On Cox Rings of Calabi-Yau hypersurfaces

Antonio Laface; Luca Ugaglia; Michela Artebani

arxiv: 2605.22184 · v1 · pith:NHBJ5EGBnew · submitted 2026-05-21 · 🧮 math.AG

On Cox Rings of Calabi-Yau hypersurfaces

Michela Artebani , Antonio Laface , Luca Ugaglia This is my paper

Pith reviewed 2026-05-22 02:56 UTC · model grok-4.3

classification 🧮 math.AG

keywords Cox ringsCalabi-Yau hypersurfacesMori dream spacesToric Fano varietiesBirational automorphism groupsMovable coneCone conjecturePrimitive pairs

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The pith

Smooth anticanonical Calabi-Yau hypersurfaces in toric Fano varieties admit explicit Cox ring presentations under certain combinatorial conditions on their polytopes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the Cox rings of smooth anticanonical Calabi-Yau hypersurfaces inside smooth toric Fano varieties. It applies the combinatorics of primitive pairs from the ambient Fano polytope together with a localization description of Cox rings for embedded varieties. This identifies several configurations where the hypersurface is a Mori dream space and supplies explicit generators and relations for its Cox ring. The work also isolates combinatorial setups that make the birational automorphism group infinite, producing in dimensions three and four a split between finite generation of the Cox ring and infinitude of birational maps. In a class of examples that fail to be Mori dream spaces, the Morrison-Kawamata conjecture is established for the movable cone.

Core claim

Using the combinatorics of primitive pairs of the ambient Fano polytope and the localization description of Cox rings of embedded varieties, several configurations are identified in which the anticanonical Calabi-Yau hypersurface is a Mori dream space and an explicit presentation of its Cox ring is obtained. Combinatorial configurations are exhibited that force the birational automorphism group to be infinite, yielding in dimensions three and four a dichotomy between finite generation of the Cox ring and an infinite birational automorphism group. For a class of non-Mori dream examples, the Morrison-Kawamata cone conjecture is proved for the movable cone.

What carries the argument

The combinatorics of primitive pairs of the ambient Fano polytope together with the localization description of Cox rings for embedded varieties.

If this is right

In the identified configurations the birational geometry of the hypersurface can be read off directly from the explicit Cox ring.
In dimensions three and four the hypersurface is either a Mori dream space or has infinitely many birational automorphisms.
The movable cone of certain non-Mori dream Calabi-Yau hypersurfaces is polyhedral and generated by finitely many classes.
The same combinatorial test distinguishes cases where the Cox ring is finitely generated from cases where it is not.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same primitive-pair test might classify which other embedded Calabi-Yau varieties have finitely generated Cox rings.
Explicit Cox ring presentations could be used to decide effectiveness of divisors or to compute the automorphism group in these geometries.
The observed dichotomy between finite generation and infinite birational automorphisms may extend to higher-dimensional Calabi-Yau varieties.

Load-bearing premise

Smoothness of the toric Fano ambient variety and of the anticanonical hypersurface is enough for the localization description of the Cox ring to apply and determine finite generation or the structure of the movable cone.

What would settle it

A concrete Fano polytope with a primitive pair whose associated smooth Calabi-Yau hypersurface has a non-finitely-generated Cox ring, or whose movable cone violates the Morrison-Kawamata conjecture.

read the original abstract

We study the Cox rings of smooth anticanonical Calabi-Yau hypersurfaces in smooth toric Fano varieties. Using the combinatorics of primitive pairs of the ambient Fano polytope and the description of Cox rings of embedded varieties via localizations, we identify several configurations for which the hypersurface is a Mori dream space and obtain explicit presentations of its Cox ring. We also exhibit combinatorial configurations forcing the birational automorphism group to be infinite, yielding in dimensions three and four a dichotomy between finite generation of the Cox ring and infinite birational automorphism group. Finally, for a class of non-Mori dream examples, we prove the Morrison-Kawamata cone conjecture for the movable cone.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives explicit Cox ring presentations for some Calabi-Yau hypersurfaces via polytope combinatorics and proves the movable cone conjecture for a subclass of non-MDS examples, plus a low-dimensional dichotomy on finite generation versus infinite birational automorphisms.

