Canonical functorial resolution of singularities in positive characteristic: extending McQuillan's characteristic-zero invariant beyond equicharacteristic zero

McQuillan (with Marzo in the preprint version) constructed, for excellent Deligne–Mumford stacks in equicharacteristic zero, a resolution functor that is canonical: étale-local, independent of the resolution history, and governed by a local numerical invariant of the ideal alone, realized by stack-theoretic ('smoothed') weighted blow-ups in regular weighted centres. The construction is confined to characteristic zero (residue fields of characteristic zero) and to excellent, J-2 hypotheses, which are used to descend centres from completions and to obtain upper semi-continuity of the invariant. The source proves its main theorem only in equicharacteristic zero. Does a canonical resolution functor of this kind — an intrinsic, history-independent invariant that strictly decreases under a sequence of weighted blow-ups and terminates — exist in positive characteristic and in mixed characteristic $(0,p)$ , where resolution of singularities itself is open beyond dimension three? The problem is open: dimension-at-most-three positive-characteristic resolution is known only by history-dependent Hironaka-style methods, documented pathologies (the kangaroo phenomenon, cycles of singularities) defeat all known monotone invariants, a February 2026 survey states that no general ranking function is known in positive characteristic, and the sole recent claimed full proof (August 2026 preprint) is unrefereed and unverified.

Background

Resolution of singularities asks for a sequence of blow-ups with controlled centres that replaces a singular Noetherian scheme by a regular one, with the exceptional divisor (the inverse image of the blown-up centres) a normal-crossings divisor, i.e. locally a union of coordinate hyperplanes. Hironaka settled the case of varieties over a field of characteristic zero in 1964, but his algorithm is history-dependent: each step depends on the previous ones, so the output is not canonical and does not commute with base change. A functorial (canonical) resolution instead assigns to every singularity a modification $M(U)$ , functorially and commuting with étale (or smooth) base change, governed by an invariant $\mathrm{inv}(U)$ in a well-ordered group that strictly decreases at every point until regularity is reached; classically this was believed impossible with smooth centres, and the resolution of that obstruction was to allow weighted centres. A weighted blow-up assigns rational weights $r_0,\dots,r_n$ to the coordinates and blows up the corresponding Rees-type graded algebra; its stack-theoretic ('smoothed') realisation, as a quotient stack of the blow-up of a suitable ideal, keeps the source regular, and Deligne–Mumford stacks (algebraic stacks locally the quotient of a smooth scheme by a finite étale group action, i.e. orbifold-type objects) provide the natural home for these centres. In this language, the source paper (preprint M. McQuillan and G. Marzo, arXiv:1906.06745; published version by McQuillan alone in Geometric and Functional Analysis 30 (2020), 858–909) proves: for reduced excellent Deligne–Mumford champs of characteristic zero there is a modification functor $M$ satisfying the canonicity axioms (M.1 new)–(M.5), built from a purely local invariant $\mathrm{inv}_A(I)\in\mathbb{Q}_{\geq 0}^{2m}$ (lexicographic order, self-bounding denominators; globally the invariant takes values in $\mathbb{Q}_{\geq 0}^{\infty}=\varinjlim\mathbb{Q}_{\geq 0}^{N}$ and attains its minimum $\underline{0}$ exactly on the regular locus) of an ideal $I$ on an $m$ -dimensional regular local ring, together with a dichotomy that either something generic happens or the induced filtrations $F^\bullet_s(I)$ converge adically; the invariant strictly decreases along the associated smoothed weighted blow-ups, giving termination. The terminal conclusion of the source's theorem (I.a/VII.h) is regularity of the final object, with the invariant zero exactly on the regular locus; it does not assert a normal-crossings exceptional divisor, the blow-ups being weighted and stack-theoretic. Here the paper's class convention differs from Grothendieck's standard one: the source defines an excellent Deligne–Mumford champ as one whose Henselian local rings are all excellent, with some (whence any) atlas J-2, so that a scheme excellent in this sense is only quasi-excellent in standard parlance, the catenary condition being systematically eschewed globally (in standard terms, quasi-excellent means the formal fibres are geometrically regular (a G-ring) and every finitely generated algebra has open singular locus (the J-2 condition), and excellent adds universal catenarity); the paper uses excellence precisely to descend the weighted centres from the completion of a local ring back to the ring itself, and uses the J-2/universally catenary hypotheses (via Dade's unpublished proof of upper semi-continuity of the Hilbert–Samuel multiplicity, the leading coefficient of the Hilbert–Samuel polynomial, which measures the order of a local singularity) to obtain upper semi-continuity of the invariant, which is what patches local blow-ups into a global functor. Two limitations of the hypotheses are explicit in the paper. First, the local theory (its §III setup) works over residue fields of characteristic $0$ , i.e. equicharacteristic zero: nothing in the work covers schemes of positive or mixed residue characteristic. Second, the paper itself observes that quasi-excellence is a necessary condition for resolution of singularities (citing EGA IV.2), so excellence cannot simply be removed; and independently of this paper, functorial desingularization of all quasi-excellent characteristic-zero schemes was already known by other, non-elementary methods (Temkin). In positive characteristic the general resolution problem — with or without canonicity — is one of the most famous open problems in algebraic geometry: it is known for excellent surfaces (Lipman) and for quasi-excellent threefolds in positive and mixed characteristic (Cossart–Piltant), but open from dimension four on, and the known proofs are history-dependent. The characteristic-zero mechanisms fail in characteristic $p$ for documented reasons: derivations cannot provide maximal-contact hypersurfaces (hypersurfaces tangent to the worst singular locus to maximal order, used to lower the dimension inductively) when $p$ divides multiplicities, Frobenius-specific pathologies such as the kangaroo phenomenon let classical numerical invariants jump back up after a blow-up, and explicit cycles of singularities in blow-up sequences show that no monotone behaviour of the standard invariants is available. A 2026 paper on constructing invariants in characteristic $3$ states outright that in positive characteristic no general ranking function of the type that powers all characteristic-zero proofs is known. Recent work on the weighted-blow-up side — the independent simultaneous proof of Abramovich–Temkin–Włodarczyk, Włodarczyk's cobordant blow-ups giving canonical functorial resolution in characteristic zero and resolution for some special classes in positive and mixed characteristic, and his 2026 normal-crossings-preserving canonical resolution — remains in characteristic zero at the functorial level. Whether the canonical-invariant paradigm itself survives in positive characteristic is the question.

