Canonical resolution of singularities in positive and mixed characteristic via cobordant blow-ups at weighted centers

Włodarczyk's 'Functorial resolution by torus actions' proves a canonical, functorial resolution of singularities in characteristic zero by sequences of cobordant blow-ups — a scheme-theoretic implementation of weighted blow-ups through a $\mathbb{G}_m$-action — at admissible weighted centers, and shows that in arbitrary (positive or mixed) characteristic a single cobordant blow-up resolves the class of 'almost homogeneous singularities', defined by nonsingularity of the weighted normal cone away from the singular locus. The paper leaves open whether the method extends to a general canonical resolution in nonzero characteristic and exactly which singularity classes it resolves; the case of Brieskorn-type hypersurfaces and quadrics over nonperfect fields is explicitly flagged as unresolved. The resolvable classes have since been broadened (locally monomial centers via Cox rings; plane curves over perfect fields via stack-theoretic blow-ups), but no general positive- or mixed-characteristic algorithm for the cobordant method is known, and nothing delineates its boundary. The problem asks for a general canonical resolution by cobordant blow-ups in positive and mixed characteristic, or a complete characterization — necessary and sufficient conditions plus explicit obstructions — of the singularity classes the method resolves, including the flagged nonperfect-field case.

Background

Resolution of singularities asks whether every algebraic variety can be replaced, by a proper birational morphism, with a smooth one; in its embedded form one blows up a smooth ambient variety $X$ along smooth centers until a given closed subvariety $Y$ has smooth strict transform (the closure of the image of $Y$ away from the center) and the exceptional divisor has simple normal crossings (SNC), i.e. its components meet locally like coordinate hyperplanes. Hironaka proved this in characteristic zero in 1964. Constructive versions proceed through a numerical invariant of the singularity that strictly drops at each blow-up, and the induction step needs a maximal contact hypersurface containing the worst locus of singularities, which exists in characteristic zero because derivations are plentiful. In positive characteristic this machinery collapses: derivations vanish on $p$-th powers, the worst locus need not lie in any smooth hypersurface (Narasimhan's characteristic-$2$ example), and substitute invariants can fail to decrease — the 'kangaroo' phenomenon. Resolution in positive characteristic is consequently a famous open problem, proved only in dimension at most $3$ by Cossart–Piltant by classical methods, and the mixed-characteristic analogue — schemes over bases such as rings of integers, where $p$ is neither invertible nor nilpotent — is likewise open beyond dimension $3$ (Cossart–Piltant for arithmetical threefolds).

A modern strategy replaces smooth centers, which implicitly carry uniform weights, with weighted centers: a $\mathbb{Q}$-ideal $\mathcal{J}=(x_1^{1/w_1},\ldots,x_k^{1/w_k})$ obtained by formally adjoining $w_i$-th roots of local parameters, so that the blow-up tracks the intrinsic non-uniform multiplicities of the singularity. Implemented on Artin stacks — geometric objects built from quotients by group actions that retain stabilizer information — (the stack-theoretic weighted blow-ups of Abramovich–Temkin–Włodarczyk, and McQuillan's resolution), such centers shorten and canonicalize the characteristic-zero algorithm. The paper gives a purely scheme-theoretic implementation: for a weighted center $\mathcal{J}$ on a smooth $X$ with SNC divisor, the cobordant blow-up is $B=\operatorname{Spec}\mathcal{O}_X[t^{-1},t^{w_1}x_1,\ldots,t^{w_k}x_k]$, a smooth variety with an action of the multiplicative group $\mathbb{G}_m=\operatorname{Spec}k[t,t^{-1}]$ (a one-chart incarnation of the birational cobordism — a $\mathbb{G}_m$-variety whose two orbit-space ends recover the two birational models — used in the proof of the weak factorization theorem, which states that any two smooth birational models of a variety are connected by sequences of blow-ups and blow-downs along smooth centers) birational to $X$, on which the transform of $Y$ is inspected through the associated graded ring of $\mathcal{J}$.

