Algebraic geometry · birational geometry · singularities in characteristic p

Resolution of Singularities in Positive Characteristic

Collaboration beta

Over a perfect field k of characteristic p>0, must every integral finite-type k-variety X of dimension at least four admit a proper birational morphism π:Y→X with Y regular?

k perfect, char (k) = p > 0, X an integral finite-type k -variety, \dim X \geq 4 \Rightarrow \exists π : Y \to X proper birational with Y regular

Known results and sources

A special radicial local model exposes codimension-at-least-two structure, but it does not supply a resolution of the general positive-characteristic problem.

Research problem

Exact mathematical statement

Let $k$ be a perfect field of characteristic $p>0$ , and let $X$ be an integral finite-type $k$ -variety with $\dim X\ge 4$ . The basic question asks whether there is a proper birational morphism

\pi:Y\longrightarrow X

with $Y$ regular. Requiring the morphism to be an isomorphism over the regular locus, or asking for embedded, simple-normal-crossings, projective, or functorial resolution, gives stronger variants that must be tracked separately.

The dimension-at-least-four positive-characteristic problem remains open. Unverified manuscripts claim all-dimensional results, but no peer-reviewed acceptance or independent validation was verified in the current external review. The retained source does not solve the problem: most of its detailed mathematics begins only after the special radicial local presentation $z^p+g=0$ has independently been constructed.

Problem infographic

Problem at a glance

Revision-v9 scientific problem map for resolution of singularities in positive characteristic: over a perfect field k of characteristic p>0, an integral finite-type k-variety X of dimension at least four asks for a proper birational map from a regular Y; scoped local source reports include a monic q=p^e presentation, graph-center transforms, the separable/radicial split, filtered Frobenius layers, and finite Newton descent in one rank-one branch, while global filtered termination, original marked-ideal compatibility, and the same-dimensional nonradicial branch remain open. — Revision v9 narrows the frontier: the local presentation and filtered formulas are source-reported advances, while global termination, compatibility with the original marked ideal, and the same-dimensional nonradicial branch remain open. Full resolution is not established.

Current mathematical picture

Where work on Resolution of Singularities in Positive Characteristic stands

Recent proof claim under review

Revision v9 supersedes the v7 presentation priority: the polynomial/local-algebra monic q=p^e presentation, graph-center classification, exact coefficient transforms, coefficient flags, and generic separable/radicial decomposition are now source-reported local results, while the completed-local transfer remains an audit item. The source also reports finite-birational rather than always-graph behavior for multiplicity components, exact arbitrary-mark filtered Frobenius coefficient objects, and finite Newton descent for one two-variable monomial rank-one branch. The full theorem remains open. The first unverified frontier is canonical smooth-center selection and strict termination for the intrinsic filtered Frobenius class, followed by compatibility with the original marked ideal, the same-dimensional nonradicial branch, general rank-one and higher-rank geometry, imperfect residue fields, and blowup-only domination. All claims remain provisional narrative evidence rather than independently checked proofs.

Strongest supported footholdLocal Frobenius and blowup identities retained

The curated baseline retains the source-asserted center criterion, current-locus detector, exact point-blowup formula, and coordinate-blowup Frobenius coefficient transform.

Evidence posture · Reported result

Leading routeGeneral high-order rank-one branch

Extend the proved two-variable monomial Newton countdown to several components and higher-dimensional transverse geometry, while preserving the exact mixed-layer object and permissibility for the original marked ideal.

Route status · Active route

Useful failureTangent-only jump invariant

Eliminated by the characteristic-two pair with identical tangent form and different transformed clean orders.

Route status · Eliminated route

Main reductionClean-order-p branch narrowed conditionally to codimension at least two

The source reports a regular contact subspace, exact mixed coefficient locus, and explicit transforms, but closure still depends on a missing Frobenius-marked lower-dimensional resolution theory.

Evidence posture · Reported reduction

Completed special caseFiltered Frobenius membership criterion

At arbitrary marks, membership in the additive filtered Frobenius class is equivalent to a finite family of ordinary coefficient and quotient-by-Q-th-power conditions.

Evidence posture · Source-reported route statement

Priority open bridgeProve filtered Frobenius termination

Construct a canonical transform-stable resolution algorithm for the intrinsic additive filtered Frobenius classes, with smooth center selection, exact recomputation in every chart, and strict finite termination.