Problem Statement

Determine whether the canonical-invariant paradigm of the source paper extends beyond equicharacteristic zero. The source proves its Theorem I.a only in equicharacteristic zero. The question is the positive-characteristic case $p$ ; the mixed-characteristic $(0,p)$ case is a further, separate question. Specifically: does there exist, for reduced excellent Deligne–Mumford stacks ('champs') of positive characteristic $p$ in the source's sense — all Henselian local rings excellent and some (whence any) atlas J-2, which for schemes means quasi-excellent in standard parlance — a resolution functor $M$ in the sense of the source paper's Theorem I.a: a modification assignment that is canonical (functorial and commuting with étale base change), independent of the resolution history, realized by a sequence of smoothed (stack-theoretic) weighted blow-ups in regular weighted centres, and governed by a purely local intrinsic invariant of the source's kind — the embedded invariant $\mathrm{inv}_A(I)$ of an ideal on a regular local ring, or the non-embedded invariant $\mathrm{inv}^{\sharp}_Y$ of the singular object — taking values in a well-ordered group (locally $\mathbb{Q}_{\geq 0}^{2m}$ in lexicographic order, globally $\mathbb{Q}_{\geq 0}^{\infty}=\varinjlim\mathbb{Q}_{\geq 0}^{N}$ ) with self-bounding denominators, equal to $\underline{0}$ exactly on the regular locus, defined without passage to a history-dependent coordinate choice, strictly decreasing at every closed point of each centre while the object is singular, and terminating after finitely many steps at a regular object — equivalently, a modification functor satisfying the source's axioms (M.1 new)–(M.5) in characteristic $p$ ? A positive answer for all quasi-excellent positive-characteristic schemes would in particular resolve the open problem of resolution of singularities in positive characteristic in dimension at least four, in its strongest (canonical, functorial) form. A negative answer, to establish a general obstruction, must show that no functor satisfying these stated axioms exists in characteristic $p$ — not merely that one attempted construction fails — and would then exhibit a structural obstruction for this canonical-invariant paradigm, leaving open whether canonicity might be achieved in characteristic $p$ by some other mechanism.