The paper builds a canonical invariant $\mathrm{inv}_p$, a lexicographic tuple of rational numbers measuring the singularity, on smooth schemes with SNC divisors, proves that admissible weighted centers exist with $\mathrm{inv}$ strictly decreasing (termination via the descending chain condition, i.e. the set of possible invariant values admits no infinite strictly decreasing sequence), and thereby obtains functorial principalization and embedded and nonembedded resolution in characteristic zero. In arbitrary characteristic the relevant object is the weighted normal cone $C_{\mathcal{J}}(Y)\subset N_{\mathcal{J}}(X)$: the limit of the strict transform of $Y$ at the negative end of the cobordism is cut out by the initial-form ideal $\mathrm{in}_{\mathcal{J}}(\mathcal{I}_Y)$ in the graded ring $\mathrm{gr}_{\mathcal{J}}(\mathcal{O}_X)$. A singularity is almost homogeneous (Definition 4.2.1 of the source) when its singular locus is a regular subscheme $Z$ and the weighted normal cone is nonsingular exactly away from $Z$ — informally, the lowest-weight part of the defining equations produces nothing worse than $Z$ itself; a single cobordant blow-up then resolves $Y$ (Theorem 4.2.2, under a hypothesis that $X$ is universally catenary — a ring-theoretic condition making codimension formulas behave under finitely generated extensions — or that every component of $C_{\mathcal{J}}(Y)\smallsetminus Z$ has codimension equal to $\operatorname{codim}_X(Y)$). The Resolution Principle (Theorem 4.3.1) adds a general reduction valid in any characteristic: if a weighted center exists whose support and whose initial-ideal locus both carry invariant at least $\Phi$, then the cobordant blow-up of that center strictly lowers the invariant. These results cover, in positive and mixed characteristic, all singularities whose worst locus can be matched with a weighted center having a tame weighted tangent cone — for instance Brieskorn hypersurfaces $f=a_1x_1^{c_1}+\cdots+a_nx_n^{c_n}$ with isolated worst locus, and mixed-characteristic equations such as $x^p+p^p+y^k$ — but the paper provides no mechanism producing such centers for arbitrary singularities, no termination guarantee for iterated blow-ups in nonzero characteristic, and no necessary conditions delimiting the resolvable classes. Over nonperfect fields (fields of characteristic $p$ in which some elements have no $p$-th root), even the Brieskorn and quadric cases are, in the paper's words, 'a challenging and unresolved problem', since the tangent cone itself can be singular (Cossart–Piltant–Schober).

Problem Statement

Determine the scope of the cobordant blow-up method in nonzero characteristic, up to and including general canonical resolution. Given an integral variety over a field of positive characteristic (perfect or not), or a scheme of finite type over an excellent mixed-characteristic base (a standard finiteness hypothesis on the base ring ensuring well-behaved dimension theory) such as a ring of integers: either (1) construct a canonical, functorial resolution of its singularities by a finite sequence of cobordant blow-ups $B=\operatorname{Spec}\mathcal{O}_X[t^{-1},t^{w_i}x_i]$ at admissible weighted centers (Definition 3.1.14 of the source) — extending the characteristic-zero algorithm of Włodarczyk's paper — together with a proof of termination by a well-founded measure decreasing along the sequence (or an equivalent induction); or (2) characterize exactly which classes of singularities in positive and mixed characteristic admit such terminating weighted-center resolution sequences, via necessary and sufficient geometric conditions on the weighted normal cone $C_{\mathcal{J}}(Y)$ and the initial-form invariant, together with explicit obstructed classes if the method does not extend generally. The classes to be classified include Brieskorn-type hypersurfaces $f=a_1x_1^{c_1}+\cdots+a_nx_n^{c_n}$, and quadrics, over nonperfect fields, where the tangent cone itself may be singular. Branch (1) is the affirmative case of branch (2), in which every singularity is resolvable; the two branches share a single scientific objective — how far the weighted-center, torus-action paradigm reaches in nonzero characteristic. This is a method-specific problem: a solution via branch (1) would in particular resolve the classical open problem of resolution of singularities in positive characteristic, of which it is a derived route rather than a restatement.

References

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