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

Resolution of Singularities in Positive Characteristic in numbers

2.4kretained lines of mathematical investigation2,062 in the current working snapshot

Argument development: 2,026 · 84%
Explored or eliminated routes: 51 · 2%
Computational analysis: 20 · 1%
Open obligations: 133 · 6%
Definitions and setup: 170 · 7%

27selected mapped statements16routes investigated6reported milestones7open questions7contribution-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.

Proposed byAlexander Grothendieck, Jean DieudonnéProposed1965Status checkedAug 6, 2026

Source recordedAug 7, 2026

Recommended next task

Prove filtered Frobenius termination

Suggested move: Define the additive filtered object invariantly, construct an upper-semicontinuous maximal locus with a canonical smooth center, and prove strict descent for the recomputed transforms against every mandatory rejection test.

What would count as progress

Select smooth centers canonically from the intrinsic additive filtered object.
Recompute every mixed layer exactly in every chart and preserve restriction-to-opens compatibility.
Prove a strict well-founded descent or a stated reduction to a smaller pair (d,N).
When the object comes from a marked ideal, prove the selected centers are permissible for that original ideal.

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

27 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.

Active routeGeneral high-order rank-one branch

Route status · Active route

Active routeHigh-order higher-rank invariant

Develop a coefficient algebra incorporating the exceptional residue and a well-founded lexicographic invariant for n>p and catalecticant rank at least two.

Route status · Active route

Active routeImperfect residue fields and generic centers

Rebuild the local Frobenius constructions with absolute p-basis residue data so that the induction can pass through generic centers.

Route status · Active route

Active routeReconnection to the original marked ideal

Use the now-available local presentation data to build the missing global elimination algebra, prove two-sided center compatibility with (J,b), and show presentation descent lowers the original marked-ideal order.

Route status · Active route

Active routeGlobal center patching and termination

Make local invariants intrinsic and upper semicontinuous, select canonical smooth permissible centers, patch them, and prove strict finite descent.

Route status · Active route

Active routeFiltered Frobenius termination

Promote the exact arbitrary-mark filtered coefficient object into an intrinsic algorithm with canonical smooth centers, exact all-chart recomputation, and strict finite descent.

Route status · Active route

Active routeSame-dimensional nonradicial normalization

Control the normalization in the maximal separable subextension as a finite generic-degree map, analyze its singular base, and retain a residual radicial presentation without attempted radicialization over the original base.

Route status · Active route

Explored alternatives

Other routes

9 recorded

Narrowed routeSuperseded v7 general-presentation priority

No longer active in its v7 form: v9 reports the local monic q=p^e presentation, graph-center transforms, coefficient flags, and generic branch decomposition. The remaining global elimination and original-marked-ideal interface is tracked separately.

Route status · Narrowed route

Narrowed routeClean order p on the pure radicial height-one branch

Narrowed to an exact local contact and coefficient transform on codimension at least two, conditional on a missing canonical resolution theory for the mixed Frobenius-marked coefficient object.

Route status · Narrowed route

Route held in reserveValuation diagnostics

Retain best-approximation and defect questions as secondary consistency and obstruction diagnostics, not as a global center-selection or termination algorithm.

Route status · Route held in reserve

Browse 6 more explored routes

Eliminated routeTangent-only jump invariant

Eliminated by the characteristic-two pair with identical tangent form and different transformed clean orders.

Route status · Eliminated route

Eliminated routePrincipal-Jacobian and differential shortcuts

Mere principality is refuted and exact-differential monomialization is circular; only the stronger principal-monomial endpoint remains in the current research map.

Route status · Eliminated route

Eliminated routeAutomatic pure-cover inference

Eliminated: intermediate Weierstrass coefficients can persist, so a Frobenius-power tangent form does not by itself yield z^(p^e)+g.

Route status · Eliminated route

Eliminated routeTransform only the old Hasse detector

Eliminated as an exact transformed replacement; the detector remains useful for the current locus, while the exact quotient class must be retained or recomputed.

Route status · Eliminated route

Eliminated routePerfect-field-only globalization

Eliminated for generic centers; absolute p-basis coefficient data is required outside the perfect-field local setting.

Route status · Eliminated route

Eliminated routeFinite normalization as embedded resolution

Eliminated without a separate domination or factorization theorem; the binomial normalization remains only a source-asserted basic-resolution endpoint.

Route status · Eliminated route

Route statements and reductions

Statements the next route can inspect and build on

Route statementSource-asserted high-order rank-one classification

For clean order n>p with catalecticant rank one, p does not divide n and, after a linear coordinate change, the degree-n initial form is H=L^sG^p with s≡n mod p and 1≤s<p.

