Resolution of Singularities in Positive Characteristic

The singularity may return; its history cannot.

Where numerical descent fails, historical descent begins.

Program Abstract

Let k be a perfect field of characteristic p > 0. This project constructs a canonical resolution of singularities in arbitrary dimension by finite sequences of ordinary blowups with regular permissible centres. More precisely, for a smooth k-scheme W equipped with an ordered simple-normal-crossings boundary E, it constructs strong principalizations of coherent ideals and strong embedded resolutions of reduced closed subschemes X ⊂ W. The construction preserves the boundary condition at every stage, is an isomorphism over the initially resolved locus, is independent of local presentations and ambient embeddings, and is functorial under open, smooth, and étale morphisms and extensions of the perfect ground field.

The local theory replaces a marked Rees algebra by its differential–integral saturation. Total Hasse operators and boundary coefficient cubes recover exact order, tangency, and multiple-incidence information without factorial denominators. Filtered Rees complexes retain the higher syzygies discarded by order functions, while semilinear Frobenius–Hasse sources, primitive–cobar complexes, and actual torsor–action data record the inseparable and additive phenomena that arise in the wild locus. Explicit transformation identities transport this complete semantic packet through permissible blowups and separate every failure of strict transport into a supported defect, a controlled change of type, or a genuinely new semilinear occurrence.

The central innovation is a termination principle that does not require any classical resolution invariant to decrease after every blowup. Positive-characteristic singularities may return, and residual order may increase, but every return is assigned a canonical address in a finite exceptional ancestry. Owner, parent, quotient, trace, and reopening data are transported across successive resolution blocks, and a no-reset theorem shows that a discharged address cannot be recreated by recomputation, change of presentation, or passage between local models. Thus each completed block replaces a nonempty multiset of active occurrences by strict descendants in a single well-founded dependent order. Canonical incidence portfolios then serialize the local resolution words and descend them to a global sequence of ordinary blowups.

This yields finite termination, terminal regularity and normal crossings, strong principalization, reduced embedded resolution, and—by stable re-embedding and descent—an intrinsic functorial resolution. Conceptually, the construction replaces numerical descent by historical descent: the singularity may return, but the mathematical obligation responsible for its return cannot.

Manuscripts

Resolution of Singularities in Positive Characteristic:

Frobenius–Hasse Towers and Exceptional-History Descent

Collected Edition, Parts I–IX · August 2026 · 798 pp.

Other Notes

Note I

A Three-Centre Clearing Block along a Narasimhan-Type Equimultiple Curve in Characteristic Two

9 pp.

A three-centre normal-flat clearing block in characteristic two: two point blowups followed by the transformed equimultiple curve clear multiplicity two above Γ \ {0}.

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Note II

Trace–Conormal Preparation of a Regular Curve against a Finite Packet of Regular Subschemes

12 pp.

Introduces a rootwise trace–conormal profile and gives a canonical finite point-preparation of a regular curve relative to a finite packet of regular subschemes and an SNC boundary.

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Acknowledgments

The author gratefully acknowledges the intellectual environment and research resources at Princeton University, and sincerely thanks the Princeton CIS Lab for access to computational-assisted resources used in this project. All explicit examples, counterexamples, and accompanying code are available upon request.

Contact

Chenxiao Tian

ct3471@alumni.princeton.edu

ct3471@princeton.edu

The Nine-Part Architecture

The nine parts form a single proof in four layers: Parts I–III construct and transport the local semantic packet; Parts IV–V close the defect calculus and globalize the local words; Part VI isolates termination and reconstruction; Parts VII–IX realize and assemble the canonical certificate.