Birational geometry · cubic fourfolds · Hodge theory · rationality

Irrationality of a Very General Cubic Fourfold

Collaboration beta

Is a very general smooth degree-three hypersurface in complex projective five-space nonrational?

X \subset P_{C}^{5} very general and smooth cubic \Rightarrow X ≁ P^{4}

Known results and sources

The question asks whether a very general smooth cubic fourfold can ever be birational to projective four-space.

Research problem

Exact mathematical statement

Let $X\subset\mathbf P^5_{\mathbf C}$ be a smooth cubic hypersurface. The target asks whether a very general such fourfold is irrational, meaning

X\not\dashsim\mathbf P^4_{\mathbf C}.

The governing source contains no unconditional proof. Current external metadata separately records a 2025 preprint claiming the result; because that claim has not been upgraded here by peer review or independent verification, this intake retains an open-with-unverified-claim posture.

Problem infographic

Problem at a glance

A problem-first moduli diagram defines a smooth cubic fourfold, explains very general, asks for absence of a birational map to projective four-space, and separately flags a recent preprint proof claim as under review. — Very general means outside a countable union of proper closed loci; the exact rationality question now has a recent preprint claim that this intake does not treat as accepted.

Current mathematical picture

Where work on Irrationality of a Very General Cubic Fourfold stands

Recent proof claim under review

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureEliminating the unique K2=4 endpoint by first-order component bounds alone

The source says the endpoint meets the first-order bound exactly and requires a stronger Petri, monodromy, or second-order obstruction. Reproducing the endpoint matrix, proving formal-plumbing domination, computing normalized local Petri defect, relative Jacobian transfer, higher-p_g control, and global carrier-family geometry remain live inside this work route.

Route status · Narrowed route

Main reductionConditional carrier picture

Under named relative-carrier and IVHS assumptions, the source reports exclusions for many p_g=2 carrier branches.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeReproduce the reported 198-by-201 endpoint matrix and wedge calculation with canonical input and complete certificates.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Irrationality of a Very General Cubic Fourfold in numbers

1.5kretained lines of mathematical investigation1,542 in the current working snapshot

Argument development: 1,326 · 86%
Explored or eliminated routes: 42 · 3%
Computational analysis: 55 · 4%
Open obligations: 31 · 2%
Definitions and setup: 88 · 6%

8selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Reproduce the reported 198-by-201 endpoint matrix and wedge calculation with canonical input and complete certificates.

Suggested move: Rebuild the triple-point and contact matching maps, then archive rank, kernel, pivot, and modular cross-check certificates.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

12 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeEliminating the unique K2=4 endpoint by first-order component bounds alone

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Reproduce the reported 198-by-201 endpoint matrix and wedge calculation with canonical input and complete certificates.Suggested move: Rebuild the triple-point and contact matching maps, then archive rank, kernel, pivot, and modular cross-check certificates.

Ready to work on

Compute and normalize the local relative-Petri cokernel length at the K2=4 endpoint.Suggested move: Construct the relative dualizing modules and multiplication map, correcting any ramified base-change length back to the original parameter.

Ready to work on

Complete the global carrier-family, family-compatibility, higher-p_g, and relative-Jacobian seams before drawing the current work's conclusion.Suggested move: Prove each named global interface separately and keep any external Hodge-atoms proof claim outside this work's dependency graph until independently assessed.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusRecent proof claim under review

A 2025 arXiv preprint by Katzarkov, Kontsevich, Pantev, and Yu claims to prove that a very general cubic fourfold is not rational, and a 2026 preprint builds on that work. This record found no peer-reviewed publication or independent formal verification supporting accepted solved status, so the claim remains explicitly unverified here.

[2][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
PreprintGuéré developed a Hodge-theoretic necessary condition for rational cubic fourfolds following the KKPY work.[3]
2025
PreprintKKPY introduced Hodge atoms and claimed as an application that a very general cubic fourfold is not rational.[2]
2017
Authoritative summaryHassett surveyed the long-standing rationality problem, special cubic loci, K3 structures, and known rational constructions.[1]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction2 of 82
lemma1 of 81
negative result3 of 83
computational claim1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included

retained route statementOutside countably many proper closed loci in moduli, a smooth cubic fourfold should admit no rational parametrization.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementPacket not a proofintermediate
retained route statementScoped v4 certificatesintermediate
retained route statementConditional carrier pictureintermediate
retained route statementEndpoint is not global completionintermediate
retained route statementConservative boundaryintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureEliminating the unique K2=4 endpoint by first-order component bounds alonereported failure
Research targetReproduce the reported 198-by-201 endpoint matrix and wedge calculation with canonical input and complete certificates.open
Research targetCompute and normalize the local relative-Petri cokernel length at the K2=4 endpoint.open
Research targetComplete the global carrier-family, family-compatibility, higher-p_g, and relative-Jacobian seams before drawing the current work's conclusion.open
Research targetPetri endpointsuperseded
Narrowed routeEliminating the unique K2=4 endpoint by first-order component bounds aloneThe source says the endpoint meets the first-order bound exactly and requires a stronger Petri, monodromy, or second-order obstruction. Reproducing the endpoint matrix, proving formal-plumbing domination, computing normalized local Petri defect, relative Jacobian transfer, higher-p_g control, and global carrier-family geometry remain live inside this work route.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeReproduce the reported 198-by-201 endpoint matrix and wedge calculation with canonical input and complete certificates.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointReproduce the reported 198-by-201 endpoint matrix and wedge calculation with canonical input and complete certificates.

Irrationality of a Very General Cubic Fourfold · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Is a very general smooth degree-three hypersurface in complex projective five-space nonrational?

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references3 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Cubic fourfolds, K3 surfaces, and rationality questionssurvey or monograph · Brendan Hassett · Springer · 2017 · ARXIV 1601.05501 · DOI 10.1007/978-3-319-46209-7_2 · accessed Aug 14, 2026
2
Birational Invariants from Hodge Structures and Quantum Multiplicationpreprint · Ludmil Katzarkov, Maxim Kontsevich, Tony Pantev, Tony Yue Yu · arXiv · 2025 · ARXIV 2508.05105 · accessed Aug 14, 2026
3
On the irrationality of cubic fourfoldspreprint · Jérémy Guéré · arXiv · 2026 · ARXIV 2603.04518 · accessed Aug 14, 2026

Important qualifications

External status was checked only against the listed primary, peer-reviewed, official-publisher, or author-hosted sources; omission is not a global nonexistence claim.
Special cases, reductions, preprints, and source-reported claims are kept at their exact scope and do not inherit proof, review, acceptance, publication, or priority authority.
The scoped formalization check found no problem-level statement or proof in current Mathlib documentation or the public Formal Conjectures repository; this is not evidence that no formalization exists elsewhere.
No source material, attachment, submitted URL, or packet computation was executed, rendered, fetched, or used as external authority.

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Irrationality of a Very General Cubic Fourfold

Exact mathematical statement

Problem at a glance

Where work on Irrationality of a Very General Cubic Fourfold stands

Irrationality of a Very General Cubic Fourfold in numbers

Reproduce the reported 198-by-201 endpoint matrix and wedge calculation with canonical input and complete certificates.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

Compare another research frontier