Source-reported route statement

Route statementSource-asserted common-factor basic-resolution endpoint

If, after translation, a complete common Frobenius factor gives g=a^pUh^s with 1≤s<p, then a finite birational variable division reduces to w^p+Uh^s=0 and étale-locally to the toric normalization of w^p+h^s=0, which is regular because gcd(p,s)=1.

Source-reported route statement

Route statementOpen transform-stable rank-one Frobenius factor extraction

For a maximal high clean-order rank-one stratum H=L^sG^p, find permissible modifications so every chart yields clean-order descent, rank or invariant progress, or a common factor of all active Frobenius coefficients.

Source-reported route statement · dependencies incomplete

Route statementOpen imperfect-residue-field extension

Reformulate clean order, Frobenius–Taylor contact, catalecticants, the coefficient multi-pair, and transforms using absolute p-basis expansions with coefficient-field residue labels.

Source-reported route statement · dependencies incomplete

Route statementOpen reconnection to the original marked ideal

Build a transform-stable elimination algebra for every Weierstrass coefficient, prove presentation-selected centers permissible for (J,b), and prove terminal presentation descent lowers ord(J) below b.

Source-reported route statement · dependencies incomplete

Route statementOpen high-order higher-rank descent invariant

For n>p and catalecticant rank r≥2, construct a coefficient algebra incorporating n mod p and a well-founded lexicographic invariant that strictly descends under permissible centers.

Source-reported route statement · dependencies incomplete

Route statementOpen global center patching and termination theorem

Construct upper-semicontinuous maximal loci and canonical smooth permissible centers, patch the local constructions, and prove strict finite descent of a well-founded invariant.

Source-reported route statement · dependencies incomplete

Route statementLocal monic q=p^e presentation bridge

From an auxiliary order-q element with tangent form z^q, the source reports sharp Weierstrass coefficient orders; in the polynomial/local-algebra model, smooth permissible centers are graphs over coefficient-base centers and relevant blowup charts retain monic degree-q presentations with exact coefficient transforms.

Source-reported route statement

Route statementFiltered Frobenius membership criterion

At arbitrary marks, membership in the additive filtered Frobenius class is equivalent to a finite family of ordinary coefficient and quotient-by-Q-th-power conditions.

Source-reported route statement

Route statementFinite monomial rank-one Newton descent

The two-variable monomial rank-one branch admits a finite Newton-slope countdown that forces order drop, rank growth, factor-multiplicity decrease, or complete factor extraction.

Source-reported route statement

Route statementNonradicial branch cannot be erased

Positive generic separable degree cannot be converted to a radicial equation by permissible birational modifications over the unchanged base function field.

Source-reported route statement

Route statementMultiplicity components are finite birational, not always graphs

For an actual reduced monic hypersurface, an irreducible multiplicity-q component need not be a graph center, but every reduced component projects finitely and birationally to a lower-dimensional image and conductor blowup graphifies that finite birational extension after strict transform.

Source-reported route statement

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

7 featured tasks

Prove filtered Frobenius termination

Ready to work on

Patch canonical centers and prove global termination

Turn the local identities into intrinsic upper-semicontinuous invariants, canonical smooth centers, compatible patching, and strict finite descent, including the high-order higher-rank branch.

Suggested move: Separate the clean-order-p dimension parameter from the high-order lexicographic candidate and prove upper semicontinuity and all-chart descent before asserting termination.

Ready to work on

Reconnect presentation descent to the original marked ideal

Build a transform-stable elimination algebra for every Weierstrass coefficient, prove height-one tower compatibility, and show every presentation-selected center is permissible for (J,b) and ultimately lowers ord(J).

Suggested move: Package the coefficient flags, inseparability ideals, terminal class, and filtered Frobenius layers into one elimination algebra, then prove both directions of center compatibility with (J,b).

Ready to work on

Extend the rank-one Newton theorem beyond the monomial branch

Extend the finite Newton theorem from the two-variable monomialized branch G=x^m to several components and higher-dimensional transverse geometry, while preserving exact mixed-layer and original-marked-ideal compatibility.

Suggested move: Develop a multidimensional Newton-polyhedron analogue of the finite slope k_* and test component intersections and changes of the distinguished linear factor in characteristics two and three.

Ready to work on

Control the same-dimensional nonradicial branch

For an actual irreducible finite cover, construct and control the normalization of the base in the maximal separable subextension without treating its finite generic-degree map as a birational base modification.

Suggested move: Construct the separable normalization scheme-theoretically, analyze its same-dimensional singularities, and track a residual radicial presentation and a total-field model birational to the original cover.

