Resolution of the monomial case of the Idealistic Filtration Program in positive characteristic in dimension four and higher

The Idealistic Filtration Program (IFP) of Kawanoue and Matsuki reduces constructive resolution of singularities in positive characteristic to two steps: Step 1, reduction of the general case to the monomial case, established in arbitrary dimension; and Step 2, resolution in the monomial case, which in positive characteristic is established only in the local setting, after reduction to a single leading generator, in ambient dimension 3. The open problem stated here is Step 2 in ambient dimension four and higher: termination, after finitely many steps, of an explicit algorithm of permissible blow-ups at the tight monomial case, where resolution is completed by a classical procedure. The problem remains open as of 2026.

Background

Resolution of singularities asks whether every integral algebraic variety over a field admits a proper birational map from a regular variety; constructive versions produce it by a finite sequence of blow-ups, each along a smooth center inside the singular locus. Hironaka proved this in characteristic zero, and modern proofs choose each center as the maximum stratum of a resolution invariant taking values in a well-ordered set, so that strict decrease after every blow-up forces termination. In positive characteristic no such algorithm is known in ambient dimension four or higher: hypersurfaces of maximal contact, the smooth hypersurfaces containing all subsequent centers locally that power the characteristic-zero induction, may fail to exist; Frobenius effects interfere with differential constructions; and the residual order, the order of the coefficient ideal of the singularity on a test hypersurface, can strictly increase under blow-up (Hauser's kangaroo phenomenon). The strongest all-dimensional substitute is de Jong's theorem on alterations, which yields only a generically finite, not birational, map from a regular scheme. In positive characteristic, resolution is settled for surfaces, notably by the invariants of Benito-Villamayor and by a recent proof of Perlega for surfaces embedded in three-dimensional ambient space, and for threefolds by Cossart-Piltant.

The Idealistic Filtration Program (IFP) of Kawanoue and Matsuki is a constructive approach to resolution in arbitrary characteristic. It replaces the ideal sheaf to be resolved by an idealistic filtration $\mathbb{R}$, a set of pairs $(f,a)$, with $f$ a regular function and $a$ a nonnegative rational level, closed under sums, multiples, and a differential saturation modeled on partial derivatives; in characteristic $p$ the program additionally closes $\mathbb{R}$ under taking $p$-th roots, which is what makes the theory characteristic-sensitive. A Leading Generator System (LGS), a minimal system of elements of pure $p$-power levels, plays the role that a hypersurface of maximal contact plays in characteristic zero. The program is split into two steps. Step 1, reduction of the general case to the monomial case, is established in arbitrary dimension: after finitely many blow-ups one reaches the monomial case, where locally $\mathbb{R}$ is generated by an LGS together with a monomial in the components of the boundary divisor $E$, the exceptional divisor already created with its natural multiplicities. Step 2, resolution in the monomial case, is immediate in characteristic zero, since the monomial itself dictates the centers, but is the deep open part in positive characteristic. Within the monomial case the first invariant is $\tau$, the number of elements of the LGS, and the authors state an inductive scheme in arbitrary ambient dimension $d$ in which the cases $\tau=j$ with $j=2,\dots,d-1$ are analyzed similarly to the case $(\dim=d-1,\tau=j-1)$, $\tau=0$ is classical, and $\tau=d$ does not occur, so that the essential remaining case is $\tau=1$, the analogue of a hypersurface singularity. The authors note, however, that this similarity reduction for $\tau\geq 2$ is "straightforward, but not trivial" and requires some argument, and they defer its details in dimension four and higher to a separate publication, which never appeared in the indexed literature.