References

M. McQuillan, G. Marzo, Very fast, very functorial, and very easy resolution of singularities, arXiv preprint arXiv:1906.06745 (2019), https://arxiv.org/abs/1906.06745
M. McQuillan, Very functorial, very fast, and very easy resolution of singularities, Geometric and Functional Analysis 30 (2020), 858–909, doi:10.1007/s00039-020-00523-7, https://doi.org/10.1007/s00039-020-00523-7
D. Abramovich, M. Temkin, J. Włodarczyk, Functorial embedded resolution via weighted blowings up, Algebra & Number Theory 18 (2024), 1557–1587, doi:10.2140/ant.2024.18.1557, https://doi.org/10.2140/ant.2024.18.1557 (preprint arXiv:1906.07106, https://arxiv.org/abs/1906.07106)
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I, Annals of Mathematics 79 (1964), 109–203, doi:10.2307/1970486, https://doi.org/10.2307/1970486
J. Lipman, Desingularization of two-dimensional schemes, Annals of Mathematics 107 (1978), 151–207, doi:10.2307/1971141, https://doi.org/10.2307/1971141
V. Cossart, O. Piltant, Resolution of singularities of threefolds in positive characteristic I, Journal of Algebra 320 (2008), 1051–1082, doi:10.1016/j.jalgebra.2008.03.032, https://doi.org/10.1016/j.jalgebra.2008.03.032
V. Cossart, O. Piltant, Resolution of singularities of threefolds in positive characteristic II, Journal of Algebra 321 (2009), 1836–1976, doi:10.1016/j.jalgebra.2008.11.030, https://doi.org/10.1016/j.jalgebra.2008.11.030
V. Cossart, O. Piltant, Resolution of singularities of arithmetical threefolds, Journal of Algebra 529 (2019), 268–535, doi:10.1016/j.jalgebra.2019.02.017, https://doi.org/10.1016/j.jalgebra.2019.02.017
M. Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Advances in Mathematics 219 (2008), 488–522, doi:10.1016/j.aim.2008.05.006, https://doi.org/10.1016/j.aim.2008.05.006
M. Temkin, Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case, Duke Mathematical Journal 161 (2012), no. 11, doi:10.1215/00127094-1699539, https://doi.org/10.1215/00127094-1699539
H. Hauser, On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for), Bulletin of the American Mathematical Society 47 (2010), 1–30, doi:10.1090/s0273-0979-09-01274-9, https://doi.org/10.1090/s0273-0979-09-01274-9
H. Hauser, S. Perlega, Cycles of singularities appearing in the resolution problem in positive characteristic, Journal of Algebraic Geometry 28 (2019), 391–403, doi:10.1090/jag/718, https://doi.org/10.1090/jag/718
J. Włodarczyk, Functorial resolution by torus actions, arXiv preprint arXiv:2203.03090 (2022, revised 2025), https://arxiv.org/abs/2203.03090
J. Włodarczyk, Resolution except for the normal-crossing locus and Galois actions, arXiv preprint arXiv:2602.14266 (2026), https://arxiv.org/abs/2602.14266
G. Bérczi, Evolving ranking functions for canonical blow-ups in positive characteristic, arXiv preprint arXiv:2602.06553 (2026), https://arxiv.org/abs/2602.06553
C. Tian, Resolution of singularities in positive characteristic: Frobenius-Hasse towers and exceptional-history descent, arXiv preprint arXiv:2608.20066 (2026), https://arxiv.org/abs/2608.20066