Ready to work on

Extend the local theory to imperfect residue fields

Replace perfect-field expansions by absolute p-basis expansions and reprove clean order, Frobenius–Taylor contact, catalecticants, coefficient transforms, and regularity of pure residue-field extensions.

Suggested move: Choose a finite local p-basis where available and add coefficient-field residue labels to the Frobenius expansion before revisiting transforms.

Ready to work on

Construct the canonical clean-order-p coefficient algebra

Make the mixed quotient-class and ordinary coefficient object independent of choices, retain its exact quotient-class transform, and obtain canonical smooth centers and conditional dimension-(d−2) termination.

Suggested move: Define the Frobenius-marked graded object while keeping [c₀′] distinct from the controlled transform of the old Hasse detector.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 6, 2026

Current statusRecent proof claim under review

The remaining positive-characteristic resolution problem is open in dimension at least four, including over perfect fields. Ordinary proper birational resolution is known through dimension three for reduced separated quasi-excellent Noetherian schemes; over F-finite fields, a stronger projective log resolution by regular-center blowups is known in dimension at most three. Hironaka's 2017 manuscript and Yi Hu's 2022 preprint claim all-dimensional results, but no peer-reviewed acceptance or independent validation was verified, and current 2026 research continues to describe the general problem as open.

[8][9][15]

External progress

What the literature has established

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

2026
Peer reviewedCastillo, Duarte, Leyton-Álvarez, and Liendo proved that iterating Nash or normalized Nash blowups does not resolve all algebraic varieties in dimensions four and higher, ruling out that universal procedure without disproving existence of resolutions by other methods.[16]
2024
Peer reviewedHauser and Perlega published a systematic embedded-resolution proof for two-dimensional hypersurface singularities over fields of arbitrary characteristic.[12]
2022
PreprintYi Hu posted an arXiv preprint claiming projective birational resolution for integral affine or projective schemes of finite presentation over perfect fields in arbitrary characteristic. This collection did not verify peer-reviewed acceptance or independent validation.[14]
2019
Peer reviewedCossart and Piltant proved ordinary resolution for reduced separated quasi-excellent Noetherian schemes of dimension at most three in arbitrary characteristic, including algebraic and arithmetical threefolds.[8]

Mathematical neighborhood

Related results and reusable starting points

Current focusResolution of singularities in positive characteristic, dimension at least four

Stronger or generalized formQuasi-excellent resolution conjecture

The Grothendieck–Dieudonné conjecture asks for resolution of all reduced quasi-excellent Noetherian schemes, including positive and mixed characteristic, and is broader than the perfect-field variety statement used as this workspace's canonical scope.

[1][2]

Solved special caseCharacteristic-zero resolution

Resolution is known in every dimension in characteristic zero: Hironaka proved the variety case, and Temkin established the quasi-excellent Noetherian Q-scheme extension.

[3][4]

Solved special caseResolution in dimensions at most three

Ordinary resolution is known in arbitrary characteristic through dimension three; the unresolved positive-characteristic frontier begins in dimension four.

[5][8]

Solved special caseProjective log resolution over F-finite fields

For reduced quasi-projective schemes of dimension at most three over F-finite fields, one has a projective log resolution by successive blowups of regular centers, isomorphic over the simple-normal-crossings locus.

[6][7]

Weaker or relaxed formDe Jong alterations

Alterations provide a proper dominant generically finite map from a regular variety in all dimensions and characteristics, but they may change the function field and therefore are not birational resolutions.

[10]

Dependency or reductionLocal uniformization

Local uniformization resolves a chosen valuation locally and is a central input to valuative approaches that globalize to resolutions in low dimension.

[5][6]

Stronger or generalized formEmbedded resolution and principalization

Embedded resolution and principalization additionally control the transform of an embedding, ideal, or boundary and seek a simple-normal-crossings total transform; these must not be silently substituted for ordinary resolution.

[11][12]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal library support · partial resource linkedLean 4 / Mathlib algebraic-geometry prerequisites
Current Mathlib documentation provides schemes, normalization, proper and smooth morphisms, and regular local/regular rings. The scoped search did not verify an exact formal statement or checked proof of positive-characteristic resolution, nor a complete scheme-level blowup/log-resolution framework for this target.
[19][20]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA scheme-level definition of resolution tying a regular source to a proper birational morphism that is an isomorphism over the regular locus.
Formalization targetBlowups along coherent ideals and regular centers, exceptional-divisor transforms, and simple-normal-crossings/log-resolution conditions at the scheme level.
Formalization targetDimension, quasi-excellence, birationality, characteristic, perfect-field, and F-finite-field infrastructure sufficient to state the exact open and solved scopes without collapsing regularity into smoothness over imperfect fields.
Formalization targetFormal statements of the low-dimensional theorems and the all-dimensional alteration comparison needed to encode the frontier faithfully.

