The source's raw rank-four replication yields m at most 4c - 4s - 1 after s root quadruples and expressly says the ratio is too weak to exhaust the source near m = 2c. Chern-root columns, Schubert penalties, and incidence estimates remain viable at excess four if the singular/nonclean codimension-three defect can be controlled at generic local points.
Route status · Narrowed routeProjective algebraic geometry · smooth subvarieties · complete intersections · Chern classes and Schubert geometry
Hartshorne Complete-Intersection Conjecture
Collaboration betaMust every smooth projective variety whose dimension is more than twice its codimension be cut out by exactly as many hypersurfaces as its codimension?
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Known results and sources 
Research problem
Exact mathematical statement
Let k be an algebraically closed field of characteristic zero and let
be smooth, connected, and nondegenerate. Hartshorne's complete-intersection conjecture asks whether X must be a complete intersection in projective space.
The full conjecture remains open. The source reports an internally dependency-checked excess-three theorem for c at least 5 and generator count mu(I_X)=c+3, modulo named standard inputs and still requiring independent expert review. Its next source-reported frontier is mu(I_X)=c+4, with the singular or nonclean codimension-three root-column defect unresolved.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Hartshorne Complete-Intersection Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
The source's weighted conormal bound forces every counterexample in Hartshorne's range to have at least c+3 generators, and its source-reported excess-three theorem raises that to c+4 when c is at least 5.
Evidence posture · Source-reported route statement · dependencies incompleteWe 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 We removed a duplicate or outdated task or route step.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.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Hartshorne Complete-Intersection Conjecture in numbers
2.3kretained lines of mathematical investigation2,307 in the current working snapshot
- Argument development
- 1,877 · 81%
- Explored or eliminated routes
- 83 · 4%
- Computational analysis
- 34 · 1%
- Open obligations
- 134 · 6%
- Definitions and setup
- 179 · 8%
8selected mapped statements2routes 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.
Recommended next task
Show that a singular or non-lci codimension-three defect either carries a usable rank-one excess quotient, forces a smaller-codimension zero locus, or yields a nonzero weight-zero four-minor.
Suggested move: Analyze the generic local algebra of a codimension-three root-column zero ideal generated by four elements in a regular local ring.
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
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
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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
The source's raw rank-four replication yields m at most 4c - 4s - 1 after s root quadruples and expressly says the ratio is too weak to exhaust the source near m = 2c. Chern-root columns, Schubert penalties, and incidence estimates remain viable at excess four if the singular/nonclean codimension-three defect can be controlled at generic local points.
Route status · Narrowed routeThe do-not-revisit list expressly closes the route that treats low height of a rank-jump locus as a contradiction. Use exact weighted minors, Chern factors, local zero ideals, and incidence bounds rather than rank-jump height alone.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Show that a singular or non-lci codimension-three defect either carries a usable rank-one excess quotient, forces a smaller-codimension zero locus, or yields a nonzero weight-zero four-minor.Suggested move: Analyze the generic local algebra of a codimension-three root-column zero ideal generated by four elements in a regular local ring.
Ready to work on
Use Rado/matroid alternatives and exact Schubert penalties to determine whether they close the finite clean-regime strips in the rank-four frontier.Suggested move: Combine local defect control with rank-four Schubert penalties and codimension-cost reduction across labeled root quadruples.
Ready to work on
Connect any finite presentation-frontier advance to the unrestricted complete-intersection conjecture before making a global claim.Suggested move: After excess four, reassess whether a general excess-rho induction is viable or whether the detector D_X provides the scalable route.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
The general small-codimension complete-intersection statement remains open. Erman, Sam, and Snowden prove a field-independent special case when ambient dimension is sufficiently large relative to codimension and degree, while current primary literature continues to use the general Hartshorne statement as a conjectural hypothesis. No reviewed source located here supplies a complete proof for every smooth X with dimension greater than twice codimension.
[1][2][3]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Peer reviewedBenoist and Voisin used the validity of Hartshorne's complete-intersection conjecture as a conjectural hypothesis in a neighboring obstruction to smoothing algebraic cycles, confirming that the general statement remains part of the current conjectural landscape.[3] Peer reviewedErman, Sam, and Snowden proved a field-independent theorem: for fixed codimension and degree, sufficiently high ambient dimension forces a projective scheme that is sufficiently nonsingular to be a complete intersection.[2] Historical sourceHartshorne formulated the small-codimension program for smooth projective subvarieties and established degree-dependent results that motivate the complete-intersection conjecture.[1]
Mathematical neighborhood
Related results and reusable starting points
Current focusHartshorne complete-intersection conjecture
For fixed codimension c and degree e, there is a bound N(c,e) such that an equidimensional projective subscheme whose singular locus has sufficiently large codimension is a complete intersection. This does not give the conjecture's degree-free m > 2c conclusion.
[2]For X of dimension m and codimension c in projective n-space, c < n/3 is algebraically equivalent to m > 2c because n = m + c.
