finite generation of one section ring or even all individual section rings does not by itself prove membership.[^ref-5a95a4135287]
Use Within the Domain¶
In birational algebraic geometry, the class makes a potentially large collection of divisor classes and birational contractions accessible through finitely many semiample cones. Hu and Keel derive termination of divisor-directed Mori programs and a finite chamber description from the definition. Those are consequences, rather than replacement membership tests.[^ref-5a95a4135287]
The projective plane \(\mathbb P^2\) gives a smooth toric rank-one case: \(\mathcal O(1)\) generates its nef and movable ray, and the identity is the only model needed for coverage. The explicit one-ray mapping is an elementary deduction from its geometry and Hu and Keel's toric setting. Batyrev and Popov's \(X_4\), the degree-five del Pezzo blow-up of \(\mathbb P^2\) at four general-position points, gives a different case. Their finite Cox-ring generators, together with Hu and Keel's conditional equivalence, certify its Mori dream space status without an enumerated list of all its nef rays or small models.[ref-5a95a4135287][ref-e4acc41e2ff0]
Clarity¶
Keep finite semiample nef generation separate from finite movable coverage by small models. Both are required. A finite section ring can coexist with a nef non-semiample divisor, as in Hu and Keel's ruled elliptic surface example, so a ring calculation must be matched to the exact Cox-ring equivalence and its carrier hypotheses.[^ref-5a95a4135287]
Keep the surface distinct from its algebraic presentation. Batyrev and Popov identify \(\operatorname{Cox}(X_4)\) with a Grassmannian Plücker coordinate ring; they do not assert \(X_4=\operatorname{Gr}(3,5)\). Under Hu and Keel's smooth-projective Picard hypothesis, the ring's non-polynomial relations show that this case is non-toric. This is a bounded deduction from both papers.[ref-5a95a4135287][ref-e4acc41e2ff0]
Manages Complexity¶
The definition turns an open-ended divisor problem into a finite audit: verify the normal projective and divisor-class carrier, identify finitely many semiample nef generators, find finitely many qualifying small models, and check that their nef cones cover movable classes. A conditional Cox-ring proof can certify all those roles without listing every cone, but should be labeled as theorem-mediated evidence.[^ref-5a95a4135287]
The plane and the del Pezzo surface fill the same requirements through different presentations. \(\mathbb P^2\) has a polynomial Cox ring in the smooth toric setting; \(X_4\) has finite degree-one generators with Plücker relations. The latter's exact relations are an implementation detail of that case, not a defining feature of every Mori dream space.[ref-5a95a4135287][ref-e4acc41e2ff0]
Abstract Reasoning¶
Given a candidate \(X\), first check that it is a normal projective Q-factorial variety and that \(\operatorname{Pic}(X)_{\mathbb Q}=N^1(X)_{\mathbb Q}\). Next test finite semiample generation of its nef cone and a finite family of small Q-factorial modifications whose pulled-back nef cones cover \(\operatorname{Mov}(X)\). If using \(\operatorname{Cox}(X)\) instead, establish its finite generation only after confirming the hypotheses of Hu and Keel's Proposition 2.9.[^ref-5a95a4135287]
Reject a candidate if one required role fails. Separate the direct definition from consequences such as chamber decompositions and terminating divisor programs. The ruled-surface counterexample shows why finitely generated individual section rings cannot substitute for semiample nef data.[^ref-5a95a4135287]
Knowledge Transfer¶
The definition transfers literally between projective varieties when all divisor and small-model roles survive, as in the toric plane and non-toric \(X_4\). A proof may use direct cone geometry in one case and the conditional Cox-ring theorem in another. Neither case's specific equations or number of chambers are inherited by the other.[ref-5a95a4135287][ref-e4acc41e2ff0]
Outside algebraic geometry, a finite family of tractable regions may resemble this organization, but it is not a Mori dream space without projective varieties, Q-factorial divisor classes, semiample nef cones, small birational models and movable coverage. The strict live parent is Projective Variety; no broader prime edge is asserted.[ref-655de237bc6e][ref-5a95a4135287]
Example¶
For \(\mathbb P^2\), smooth projectivity supplies the normal projective carrier, rank-one divisor classes give the Picard/numerical condition, \(\mathcal O(1)\) supplies a semiample nef generator, and the identity model covers the movable ray. For the unlike \(X_4\) case, Batyrev and Popov's finite Cox ring and Hu and Keel's equivalence certify the same membership conditions through a different proof route. The original source passages do not enumerate \(X_4\)'s full nef fan or SQM list.[ref-5a95a4135287][ref-e4acc41e2ff0]
Relationships to Other Abstractions¶
Current abstraction Mori dream space Domain-specific
Parents (1) — more general patterns this builds on
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Mori dream space is a kind of Projective variety Domain-specific
Every Mori dream space is a projective variety with further finite nef and movable-cone conditions.
Hierarchy path (1) — routes to 1 parentless root
- Mori dream space → Projective variety → Algebraic Variety
Neighborhood in Abstraction Space¶
Mori dream space sits in a sparse region of the domain-specific corpus (64th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Sheaves & Birational Geometry (23 abstractions)
Nearest neighbors
- Projective variety — 0.85
- Mordellic Variety — 0.85
- Quasi-projective variety — 0.85
- Zero Divisor — 0.84
- Classifying space for SO(n) — 0.83
Computed from structural-signature embeddings · 2026-10-17
Not to Be Confused With¶
Projective Variety is the broader carrier and need not satisfy the finite divisor conditions. Toric Variety covers one family of examples but not non-toric \(X_4\). A Cox ring is an algebraic object used in a conditional equivalence, not the variety itself. A Mori dream region is Hu and Keel's separate local cone notion. An individual finite section ring is too weak a test.[ref-5a95a4135287][ref-e4acc41e2ff0]
References¶
[^ref-5a95a4135287]: Yi Hu and Sean Keel, “Mori Dream Spaces and GIT,” Michigan Mathematical Journal 48 (2000), 331–348, original author version https://arxiv.org/pdf/math/0004017. Definition 1.10; Proposition 1.11; Remark 1.12; Proposition 2.9; Corollaries 2.10 and 2.16. The log-Fano result in this version is limited to dimension at most three. [^ref-e4acc41e2ff0]: Victor V. Batyrev and Oleg N. Popov, “The Cox Ring of a Del Pezzo Surface,” original author version https://arxiv.org/pdf/math/0309111. Section 2, Theorem 3.2 and Proposition 4.1. The Cox-ring/Grassmannian identification does not identify the surface with the Grassmannian. [^ref-655de237bc6e]: The Stacks Project, “Types of Varieties,” Definition 33.26.1, https://stacks.math.columbia.edu/tag/04L0. Projective varieties admit closed immersion into finite-dimensional projective space over the field.