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

v9 closes the local presentation gap and resets the frontierRevision v9 supersedes the v7 presentation priority with source-reported local presentation, branch-separation, filtered-coefficient, and monomial Newton results, while canonical filtered termination and original-marked-ideal compatibility become the leading open interface.

Changed the research frontierLater mathematical revision

Research stage 7

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

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 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 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 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

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

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

Open resolution target and conditional local frontierThe source reports a conditional codimension-at-least-two clean-order-p reduction and a high-order rank-one classification inside a special radicial model, while the full dimension-at-least-four resolution problem and its general bridges remain open.

Mapped research milestoneInitial research sequence

Research stage 1

Detailed research inventory

Claims, milestones, and routes in the current map

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

19 standing statements8 proposed statements6 mathematical milestones7 open questions2 narrowed routes2 conditional results16 completed special cases

Statements by mathematical role27 selected mapped statements

theorem candidate11 of 2711
definition1 of 271
equivalence5 of 275
reduction6 of 276
lemma3 of 273
negative result1 of 271

Selected mathematical clusters6 mathematical clusters

Problem scope and trust boundaryThe unresolved basic and strong targets, separated from the special perfect-field radicial local model.5 displayed rows

retained route statementBasic positive-characteristic resolution target
retained route statementStronger embedded, SNC, local, and functorial targets
retained route statementRetained radicial height-one local scopespecial case
Recorded relationshipThe embedded, SNC, locality, and functoriality requirements strengthen rather than redefine the basic proper-birational regularization target.generalizes · reported by source
Recorded relationshipThe retained calculations concern a special local radicial height-one model inside the much broader resolution problem.specializes · reported by source

Frobenius, contact, and blowup identitiesSource-asserted current-locus, centerwise cleaning, point-blowup, coefficient-transform, monomial-Jacobian, and moment-support mathematics.8 displayed rows

retained route statementSource-asserted Frobenius–Taylor center criterionspecial case
retained route statementSource-asserted finite detector of the current clean-order-p locusspecial case
retained route statementSource-asserted one-step clean-order blowup formulaspecial case
retained route statementSource-asserted coordinate-blowup transform of Frobenius coefficientsspecial case
retained route statementSource-asserted monomial-Jacobian endpointspecial case
retained route statementSource-asserted minimum moment-code supportspecial case
retained route statementSource-asserted equality classification for minimum supportspecial case
DerivationThe current work specializes the centerwise Frobenius–Taylor equivalence to I=m_x, identifying membership in the current Hasse differential singular locus with clean order at least p.active reported

Clean order pCodimension-at-least-two contact, exact coefficient locus and transform, and the remaining conditional coefficient-algebra closure.8 displayed rows · 1 route included

retained route statementSource-asserted codimension-at-least-two contact at clean order pspecial case
retained route statementSource-asserted exact coefficient multi-pair locusspecial case
retained route statementSource-asserted clean-order-p contact transformspecial case
retained route statementConditional codimension-at-least-two reduction of the clean-order-p branchconditional
retained route statementOpen canonical Frobenius-marked coefficient algebraconditional
DerivationThe reported contact and transform calculations place the persistent locus on a regular codimension-at-least-two subspace. The conclusion remains conditional because the canonical resolution theory for its mixed quotient-class and ordinary coefficient object is open.proposed
Research targetConstruct the canonical clean-order-p coefficient algebraopen
Narrowed routeClean order p on the pure radicial height-one branchNarrowed to an exact local contact and coefficient transform on codimension at least two, conditional on a missing canonical resolution theory for the mixed Frobenius-marked coefficient object.

High clean orderRank-one classification and conditional common-factor endpoint, plus open rank-one extraction and higher-rank descent programs.8 displayed rows · 2 routes included

retained route statementSource-asserted high-order rank-one classificationspecial case
retained route statementSource-asserted common-factor basic-resolution endpointspecial case
retained route statementOpen transform-stable rank-one Frobenius factor extractionintermediate
retained route statementOpen high-order higher-rank descent invariantintermediate
DerivationThe source uses the rank-one leading form, weighted residue-coefficient transforms, and the known endpoint after complete factor extraction to define a proposed descent-or-extraction theorem. No universal extraction theorem is claimed.proposed
Research targetExtend the rank-one Newton theorem beyond the monomial branchopen
Active routeGeneral high-order rank-one branchExtend the proved two-variable monomial Newton countdown to several components and higher-dimensional transverse geometry, while preserving the exact mixed-layer object and permissibility for the original marked ideal.
Active routeHigh-order higher-rank invariantDevelop a coefficient algebra incorporating the exceptional residue and a well-founded lexicographic invariant for n>p and catalecticant rank at least two.

