Kawanoue and Matsuki established, at the local level and over an algebraically closed base field, an algorithm for resolution of singularities of an idealistic filtration in ambient dimension 3 in positive characteristic, built on their Idealistic Filtration Program (IFP). The mechanism has two steps: reduction of the general case to the so-called monomial case (which works in arbitrary dimension), and solution of the monomial case, which the authors completed only in dimension 3 by adapting the method of Benito–Villamayor, with an explicit invariant that strictly drops after each blow-up. The authors state that establishing the algorithm in ambient dimension 4 or above remains an open problem. This problem asks for that extension: a center-choice rule with invariants and a proven termination mechanism solving the monomial case (and with it the full algorithm) for idealistic filtrations on smooth varieties of dimension at least 4 in positive characteristic, at the local level at which the source states the algorithm and its open problem. A solution in dimension 4 would in particular yield embedded resolution of singularities of 3-folds in nonsingular ambient 4-folds in positive characteristic, which is open by any known method. The problem remains open: the authors' own 2018 follow-up stops at dimension 3, and no other route has reached embedded resolution of 3-folds in positive characteristic.

Background

Resolution of singularities asks for a proper birational map from a nonsingular variety onto a given algebraic variety; embedded resolution asks, for a variety XX embedded as a closed subvariety of a smooth ambient variety WW, for a sequence of blow-ups of WW along smooth centers inside the singular locus that makes the strict transform of XX smooth and transversal to the exceptional divisor. In characteristic zero this is Hironaka's classical theorem, and the modern algorithms (e.g. Bierstone–Milman) are constructive and functorial: they attach to the pair an upper-semicontinuous invariant whose maximum locus is a smooth permissible center, and which strictly drops after each blow-up, so the process terminates. The induction runs on dimension via a hypersurface of maximal contact — a smooth hypersurface containing all the relevant data, which exists in characteristic zero because Taylor expansions allow reducing the order of generators.

In positive characteristic the hypersurface of maximal contact generally fails to exist (derivatives of pp-th powers vanish), and resolution of singularities remains one of the main open problems of algebraic geometry: non-embedded resolution is known only up to dimension 3 (Cossart–Piltant in two papers, independently Cutkosky, extended by Cossart–Piltant to all quasi-excellent schemes of dimension three and arbitrary characteristic), and embedded resolution is known only for surfaces in smooth 3-folds (classically due to Abhyankar; a canonical version for excellent two-dimensional schemes was given by Cossart–Jannsen–Saito). Embedded resolution of 3-folds in a nonsingular ambient 4-fold in positive characteristic is open, even by methods outside the program discussed here.

The Idealistic Filtration Program (IFP) of Kawanoue and Matsuki is a characteristic-free reformulation of the resolution algorithm. An idealistic filtration R=⋃a≥0(Ia,a)R=\bigcup_{a\ge0}(I_a,a) on a smooth variety WW is a nested family of coherent ideal sheaves I0=OW⊃I1⊃I2⊃⋯I_0=\mathcal{O}_W\supset I_1\supset I_2\supset\cdots in which aa records a 'level'; it is said to be of i.f.g. (idealistically finitely generated) type when finitely many pairs (f,a)(f,a) generate it under sums, products and powers, and it is studied through its closure under differential operators (D\mathcal{D}-saturation) and under taking radicals (R\mathcal{R}-saturation). Its singular locus is Sing(R)={P∈W∣ordP(Ia)≥a ∀a}\mathrm{Sing}(R)=\{P\in W\mid \mathrm{ord}_P(I_a)\ge a\ \forall a\}, where ordP\mathrm{ord}_P is the order of an ideal at a point (the largest nn with I⊂mPnI\subset\mathfrak{m}_P^n). Resolution for the triplet (W,R,E)(W,R,E), where EE is a simple normal crossing divisor (the boundary), means a finite sequence of blow-ups with smooth centers contained in Sing(R)\mathrm{Sing}(R) and transversal to EE, with the natural transform rule for RR and EE, ending with empty singular locus; solving this in a functorial way implies embedded resolution (take RR generated by the ideal of XX at level 1). In place of hypersurfaces of maximal contact the IFP uses leading generator systems — systems of elements of minimal order extracted from the differentially saturated filtration — and the algorithm proceeds by induction on the invariant σ\sigma (essentially the order of the filtration). The mechanism splits into two steps: Step 1 reduces the general case to the monomial case, the situation where the filtration is generated by monomials in parameters defining components of the boundary (precisely: the SITUATION of §4.4 of the Kawanoue–Matsuki dimension-3 paper, arXiv:1205.4556, in the terminology of their Idealistic Filtration Program foundation papers); Step 1 works in arbitrary dimension. Step 2, solving the monomial case, is easy in characteristic zero but subtle in positive characteristic, because invariants built from orders can strictly increase after a blow-up (the Abhyankar–Moh pathology of the residual order, and the 'Kangaroo' jumping phenomenon studied by Hauser and Moh). Kawanoue–Matsuki solved Step 2 in ambient dimension 3 by incorporating ideas of Benito–Villamayor: they introduced an explicit invariant invMON\mathrm{inv}_{\mathrm{MON}} that strictly drops after each blow-up, making the termination argument effective. This established the local algorithm in dimension 3 over an algebraically closed field (a global version, and descent to non-perfect base fields, were deferred to future work by the authors).