read the letter

The main thing to know is that the authors use the combinatorics of primitive pairs in the Fano polytope to find configurations where smooth anticanonical Calabi-Yau hypersurfaces are Mori dream spaces and to write down explicit generators and relations for their Cox rings. They also locate combinatorial setups that force the birational automorphism group to be infinite, which produces a clean split in dimensions three and four, and they prove the Morrison-Kawamata movable cone conjecture for a designated class of the non-MDS examples. The work turns the localization description of Cox rings for embedded varieties into concrete, checkable statements under the stated smoothness hypotheses on the toric ambient space and the hypersurface. This supplies verifiable instances rather than pure existence results, which is the useful part for classification work or for testing mirror symmetry ideas. The arguments stay within the combinatorial conditions that decide finite generation or infinitude of automorphisms, and the restriction of the cone conjecture proof to a specific subclass avoids overreaching. The soft spots are limited. The central claims rest on applying the localization description correctly to these hypersurfaces, which looks reasonable but would need explicit checks in the examples during review. There is no visible circularity or unverified load-bearing step in the high-level structure, and the paper does not claim to handle the general case. This paper is for algebraic geometers working on toric varieties, Calabi-Yau hypersurfaces, and birational geometry of threefolds and fourfolds. Readers who want explicit computations or new instances of the cone conjecture will get direct value from it. It shows clear combinatorial thinking and honest use of prior results, so it deserves a serious referee. I recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper studies Cox rings of smooth anticanonical Calabi-Yau hypersurfaces in smooth toric Fano varieties. Using the combinatorics of primitive pairs of the ambient Fano polytope and the localization description of Cox rings of embedded varieties, the authors identify configurations where the hypersurface is a Mori dream space and give explicit presentations of its Cox ring. They exhibit combinatorial configurations forcing the birational automorphism group to be infinite, yielding a dichotomy in dimensions three and four between finite generation of the Cox ring and infinite birational automorphism group. For a class of non-Mori dream examples they prove the Morrison-Kawamata cone conjecture for the movable cone.

Significance. If the results hold, the paper supplies combinatorial criteria and explicit examples that clarify when anticanonical Calabi-Yau hypersurfaces in toric Fano varieties have finitely generated Cox rings. The low-dimensional dichotomy and the proof of the cone conjecture for a designated subclass of non-MDS cases are concrete advances that can be used to test broader conjectures on birational geometry of Calabi-Yau threefolds and fourfolds. The explicit presentations constitute a strength for future computational or classification work.

minor comments (3)

[§2.3] §2.3: the statement that the localization description applies directly to the anticanonical hypersurface would benefit from a short reminder of the precise hypotheses on the primitive pairs that guarantee the embedding is regular.
[Table 1] Table 1: the column headings for the listed polytopes could include a brief indication of the dimension of the ambient toric variety to avoid ambiguity when the table is read in isolation.
[§4.2] §4.2, last paragraph: the sentence claiming the birational automorphism group is infinite would be clearer if it explicitly referenced the combinatorial condition (e.g., the existence of a certain pair of rays) that forces the infinitude.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, which recommends minor revision. We appreciate the recognition of the combinatorial criteria for Mori dream spaces, the explicit Cox ring presentations, the low-dimensional dichotomy, and the proof of the Morrison-Kawamata cone conjecture for the designated non-MDS subclass. Since the report contains no specific major comments requiring clarification or correction, we have no point-by-point revisions to address at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external localization and combinatorial tools to new hypersurface cases

full rationale

The paper's central results follow from applying the localization description of Cox rings for embedded varieties (invoked via primitive pairs in the Fano polytope) together with explicit combinatorial checks on configurations. These steps rest on independently established external theorems and toric combinatorics rather than reducing by construction to the paper's own fitted parameters, self-definitions, or unverified self-citations. The smoothness hypotheses and restriction of the cone-conjecture proof to a designated subclass of non-MDS examples are stated explicitly and do not create internal circularity. The work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard domain assumptions in toric algebraic geometry; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (1)

domain assumption Description of Cox rings of embedded varieties via localizations applies to smooth anticanonical hypersurfaces in smooth toric Fano varieties
Invoked to identify Mori dream space configurations from primitive pairs of the Fano polytope.

pith-pipeline@v0.9.0 · 5640 in / 1377 out tokens · 45119 ms · 2026-05-22T02:56:40.886529+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4. ... the Morrison–Kawamata cone conjecture holds for X.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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[1] [1]

Math.146(2010), no

Michela Artebani, J¨ urgen Hausen, and Antonio Laface,On Cox rings of K3 surfaces, Compos. Math.146(2010), no. 4, 964–998. MR2660680

work page 2010

[2] [2]

144, Cambridge University Press, Cambridge, 2015

Ivan Arzhantsev, Ulrich Derenthal, J¨ urgen Hausen, and Antonio Laface,Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR3307753

work page 2015

[3] [3]