Scoped failures and corrected narrativesReusable counterexamples and circularity boundaries that narrow the viable presentation, coefficient, valuation, and normalization routes.26 displayed rows · 7 routes included

refuted in scopeTangent form alone controls clean-order jumpsspecial case
refuted in scopeA merely principal Jacobian forces multiplicity dropspecial case
refuted in scopeFrobenius-power tangent automatically yields a pure radicial cover
refuted in scopeThe Hasse detector transforms exactly as the quotient classspecial case
ChallengeIn characteristic two, xy^3 and xy^3+x^5 have the same clean order and tangent form but transform to clean orders five and three in the x-chart.counterexample · reported resolved
ChallengeThe characteristic-three coefficient G=2x^3y+y^2z^3 has principal Jacobian (x^3+yz^3) but clean order four, so its p-cover retains multiplicity three.counterexample · reported resolved
ChallengeThe source's z^p+x^(p+1)z+x^(2p+1) example has tangent z^p but retains a nonzero linear coefficient after translation and is generically separable.counterexample · reported resolved
ChallengeFor f=u^py blown up along (u,v), the recomputed coefficient is y of clean order one while the controlled transform of the old first derivative is e and remains singular on the exceptional divisor.counterexample · reported resolved
Useful failureControl transformed clean-order jumps using only the degree-n tangent form H_n.reported failure
Useful failureUse principality of J(g), without monomiality in regular parameters, to force multiplicity drop.reported failure
Useful failureReplace principalization or resolution by monomializing the coefficients of the exact differential dg.reported failure
Useful failureInfer a radicial equation z^(p^e)+g directly from a p^e-power tangent form.reported failure
Useful failureDiscard the Frobenius quotient class and transform only its finite positive-Hasse-derivative detector.reported failure
Useful failureTreat immediate valuation defect as the only remaining obstruction after best p-th-power approximation.reported failure
Useful failureUse valuation-local resolution as a complete global resolution procedure.reported failure
Useful failureAssume divisor monomialization of a selected coefficient in high dimension as an input to the local branch.reported failure
Useful failureGlobalize the perfect-coefficient-field formulas unchanged through generic points of positive-dimensional centers.reported failure
Useful failurePromote a field-theoretic splitting of a p^e-extension directly to a ring-level height-one induction.reported failure
Useful failureTreat finite normalization at a binomial endpoint as an embedded blowup resolution.reported failure
Route held in reserveValuation diagnosticsRetain best-approximation and defect questions as secondary consistency and obstruction diagnostics, not as a global center-selection or termination algorithm.
Eliminated routeTangent-only jump invariantEliminated by the characteristic-two pair with identical tangent form and different transformed clean orders.
Eliminated routePrincipal-Jacobian and differential shortcutsMere principality is refuted and exact-differential monomialization is circular; only the stronger principal-monomial endpoint remains in the current research map.
Eliminated routeAutomatic pure-cover inferenceEliminated: intermediate Weierstrass coefficients can persist, so a Frobenius-power tangent form does not by itself yield z^(p^e)+g.
Eliminated routeTransform only the old Hasse detectorEliminated as an exact transformed replacement; the detector remains useful for the current locus, while the exact quotient class must be retained or recomputed.
Eliminated routePerfect-field-only globalizationEliminated for generic centers; absolute p-basis coefficient data is required outside the perfect-field local setting.
Eliminated routeFinite normalization as embedded resolutionEliminated without a separate domination or factorization theorem; the binomial normalization remains only a source-asserted basic-resolution endpoint.

