Embedded resolution of singularities in positive characteristic: stability of the residual order under permissible blowups and the existence of a terminating resolution invariant

Over fields of characteristic zero, Hironaka's embedded resolution of singularities is proved by a local invariant — local order first, then the residual order (the order of the coefficient ideal on a hypersurface of maximal contact, minus exceptional multiplicities) — that decreases lexicographically under blowups in regular permissible centers. In positive characteristic the same scheme breaks down: hypersurfaces of maximal contact need not exist, and the residual order can increase under permissible blowups (the kangaroo phenomenon). Hauser and Perlega (2019) characterized exactly when such an increase occurs and bounded it by $c!/p$, but left open whether the characteristic-free toolkit (weak maximal contact, modified coefficient ideals, permissible blowups) can be made to terminate. The problem asks, in the source's setting of complete regular local rings over an algebraically closed field of characteristic $p>0$: (i) whether an infinite sequence of permissible blowups with centers of maximal dimension can force unbounded growth of the residual order at constant multiplicity, and (ii) whether an efficient local resolution invariant decreasing under all permissible blowups exists in embedded dimension three and higher; a positive answer to (ii) would yield embedded resolution of singularities in positive characteristic in dimension at least three. As of 2026 both questions are open: embedded resolution is settled only up to dimension two, and the known example of unbounded residual-order growth uses zero-dimensional centers.

Background

Resolution of singularities asks for a sequence of blowups of a variety along regular centers that renders it smooth; the embedded version asks this for a subscheme $X$ of a regular ambient scheme $W$ so that additionally the total transform of $X$ becomes a normal crossings divisor. In characteristic zero this was proved by Hironaka (1964): one attaches to each point a local invariant, a string of non-negative integers whose first component is the order $\operatorname{ord}_a J$ of the defining ideal $J$; the locus where the invariant is maximal is a regular center, and blowing it up makes the invariant drop lexicographically, so the process terminates. The second component is built from the coefficient ideal: if $V = V(z)$ is a regular hypersurface and elements of $J$ are expanded as power series $f = \sum_i f_i z^i$, the coefficient ideal is $\operatorname{coeff}_V(J) = \sum_{i<c} (f_i, f \in J)^{c!/(c-i)}$ with $c = \operatorname{ord} J$. On a hypersurface of maximal contact — one containing all points of worst singularity, which always exists in characteristic zero — the order of the coefficient ideal minus the exceptional multiplicities (the multiplicities recording past blowups in a normal crossings divisor $D$) is called the residual order, and it does not increase under permissible blowups (blowups whose regular center has normal crossings with $D$ and lies in the equimultiple loci). In positive characteristic each step of this construction can fail. Narasimhan (1983) showed the equimultiple locus (the locus of points where the order of the defining ideal attains its maximal value) need not be hyperplanar; Cossart–Jannsen–Saito (2020) showed hypersurfaces of maximal contact may fail to exist already for surfaces. The substitute notion of weak maximal contact (a hypersurface merely maximizing the order of the coefficient ideal) still yields a well-defined residual order, but Hauser's theory of kangaroo points (2008) shows this residual order can jump upward at points of the new exceptional divisor where the multiplicity has not dropped. Moh (1987) proved a stability bound for purely inseparable hypersurface singularities (hypersurfaces defined by a purely inseparable power series, i.e. one that is a power series in the $p^e$-th powers of the coordinates) and claimed a general stability theorem under blowup sequences; Hauser–Perlega (2019) disproved the claimed theorem with an explicit purely inseparable hypersurface singularity and a cycle of point blowups whose iteration drives the residual order to infinity — though these centers have dimension zero, not the maximal dimension that a resolution algorithm would choose, so the example does not obstruct resolution itself. Their companion paper (2019) then characterized structurally when a single permissible blowup can increase the residual order: the order $c$ must be a multiple $m p^e$ of a power of the characteristic, the coefficient-ideal order a multiple of $c!$, the weighted initial forms (initial forms with respect to a weighted degree) of minimal elements of $J$ must be powers $\alpha(z^{p^e} + F(x))^m$ of purely inseparable forms with $F$ not a $p^e$-th power and with a specific shape of logarithmic Hasse derivatives (the positive-characteristic analogue of ordinary derivatives, normalized coefficient-wise so that they do not degenerate on $p$-th powers), the exceptional exponents must satisfy an arithmetic inequality $\sum \bar{r}_i \le (b-1)p^{\ell+1}$, and the increase is at most $c!/p$. What is known dimensionwise: non-embedded resolution in positive characteristic is settled up to dimension three (Cossart–Piltant for quasi-excellent threefolds in arbitrary characteristic, Cutkosky for algebraic threefolds), and embedded resolution up to dimension two (Cossart–Jannsen–Saito; Hauser–Wagner for purely inseparable surface singularities; a new systematic proof for surface hypersurface singularities by Hauser–Perlega 2024 using a resolution function with finer, characteristic-sensitive components). Embedded resolution in positive characteristic remains open from dimension three upward, and no resolution invariant playing the role of the characteristic-zero string invariant is known there.

Problem Statement

In the setting of Hauser–Perlega (2019) — complete regular local noetherian rings over an algebraically closed field of positive characteristic, ideals admitting a hypersurface of weak maximal contact compatible with a normal crossings divisor, and completed local blowups permissible with respect to them — decide whether the characteristic-free version of the characteristic-zero resolution toolkit can be made to terminate, concretely along two linked questions forming one objective, making the Hironaka-style inductive strategy work in positive characteristic. First (stability of the key invariant): determine whether there exist an ideal, a divisor, and an infinite sequence of permissible blowups whose centers have maximal dimension among the permissible centers at each stage, such that the local order of the ideal remains constant along the sequence while its residual order grows without bound; a negative answer (nonexistence of such a sequence) would remove the main known obstruction to the strategy, while a positive answer (an explicit construction) would show that the residual order cannot serve as a resolution invariant and would critically obstruct the coefficient-ideal strategy. Second (the target the strategy aims at): construct, or prove unattainable, an efficient local resolution invariant for embedded resolution of singularities in positive characteristic in ambient dimension three and higher — an upper semicontinuous invariant defining regular centers with normal crossings such that each permissible blowup of the maximal stratum makes it decrease lexicographically in a well-ordered value set — thereby proving embedded resolution of singularities over algebraically closed fields of positive characteristic at least for threefolds (it is known for surfaces and open from dimension three up). A complete answer of the second, positive kind subsumes the first; the first is the decisive structural subquestion on the path. The source's further extension directions (from complete local rings with algebraically closed residue fields to non-complete rings, non-closed residue fields, and global schemes) are recorded as part of the problem's eventual scope, not as separate objectives.

References

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