For $\tau=1$ the single LGS element can be put in Weierstrass form $h=x_1^{p^e}+a_1x_1^{p^e-1}+\cdots+a_{p^e}$ with coefficients $a_i$ in the remaining variables of order greater than $i$ at the point. The difficulty concentrates in the last coefficient $a_{p^e}$: because the initial form of $a_{p^e}$ may be a $p$-th power, its order depends on the choice of coordinates, and the cleaning process removes this ambiguity. Kawanoue-Matsuki define the invariant $H(P)=\min\{\operatorname{ord}_P(a_{p^e})/p^e,\operatorname{ord}_P(M_{\mathrm{usual}})\}$ and the tight monomial $M_{\mathrm{tight}}$; when $H(P)=\operatorname{ord}_P(M_{\mathrm{tight}})$ one is in the tight monomial case, where resolution is finished by a classical easy procedure, so the whole problem is to reach the tight monomial case. In ambient dimension 3, their earlier work following Benito-Villamayor produced a strictly decreasing invariant, and the source paper, published as Rev. Mat. Iberoam. 34 (2018), 1229-1276, constructs a new invariant $\mathrm{inv}_{\mathrm{MON},\spadesuit}$ from the Newton polygon of the last coefficient, more faithful to Villamayor's original philosophy. This invariant strictly increases in some configurations, the Moh-Hauser jumping phenomenon, but the authors prove it eventually decreases below its earlier value, which terminates the algorithm.

The obstacle in ambient dimension four is structural. After cleaning, $a_{p^e}=x^r\{g_x(y,z)+x\,\omega(x,y,z)\}$ with respect to parameters $(w,x,y,z)$, so the residual data $g_x(y,z)$ lives on the surface $\{w=x=0\}$: the usual order on a locus of dimension at least two can strictly increase after a point blow-up, and the order-based mechanism that made the dimension-3 analysis work on a curve stops functioning. The authors' proposed remedy is to use weak versions of the residual invariants, subtracting the contributions of the boundary divisor, at the price of reintroducing the jumping phenomenon in a setting where the jump configurations are far less controllable and where the permissible centers are expected to be more intricate combinatorial loci. The authors state that the remaining task for their approach is precisely to establish resolution in the monomial case, i.e. Step 2 beyond dimension 3, and that the dimension-4 analysis and the global version of the local procedure will be discussed elsewhere.

Problem Statement

In the Idealistic Filtration Program over a perfect field of characteristic $p>0$ (stated for algebraically closed fields and extended by Galois descent, as in the source), Step 1, the reduction of the general case to the monomial case, is established in arbitrary dimension, and Step 2, resolution in the monomial case, is established only in the local, $\tau=1$ setting of ambient dimension 3. Establish Step 2 in ambient dimension four and higher, ultimately in arbitrary dimension: given a triplet $(W,\mathbb{R},E)$ in the monomial case in the local analytic setting of the source, construct an explicit algorithm of permissible blow-ups that terminates after finitely many steps and reaches the tight monomial case, where resolution is completed by the classical procedure. Following the authors' inductive scheme in terms of $\tau$ (Section 3.2 of the source), the essential core is the case $\tau=1$: their own "final remaining task" is the procedure that reaches the tight monomial case after reaching the monomial case with $\tau=1$, in ambient dimension four and higher. Since the details of the similarity reduction of the cases $\tau\geq 2$ in dimension four and higher were deferred by the authors to a separate publication that never appeared, a complete establishment of Step 2 in dimension four and higher must in addition supply these deferred details of the $\tau\geq 2$ reduction (or invoke them if established elsewhere).

Concretely, the difficulty of Step 2 in dimension four and higher concentrates in four points: (a) the weak, boundary-corrected versions of the residual invariants built from the cleaned last coefficient $a_{p^e}=x^r\{g_x(y,z)+x\,\omega(x,y,z)\}$ of the Weierstrass form of the single leading generator, whose usual-order counterparts can increase after blow-up on loci of dimension at least two; (b) the permissible centers of blow-up, which are smooth and compatible with the boundary divisor and which in dimension four are expected to be more intricate combinatorial loci than in dimension three; (c) the Moh-Hauser jumping phenomenon, in all configurations in which the invariant can increase; and (d) eventual decrease, the invariant falling strictly below its earlier value after finitely many further transformations, whence termination. The object of study is the local procedure in the monomial case; its globalization, explicitly deferred by the authors to separate work, is a separate direction.

References

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