Math Conjectures

Resolution of Singularities in Positive Characteristic

OPENLandmarkOpen problemProposed 1964 · Full conjecture

Canonical statement

Let kk be a perfect field of positive characteristic and XX an integral separated scheme of finite type over kk. There exist a smooth integral kk-scheme YY and a proper birational morphism f:Y→Xf:Y\to X that is an isomorphism over the regular locus XregX_{\mathrm{reg}}.
View source LaTeX
Let \(k\) be a perfect field of positive characteristic and \(X\) an integral separated scheme of finite type over \(k\). There exist a smooth integral \(k\)-scheme \(Y\) and a proper birational morphism \(f:Y\to X\) that is an isomorphism over the regular locus \(X_{\mathrm{reg}}\).
AG-SING-013:[Hironaka1964Resolution][CossartPiltant2009Threefolds][Temkin2018Desingularization]
Algebraic geometryAlgebra and representation theory

Notes

Hironaka proved in 1964 that every variety over a field of characteristic zero admits a resolution of singularities [Hironaka1964Resolution]. The corresponding problem in positive characteristic, dating from the same moment, asks: given an integral separated scheme XX of finite type over a perfect field kk of characteristic p>0p>0, find a proper birational morphism f:Y→Xf:Y\to X from a smooth kk-scheme that is an isomorphism over the regular locus XregX_{\mathrm{reg}}.

Low dimensions are settled: curves are classical, surfaces have long been known, and Cossart and Piltant resolved threefolds in positive characteristic, with refinements depending on the precise category considered [CossartPiltant2009Threefolds]. On the characteristic-zero side the theory has meanwhile been sharpened into functorial and embedded forms [Temkin2018Desingularization], but its key inductive tools, such as hypersurfaces of maximal contact, break down when the characteristic is positive.

No proof covers arbitrary dimension in characteristic pp; even dimension four is open, and a solution is expected to require genuinely new ideas beyond the characteristic-zero algorithms.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.