Resolution of singularities

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

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