REVIEW 2 minor 12 references
Bounds on the F-Pure Threshold of Isolated Hypersurface Singularities
T0 review · 0 major / 2 minor · reviewed 2026-06-28 · grok-4.3
Pith's one-line read The F-pure threshold of isolated hypersurface singularities is bounded in terms of their Milnor and Tjurina numbers.
desk verdict This note gives bounds on F-pure threshold via Milnor/Tjurina numbers and semigroup generators for curves, as a positive-char Igusa analogue, with applications to complex LCT and Briancon-Skoda. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The relations between the F-pure threshold and the Milnor and Tjurina numbers for hypersurface singularities, together with the value semigroup and its generators for the curve case.
What would settle it
Compute the F-pure threshold and the Milnor number for a specific isolated hypersurface singularity, such as the cusp defined by x squared plus y cubed equals zero in characteristic p, and check whether the computed F-pure threshold lies within the claimed bounds from the Milnor number.
Extended reading notes
Core claim
For an isolated hypersurface singularity over an algebraically closed field of positive characteristic, the F-pure threshold satisfies bounds expressed using the Milnor number and the Tjurina number. In the case of curve singularities, the F-pure threshold is bounded, and frequently computed exactly, by the minimal generators of the associated value semigroup. This construction yields a positive-characteristic analogue of Igusa's formula for the log canonical threshold and produces bounds on the log canonical threshold and Briancon-Skoda exponent for the corresponding complex singularities.
Load-bearing premise
The singularities under consideration are isolated hypersurface singularities over an algebraically closed field of positive characteristic.
Editorial extensions
If this is right
- The F-pure threshold admits explicit computations for many curve singularities via their value semigroups.
- A positive-characteristic analogue of Igusa's formula holds for the log canonical threshold.
- Bounds are obtained on the log canonical threshold of complex isolated hypersurface singularities.
- Bounds are obtained on the Briancon-Skoda exponent of complex isolated hypersurface singularities.
Reading between the lines
- These bounds might allow lifting computations from positive characteristic to characteristic zero for certain invariants.
- The approach could extend to non-isolated singularities or other types of singularities if similar relations hold.
- Comparisons between F-pure threshold and other thresholds like F-threshold could be explored using these bounds.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript establishes bounds on the F-pure threshold of isolated hypersurface singularities over an algebraically closed field of positive characteristic, expressed in terms of the Milnor and Tjurina numbers. For curve singularities it further provides bounds (and often explicit values) in terms of the generators of the value semigroup, yielding a positive-characteristic analogue of Igusa's formula; applications to bounds on the log canonical threshold and Briançon-Skoda exponent of complex isolated hypersurface singularities are derived.
Significance. If the stated bounds are valid, the work supplies concrete relations between the F-pure threshold and classical numerical invariants, with the semigroup description for curves offering a route to explicit calculations. The reduction-mod-p applications to characteristic-zero invariants are a clear strength when the inequalities are effective. The paper rests on the standard hypotheses under which the cited invariants are defined and the F-pure threshold is known to behave well.
minor comments (2)
- [Abstract] Abstract and introduction: the phrase 'often explicit computations' for the curve case should be clarified by indicating the precise conditions on the value-semigroup generators under which an explicit value is obtained rather than a bound.
- [Applications] The applications section should include a short remark on how the positive-characteristic bounds translate to effective statements for the log canonical threshold and Briançon-Skoda exponent after reduction mod p, including any dependence on the choice of model.
Simulated Author's Rebuttal
We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were raised in the report, so we have no specific points requiring rebuttal or clarification. We will incorporate any minor editorial suggestions in the revised version.
Circularity Check
No significant circularity; derivation self-contained
full rationale
The paper derives bounds on the F-pure threshold of isolated hypersurface singularities from the Milnor and Tjurina numbers (standard external invariants) and, for curves, from generators of the value semigroup as a positive-characteristic analogue of Igusa's formula. These relations rest on the well-defined behavior of the F-pure threshold under the stated hypotheses (isolated hypersurface over algebraically closed field of positive characteristic) without any reduction of the claimed bounds to fitted parameters, self-definitions, or load-bearing self-citations within the paper. The central claims remain independent of the paper's own equations by construction.
Assumptions & free parameters
Cite this review
Pith. "Pith review of Bounds on the F-Pure Threshold of Isolated Hypersurface Singularities." pith.science (2026). https://pith.science/paper/FPD3T5US
@misc{pith2026260604014,
author = {Pith},
title = {Pith review of: Bounds on the F-Pure Threshold of Isolated Hypersurface Singularities},
year = {2026},
howpublished = {\url{https://pith.science/paper/FPD3T5US}},
note = {Machine review of arXiv:2606.04014}
}
abstract
In this note, we obtain bounds for the $F$-pure threshold of isolated hypersurface singularities over an algebraically closed field of positive characteristic in terms of classical singularity invariants, notably the Milnor and Tjurina numbers. For curve singularities, we show that the $F$-pure threshold admits bounds, and often explicit computations, in terms of the generators of the associated value semigroup, yielding a positive-characteristic analogue of Igusa's formula for the log canonical threshold. As applications, we derive bounds on the log canonical threshold and the Briancon-Skoda exponent of complex isolated hypersurface singularities.
Reference graph
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Reviewed June 28, 2026 · model on record in the stance chip above.
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