Current v9 open programThe current portfolio starts with filtered Frobenius termination and original-marked-ideal compatibility, then separately tracks nonradicial normalization, general rank-one and higher-rank descent, imperfect residue fields, and global or blowup-only packaging.25 displayed rows · 8 routes included

retained route statementOpen canonical Frobenius-marked coefficient algebraconditional
retained route statementOpen imperfect-residue-field extension
retained route statementOpen reconnection to the original marked ideal
retained route statementOpen high-order higher-rank descent invariantintermediate
retained route statementOpen global center patching and termination theorem
retained route statementLocal monic q=p^e presentation bridgeintermediate
retained route statementFiltered Frobenius membership criterionspecial case
retained route statementFinite monomial rank-one Newton descentspecial case
retained route statementNonradicial branch cannot be erasedintermediate
retained route statementMultiplicity components are finite birational, not always graphsspecial case
Research targetExtend the rank-one Newton theorem beyond the monomial branchopen
Research targetConstruct the canonical clean-order-p coefficient algebraopen
Research targetExtend the local theory to imperfect residue fieldsopen
Research targetReconnect presentation descent to the original marked idealopen
Research targetPatch canonical centers and prove global terminationopen
Research targetProve filtered Frobenius terminationopen
Research targetControl the same-dimensional nonradicial branchopen
Narrowed routeClean order p on the pure radicial height-one branchNarrowed to an exact local contact and coefficient transform on codimension at least two, conditional on a missing canonical resolution theory for the mixed Frobenius-marked coefficient object.
Active routeGeneral high-order rank-one branchExtend the proved two-variable monomial Newton countdown to several components and higher-dimensional transverse geometry, while preserving the exact mixed-layer object and permissibility for the original marked ideal.
Active routeHigh-order higher-rank invariantDevelop a coefficient algebra incorporating the exceptional residue and a well-founded lexicographic invariant for n>p and catalecticant rank at least two.
Active routeImperfect residue fields and generic centersRebuild the local Frobenius constructions with absolute p-basis residue data so that the induction can pass through generic centers.
Active routeReconnection to the original marked idealUse the now-available local presentation data to build the missing global elimination algebra, prove two-sided center compatibility with (J,b), and show presentation descent lowers the original marked-ideal order.
Active routeGlobal center patching and terminationMake local invariants intrinsic and upper semicontinuous, select canonical smooth permissible centers, patch them, and prove strict finite descent.
Active routeFiltered Frobenius terminationPromote the exact arbitrary-mark filtered coefficient object into an intrinsic algorithm with canonical smooth centers, exact all-chart recomputation, and strict finite descent.
Active routeSame-dimensional nonradicial normalizationControl the normalization in the maximal separable subextension as a finite generic-degree map, analyze its singular base, and retain a residual radicial presentation without attempted radicialization over the original base.

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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 bridgeConstruct a canonical transform-stable resolution algorithm for the intrinsic additive filtered Frobenius classes, with smooth center selection, exact recomputation in every chart, and strict finite termination.

2 approaches have 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.

Select smooth centers canonically from the intrinsic additive filtered object.
Recompute every mixed layer exactly in every chart and preserve restriction-to-opens compatibility.
Prove a strict well-founded descent or a stated reduction to a smaller pair (d,N).
When the object comes from a marked ideal, prove the selected centers are permissible for that original ideal.

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Prepared starting pointProve filtered Frobenius termination

Resolution of Singularities in Positive Characteristic · ready to start

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Over a perfect field k of characteristic p>0, must every integral finite-type k-variety X of dimension at least four admit a proper birational morphism π:Y→X with Y regular?

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Sources and references22 cited works · next context review by Nov 6, 2026

The mathematical context was checked on Aug 6, 2026. Status can be refreshed sooner after a material result or claim.