V. V. Batyrev,On the classification of toric Fano4-folds, 1999, pp. 1021–1050. Algebraic geometry, 9. MR1703904

work page 1999

[4] [4]

Hacon, and James McKernan,Existence of minimal models for varieties of log general type, J

Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan,Existence of minimal models for varieties of log general type, J. Amer. Math. Soc.23(2010), no. 2, 405–468. MR2601039

work page 2010

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II, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 1991, pp

Ciprian Borcea,Homogeneous vector bundles and families of Calabi-Yau threefolds. II, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 1991, pp. 83–91. MR1128537

work page 1989

[6] [6]

Translated from the 1981 Ro- manian original by Vladimir Ma¸ sek and revised by the author

Lucian B˘ adescu,Algebraic surfaces, Universitext, Springer-Verlag, New York, 2001. Translated from the 1981 Ro- manian original by Vladimir Ma¸ sek and revised by the author. MR1805816

work page 2001

[7] [7]

Buchsbaum and David Eisenbud,Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, American Journal of Mathematics99(1977), no

David A. Buchsbaum and David Eisenbud,Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, American Journal of Mathematics99(1977), no. 3, 447–485

work page 1977

[8] [8]

Serge Cantat and Keiji Oguiso,Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups, Amer. J. Math.137(2015), no. 4, 1013–1044. MR3372313 37

work page 2015

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Z.243(2003), no

Cinzia Casagrande,Contractible classes in toric varieties, Math. Z.243(2003), no. 1, 99–126. MR1953051

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MR1786869

Pierre de la Harpe,Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR1786869

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Crist´ obal Herrera, Antonio Laface, and Luca Ugaglia,The Cox ring of an embedded variety, 2024

work page 2024

[12] [12]

Yi Hu and Sean Keel,Mori dream spaces and GIT, 2000, pp. 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR1786494

work page 2000

[13] [13]

Yujiro Kawamata,On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math.8(1997), no. 5, 665–687. MR1468356

work page 1997

[14] [14]

J´ anos Koll´ ar,Higher direct images of dualizing sheaves. I, Ann. of Math. (2)123(1986), no. 1, 11–42. MR825838

work page 1986

[15] [15]

Vladimir Lazi´ c, Keiji Oguiso, and Thomas Peternell,The Morrison-Kawamata cone conjecture and abundance on Ricci flat manifolds, Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds & Picard-Fuchs equa- tions, 2018, pp. 157–185. MR3822906

work page 2018

[16] [16]

Makiko Mase,Families ofK3surfaces in smooth Fano 3-folds, Comment. Math. Univ. St. Pauli61(2012), no. 2, 103–114. MR3052987

work page 2012

[17] [17]

James McKernan,Mori dream spaces, Jpn. J. Math.5(2010), no. 1, 127–151. MR2609325

work page 2010

[18] [18]

Morrison,Compactifications of moduli spaces inspired by mirror symmetry, 1993, pp

David R. Morrison,Compactifications of moduli spaces inspired by mirror symmetry, 1993, pp. 243–271. Journ´ ees de G´ eom´ etrie Alg´ ebrique d’Orsay (Orsay, 1992). MR1265317

work page 1993

[19] [19]

V. V. Nikulin,Finite groups of automorphisms of K¨ ahlerianK3surfaces, Trudy Moskov. Mat. Obshch.38(1979), 75–137. MR544937

work page 1979

[20] [20]

Keiji Oguiso,Smooth quartic K3 surfaces and Cremona transformations, ii, 2012

work page 2012

[21] [21]

Z.280(2015), no

John Christian Ottem,Birational geometry of hypersurfaces in products of projective spaces, Math. Z.280(2015), no. 1-2, 135–148. MR3343900

work page 2015

[22] [22]

The Stacks project authors,The stacks project, 2024

work page 2024

[23] [23]

Pure Appl

Long Wang,Remarks on nef and movable cones of hypersurfaces in Mori dream spaces, J. Pure Appl. Algebra226 (2022), no. 11, Paper No. 107101, 22. MR4412229

work page 2022

[24] [24]

Keiichi Watanabe and Masayuki Watanabe,The classification of Fano3-folds with torus embeddings, Tokyo J. Math. 5(1982), no. 1, 37–48. MR670903

work page 1982

[25] [25]

Pure Appl

Jos´ e Ignacio Y´ a˜ nez,Birational automorphism groups and the movable cone theorem for Calabi-Yau complete inter- sections of products of projective spaces, J. Pure Appl. Algebra226(2022), no. 10, Paper No. 107093, 22. MR4397131 38

work page 2022