[1][2]The validity of Hartshorne's conjecture would conflict with a proposed universal smoothability statement for rational algebraic-cycle classes, revealing a current topological and vector-bundle consequence without resolving the conjecture.
[3]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA proof-assistant development of projective schemes and varieties, smoothness, connectedness, nondegeneracy, codimension, homogeneous ideals, and complete intersections at the needed generality.
- Formalization targetFormal conormal bundles, Chern classes, Chow or cohomology rings, Grassmannians, Schubert varieties, and Lefschetz/Barth–Larsen restriction theorems with exact endpoint hypotheses.
- Formalization targetFormal projective dimension, unmixedness, generic reducedness, and descent of complete-intersection status under characteristic-zero spreading out and faithfully flat base change.
- Formalization targetA formal statement separating the degree-free full conjecture from high-degree, bounded-degree, fixed-codimension, and positive-characteristic neighboring results.
Research-record corrections
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 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.
Correction detailsCorrected the research recordCorrection note
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.
4 standing statements4 proposed statements3 open questions2 narrowed routes
- theorem candidate
1 of 8 1 - reduction
3 of 8 3 - lemma
2 of 8 2 - equivalence
1 of 8 1 - negative result
1 of 8 1
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.23 displayed rows · 2 routes included
- retained route statementDoes sufficiently small codimension force a complete intersection?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementSmall-codimension complete-intersection questionintermediate
- retained route statementGlobal complete-intersection detectorintermediate
- retained route statementWeighted conormal boundintermediate
- retained route statementSource-reported excess-three closureintermediate
- retained route statementClean root-defect exclusionintermediate
- 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 failureRaw replication of the rank-three root-column argument in rank fourreported failure
- Useful failureTreating low height of a rank-jump locus as an immediate contradictionreported failure
- Research targetShow that a singular or non-lci codimension-three defect either carries a usable rank-one excess quotient, forces a smaller-codimension zero locus, or yields a nonzero weight-zero four-minor.open
- Research targetUse Rado/matroid alternatives and exact Schubert penalties to determine whether they close the finite clean-regime strips in the rank-four frontier.open
- Research targetConnect any finite presentation-frontier advance to the unrestricted complete-intersection conjecture before making a global claim.open
- Research targetSingular root-defect local algebrasuperseded
- Narrowed routeRaw replication of the rank-three root-column argument in rank fourThe source's raw rank-four replication yields m at most 4c - 4s - 1 after s root quadruples and expressly says the ratio is too weak to exhaust the source near m = 2c. Chern-root columns, Schubert penalties, and incidence estimates remain viable at excess four if the singular/nonclean codimension-three defect can be controlled at generic local points.
- Narrowed routeTreating low height of a rank-jump locus as an immediate contradictionThe do-not-revisit list expressly closes the route that treats low height of a rank-jump locus as a contradiction. Use exact weighted minors, Chern factors, local zero ideals, and incidence bounds rather than rank-jump height alone.
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
2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.
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.
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Prepared starting pointShow that a singular or non-lci codimension-three defect either carries a usable rank-one excess quotient, forces a smaller-codimension zero locus, or yields a nonzero weight-zero four-minor.
Hartshorne Complete-Intersection Conjecture · ready to start
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Research contextPrepared context for any AI agent
Must every smooth projective variety whose dimension is more than twice its codimension be cut out by exactly as many hypersurfaces as its codimension?
- 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.
- 1Varieties of small codimension in projective spaceoriginal source · Robin Hartshorne · Bulletin of the American Mathematical Society · 1974 · DOI 10.1090/S0002-9904-1974-13612-8 · accessed Aug 14, 2026
- 2Strength and Hartshorne's Conjecture in high degreepeer reviewed result · Daniel Erman, Steven V. Sam, Andrew Snowden · Mathematische Zeitschrift / Springer · 2021 · ARXIV 1804.09730 · DOI 10.1007/s00209-020-02564-y · accessed Aug 14, 2026
- 3On the smoothability problem with rational coefficientspeer reviewed result · Olivier Benoist, Claire Voisin · Rendiconti Lincei. Matematica e Applicazioni / EMS Press · 2026 · DOI 10.4171/RLM/1076 · accessed Aug 14, 2026
Important qualifications
- The review is scoped to the smooth projective small-codimension complete-intersection statement and representative degree-bounded progress; it is not an exhaustive bibliography of Babylonian tower theorems, vector-bundle splitting, or complete intersections.
- The characteristic-zero formulation used by the intake candidate is narrower than some field-independent special-case results and must remain explicitly distinguished.
- No intake-packet theorem, packet status label, source-reported computation, embedded bibliography, or submitted URL was used as external-status authority.
- The 2026 Benoist–Voisin paper is neighboring evidence that the statement remains conjectural; it is not itself a status registry or a proof of the conjecture.
- No exact problem-level formalization was identified in the scoped review, but this does not establish nonexistence in every proof assistant or private project.
- This metadata has no proof, novelty, review, acceptance, credit, visibility, publication, or deployment authority.
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