Hironaka's 1964 Annals papers proved that every algebraic variety over a field of characteristic zero admits resolution of singularities by a finite sequence of blowings-up along nonsingular centers, and the introduction of that work explicitly left the positive-characteristic (and unequal/mixed-characteristic) case open, together with adjacent questions on factorization of birational maps and analytic generality. Later literature settled the characteristic-zero analytic and functorial forms and the positive/mixed-characteristic cases up to dimension three (Abhyankar and Lipman in dimension at most two; Cossart-Piltant for three-dimensional quasi-excellent schemes of arbitrary characteristic), while de Jong's alterations provide a weaker non-birational substitute in all dimensions. The surviving open core is: prove, or refute, resolution of singularities for varieties and quasi-excellent noetherian schemes of dimension at least four in positive or mixed characteristic. No accepted resolution is known; two unrefereed preprints (2022 and 2026) claim proofs, but neither has a journal publication or expert-acceptance record.

Background

A noetherian scheme or algebraic variety is singular at a point where its local ring (the ring of germs of algebraic functions at that point) fails to be regular; resolution of singularities asks for a birational replacement of a singular object by a regular (nonsingular) one. The strong form of the problem, established by Hironaka in 1964 for schemes of finite type over fields of characteristic zero, asks that every reduced variety X (reduced meaning that its rings of functions contain no nilpotent elements) admit a proper birational morphism (properness being the algebraic analogue of compactness) from a regular variety that is an isomorphism over the regular locus of X and that is a finite composition of blowings-up along regular centers, where a blowing-up is the basic birational operation replacing X by the Proj of the Rees algebra (the graded algebra built from the powers) of a coherent sheaf of ideals. Hironaka's characteristic-zero proof descends in dimension through hypersurfaces of maximal contact, that is, local hypersurfaces containing all the relevant singular data and so allowing induction on dimension, constructed from derivatives of defining equations, together with a numerical resolution invariant that strictly decreases at each step, guaranteeing termination.

Grothendieck later reformulated the problem for arbitrary quasi-excellent noetherian schemes (noetherian schemes whose formal fibers, the fibers of the passage from a local ring to its completion, are geometrically regular, a class covering varieties over fields and schemes of finite type over the integers or Dedekind rings); schemes over Z or its localizations combine characteristic zero at the generic point with positive characteristic at special points, the mixed characteristic (0,p) case. In positive characteristic the characteristic-zero strategy collapses: derivations may vanish identically, imperfect fields lack coefficient fields (subfields of the local ring mapping isomorphically onto its residue field, which the characteristic-zero construction needs), wild ramification (ramification of degree divisible by the residue characteristic) intervenes, and Hauser's examples (analyzed further by Hauser-Perlega) show that the standard numerical invariants of a singularity, such as the residual order (the order of the residual equation after elimination along a hypersurface of maximal contact), can increase rather than decrease under permissible blowings-up. What is known: local uniformization (making the local ring of the variety regular at the center of a chosen valuation of its function field) and resolution for surfaces (Abhyankar 1956; Lipman 1978 for arbitrary excellent two-dimensional schemes); resolution for threefolds in positive and mixed characteristic (Cossart-Piltant 2008-2019, proving Grothendieck's conjecture in dimension three for arbitrary characteristic); and de Jong's alterations (1996), which dominate every variety by a smooth variety through a proper surjective generically finite map in any characteristic but sacrifice birationality.

The problem posed here is the remaining case: dimension at least four in positive or mixed characteristic. The 1964 introduction also states sibling questions (analytic generality, factorization of birational maps, domination of meromorphic maps by projective modifications, canonicality); these are out of scope for this task, which covers only the positive- and mixed-characteristic resolution statement, the direct continuation of the paper's own main theorem.

Problem Statement

Prove, or refute, resolution of singularities in positive and mixed characteristic in the dimensions where it remains open. Target statement (Grothendieck's conjecture on resolution of singularities, in the strong form established in characteristic zero by Hironaka for varieties and later extended to quasi-excellent schemes): every reduced quasi-excellent noetherian scheme X of dimension at least four and of positive or mixed characteristic — positive characteristic meaning that all its points have residue fields of positive characteristic, and mixed characteristic (0,p) meaning that the generic point has characteristic zero while the special points have characteristic p, as for schemes over the integers or the p-adic integers — admits a resolution, that is, a proper birational morphism from a regular scheme that is an isomorphism over the regular locus of X, given by a finite sequence of blowings-up whose centers are regular subschemes. In particular, every integral variety (reduced and irreducible) of dimension at least four over a field of positive characteristic, perfect or imperfect (an imperfect field being one of positive characteristic in which not every element is a p-th power), must admit such a resolution. A refutation is an explicit variety or quasi-excellent scheme of dimension at least four in positive or mixed characteristic, together with a rigorous proof that it admits no resolution of the stated form. The ground field is not required to be perfect, and the centers of the blowings-up are required only to be regular subschemes.

References

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