Resolution of singularities

Resolution of singularities in positive characteristic. Let k be a perfect field of characteristic p > 0 and let X be an integral scheme that is separated and of finite type over k. Then there is an integral scheme Y that is smooth over k together with a proper birational morphism Y → X.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement resolution_of_singularities as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `resolution_of_singularities`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 2 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-resolution-of-singularities,
  title        = {Resolution of singularities},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/resolution-of-singularities}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Resolution of singularities in positive characteristic. Let $k$ be a perfect field of characteristic $p > 0$ and let $X$ be an integral scheme that is separated and of finite type over $k$ . Then there is an integral scheme $Y$ that is smooth over $k$ together with a proper birational morphism $Y \to X$ .

This is open from dimension four on; see [Hau2010]. Perfectness of $k$ is needed for the conclusion as stated; see exists_not_hasResolution_of_not_perfectField.

A variety $X$ over a field $k$ admits a resolution of singularities if there is a smooth $k$ -variety $Y$ and a proper birational morphism $Y \to X$ . Hironaka proved that every variety over a field of characteristic zero admits one. In positive characteristic this is known over a perfect field in dimension at most three, and is open from dimension four on. Perfectness of $k$ cannot be dropped from the statement in this form: over an imperfect field it fails already in dimension zero, see exists_not_hasResolution_of_not_perfectField below. Over an arbitrary field one asks instead that $Y$ be regular.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.ResolutionOfSingularities.

theorem resolution_of_singularities (k : Type u) [Field k] [PerfectField k] (p : ℕ)
    [Fact p.Prime] [CharP k p] {X : Scheme.{u}} (sX : X ⟶ Spec (.of k)) [IsIntegral X]
    [LocallyOfFiniteType sX] [QuasiCompact sX] [IsSeparated sX] :
    Scheme.HasResolution sX

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
[Hir1964] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I and II, Ann. of Math. 79 (1964), 109--203 and 205--326.
[Kol2007] J. Kollár, [Resolution of singularities -- Seattle lecture](https://arxiv.org/abs/math/0508332), Theorem 36.
[Hau2010] H. Hauser, [On the problem of resolution of singularities in positive characteristic (or: a proof we are still waiting for)](https://doi.org/10.1090/S0273-0979-09-01274-9), Bull. Amer. Math. Soc. 47 (2010), 1--30.
[Lip1978] J. Lipman, [Desingularization of two-dimensional schemes](https://doi.org/10.2307/1971141), Ann. of Math. 107 (1978), 151--207.
[CP2008] V. Cossart and O. Piltant, [Resolution of singularities of threefolds in positive characteristic I](https://doi.org/10.1016/j.jalgebra.2008.03.032), J. Algebra 320 (2008), 1051--1082.
[CP2009] V. Cossart and O. Piltant, [Resolution of singularities of threefolds in positive characteristic II](https://doi.org/10.1016/j.jalgebra.2008.11.030), J. Algebra 321 (2009), 1836--1976.
[CP2019] V. Cossart and O. Piltant, [Resolution of singularities of arithmetical threefolds](https://doi.org/10.1016/j.jalgebra.2019.02.017), J. Algebra 529 (2019), 268--535.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.