1
Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas, seconde partieoriginal source · Alexander Grothendieck · Publications Mathématiques de l'IHÉS · 1965 · DOI 10.1007/BF02684322 · accessed Aug 6, 2026
2
Relative vanishing theorems for Q-schemespeer reviewed result · Takumi Murayama · Algebraic Geometry · 2025 · DOI 10.14231/AG-2025-003 · accessed Aug 6, 2026
3
Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero, I–IIpeer reviewed result · Heisuke Hironaka · Annals of Mathematics · 1964 · DOI 10.2307/1970486 · accessed Aug 6, 2026
4
Desingularization of quasi-excellent schemes in characteristic zeropeer reviewed result · Michael Temkin · Advances in Mathematics · 2008-10-01 · ARXIV math/0703678 · DOI 10.1016/j.aim.2008.05.006 · accessed Aug 6, 2026
5
Local uniformization on algebraic surfaces over ground fields of characteristic p ≠ 0peer reviewed result · Shreeram S. Abhyankar · Annals of Mathematics · 1956-05-01 · DOI 10.2307/1970014 · accessed Aug 6, 2026
6
Resolution of singularities of threefolds in positive characteristic I: Reduction to local uniformization on Artin–Schreier and purely inseparable coveringspeer reviewed result · Vincent Cossart, Olivier Piltant · Journal of Algebra · 2008 · DOI 10.1016/j.jalgebra.2008.03.032 · accessed Aug 6, 2026
7
Resolution of singularities of threefolds in positive characteristic IIpeer reviewed result · Vincent Cossart, Olivier Piltant · Journal of Algebra · 2009-04-01 · DOI 10.1016/j.jalgebra.2008.11.030 · accessed Aug 6, 2026
8
Resolution of singularities of arithmetical threefoldspeer reviewed result · Vincent Cossart, Olivier Piltant · Journal of Algebra · 2019-07-01 · ARXIV 1412.0868 · DOI 10.1016/j.jalgebra.2019.02.017 · accessed Aug 6, 2026
9
On the log minimal model program for threefolds over imperfect fields of characteristic p > 5peer reviewed result · Omprokash Das, Joe Waldron · Journal of the London Mathematical Society · 2022 · DOI 10.1112/jlms.12677 · accessed Aug 6, 2026
10
Smoothness, semi-stability and alterationspeer reviewed result · A. J. de Jong · Publications Mathématiques de l'IHÉS · 1996 · DOI 10.1007/BF02698644 · accessed Aug 6, 2026
11
On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for)survey or monograph · Herwig Hauser · Bulletin of the American Mathematical Society · 2010 · DOI 10.1090/S0273-0979-09-01274-9 · accessed Aug 6, 2026
12
Resolving Surface Singularities in Positive Characteristicpeer reviewed result · Herwig Hauser, Stefan Perlega · Publications of the Research Institute for Mathematical Sciences · 2024 · DOI 10.4171/PRIMS/60-4-5 · accessed Aug 6, 2026
13
Resolution of Singularities in Positive Characteristicspreprint · Heisuke Hironaka · Author manuscript · 2017-03-23 · accessed Aug 6, 2026
14
Resolution of Singularities in Arbitrary Characteristicpreprint · Yi Hu · arXiv · 2022-03-08 · ARXIV 2203.03842 · accessed Aug 6, 2026
15
Evolving Ranking Functions for Canonical Blow-Ups in Positive Characteristicpreprint · Gergely Bérczi · arXiv · 2026-02-06 · ARXIV 2602.06553 · accessed Aug 6, 2026
16
Nash blowup fails to resolve singularities in dimensions four and higherpeer reviewed result · Federico Castillo, Daniel Duarte, Maximiliano Leyton-Álvarez, Alvaro Liendo · Annals of Mathematics · 2026-03-01 · DOI 10.4007/annals.2026.203.2.7 · MR 5037842 · accessed Aug 6, 2026
17
Resolution of singularitiesencyclopedia · Wikipedia · accessed Aug 6, 2026
18
List of unsolved problems in mathematicsencyclopedia · Wikipedia · accessed Aug 6, 2026
19
Mathlib.AlgebraicGeometry.Morphisms.Smoothformalization · Lean community / Mathlib · accessed Aug 6, 2026
20
Mathlib.AlgebraicGeometry.Morphisms.Properformalization · Lean community / Mathlib · accessed Aug 6, 2026
21
Mathlib.RingTheory.RegularLocalRing.Defsformalization · Lean community / Mathlib · accessed Aug 6, 2026
22
Mathlib.AlgebraicGeometry.Normalizationformalization · Lean community / Mathlib · accessed Aug 6, 2026

Important qualifications

The canonical workspace scope is the remaining dimension-at-least-four problem over perfect fields of positive characteristic. The stronger quasi-excellent-scheme, projective, embedded, log-resolution, and principalization formulations are recorded separately rather than treated as interchangeable.
For arbitrary quasi-excellent schemes of dimension at most three, the cited 2019 theorem gives a proper birational resolution that may not be projective. The stronger projective log-resolution statement cited here is for quasi-projective schemes over F-finite fields.
Hironaka's 2017 manuscript and Yi Hu's 2022 arXiv preprint claim all-dimensional positive-characteristic resolution results. No peer-reviewed acceptance or independent validation was verified in this scoped review, while current 2026 research still calls the general problem open.
De Jong alterations are a weaker substitute because they are generically finite and need not be birational; they must not be presented as solutions of the canonical resolution problem.
Wikipedia recognition entries are broad reference links only, not mathematical status authorities, prize distinctions, or maintained selective-list badges.
The formalization search was scoped to current public Mathlib documentation and broad Lean, Rocq, and Isabelle web queries. No exact formal statement or checked proof was verified; this negative search does not establish nonexistence.
No computation, certificate, software package, or dataset resolving a bounded instance of the existential theorem was verified. The Nash-blowup counterexample is a mathematical obstruction to one procedure, not a computation resource or a counterexample to resolution itself.

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