Problem Statement

Over an algebraically closed base field of positive characteristic, establish the Idealistic Filtration Program's algorithm for resolution of singularities of an idealistic filtration in ambient dimension 4 and above — completing, for smooth varieties WW of dimension n≥4n\ge4 and triplets (W,R,E)(W,R,E) with RR an idealistic filtration of i.f.g. type and EE a simple normal crossing divisor, the step that the 2012 Kawanoue–Matsuki dimension-3 paper leaves open. Since Step 1 (reduction of the general case to the monomial case) already works in arbitrary dimension, the requested core is Step 2 in dimension n≥4n\ge4: a center-choice rule for the monomial case — in the sense of the SITUATION of §4.4 of the dimension-3 paper, carried over to n≥4n\ge4 in the terminology of the Idealistic Filtration Program — together with a proof of termination: a well-founded argument that the process cannot continue indefinitely, so that every such triplet is resolved by a finite sequence of permissible blow-ups (smooth centers inside Sing(R)\mathrm{Sing}(R), transversal to the boundary, with the program's transform rules), ending with Sing(R)=∅\mathrm{Sing}(R)=\emptyset. As in the source, the algorithm is requested at the local level; the authors describe the local-to-global adjustments as 'not so difficult, though technical and rather lengthy' and announced a detailed discussion that never appeared, so globalization — like the separately deferred descent to non-perfect base fields — is not part of the open core. A solution in ambient dimension 4 would, after globalization, yield embedded resolution of singularities of 3-folds in nonsingular ambient 4-folds in positive characteristic; a solution in arbitrary dimension would give a resolution algorithm uniform in the characteristic.

References

  1. Hiraku Kawanoue and Kenji Matsuki, Resolution of singularities of an idealistic filtration in dimension 3 after Benito–Villamayor, Advanced Studies in Pure Mathematics 70 (Minimal Models and Extremal Rays, Kyoto 2011), Mathematical Society of Japan, pp. 115–214 (2016); arXiv:1205.4556; DOI 10.2969/aspm/07010115. https://arxiv.org/abs/1205.4556
  2. Hiraku Kawanoue and Kenji Matsuki, A new strategy for resolution of singularities in the monomial case in positive characteristic, Revista Matemática Iberoamericana 34 (2018), 1229–1276; arXiv:1507.05195; DOI 10.4171/rmi/1023. https://arxiv.org/abs/1507.05195
  3. Hiraku Kawanoue, Toward resolution of singularities over a field of positive characteristic. Part I. Foundation; the language of the idealistic filtration, Publications of the Research Institute for Mathematical Sciences 43 (2007), no. 3, 819–909; DOI 10.2977/prims/1201012043. https://doi.org/10.2977/prims/1201012043
  4. Hiraku Kawanoue and Kenji Matsuki, Toward resolution of singularities over a field of positive characteristic (The idealistic filtration program). Part II. Basic invariants associated to the idealistic filtration and their properties, Publications of the Research Institute for Mathematical Sciences 46 (2010), no. 2, 359–422; DOI 10.2977/prims/12. https://doi.org/10.2977/prims/12
  5. Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings, Journal of Algebra 320 (2008), no. 3, 1051–1082; DOI 10.1016/j.jalgebra.2008.03.032. https://doi.org/10.1016/j.jalgebra.2008.03.032
  6. Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. II, Journal of Algebra 321 (2009), no. 7, 1836–1976; DOI 10.1016/j.jalgebra.2008.11.030. https://doi.org/10.1016/j.jalgebra.2008.11.030
  7. Vincent Cossart and Olivier Piltant, Resolution of singularities of arithmetical threefolds, Journal of Algebra 529 (2019), 268–535; arXiv:1412.0868; DOI 10.1016/j.jalgebra.2019.02.017. https://arxiv.org/abs/1412.0868
  8. Steven Dale Cutkosky, Resolution of singularities for 3-folds in positive characteristic, American Journal of Mathematics 131 (2009), no. 1, 59–127; DOI 10.1353/ajm.0.0036. https://doi.org/10.1353/ajm.0.0036
  9. Angélica Benito and Orlando Villamayor U., Techniques for the study of singularities with applications to resolution of 2-dimensional schemes, Mathematische Annalen 353 (2012), no. 3, 1037–1068; DOI 10.1007/s00208-011-0709-5. https://doi.org/10.1007/s00208-011-0709-5
  10. Angélica Benito and Orlando Villamayor U., Monoidal transforms and invariants of singularities in positive characteristic, Compositio Mathematica 149 (2013), no. 8, 1267–1311; DOI 10.1112/s0010437x1200084x. https://doi.org/10.1112/s0010437x1200084x
  11. Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Inventiones Mathematicae 128 (1997), no. 2, 207–302; DOI 10.1007/s002220050141. https://doi.org/10.1007/s002220050141
  12. Herwig 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), no. 1, 1–30; DOI 10.1090/S0273-0979-09-01274-9. https://doi.org/10.1090/S0273-0979-09-01274-9
  13. Vincent Cossart, Uwe Jannsen, and Shuji Saito, Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes, arXiv:0905.2191 (2009; latest version 2013). https://arxiv.org/abs/0905.2191
  14. Michael Temkin, Relative and logarithmic resolution of singularities, in: D. Abramovich, A. Frühbis-Krüger, M. Temkin, J. Włodarczyk (eds.), New Techniques in Resolution of Singularities, Oberwolfach Seminars 55, Birkhäuser/Springer, Cham, 2023; arXiv:2303.00407; DOI 10.1007/978-3-031-32115-3_5. https://arxiv.org/abs/2303.00407
  15. Vincent Cossart, Olivier Piltant, and Bernd Schober, Constancy of the Hilbert–Samuel function, Nagoya Mathematical Journal 256 (2024), 938–952; arXiv:2305.09242; DOI 10.1017/nmj.2024.13. https://arxiv.org/abs/2305.09242
  16. Yi Hu, Universal characteristic-free resolution of singularities, I, arXiv:2507.21400 (2025). https://arxiv.org/abs/2507.21400