Bounds on the F-Pure Threshold of Isolated Hypersurface Singularities

Abstract

We obtain bounds for the F-pure threshold of isolated hypersurface singularities over an algebraically closed field of positive characteristic p in terms of classical singularity invariants. For an irreducible plane curve whose value semigroup has first two generators β₀,β₁, we show that the F-pure threshold is at most 1/β₀+1/β₁ and differs from it by less than 1/gcd(β₀,β₁)p, with equality when certain base-p digits add without carrying, and we give an example showing that higher weighted-order terms can change the F-pure threshold. We also conjecture a lower bound in terms of the Milnor number, and prove it for semi-quasi-homogeneous singularities, for all plane curves, and whenever the number of variables exceeds p. In particular, it holds in characteristic $2$. As applications, we reprove Igusa's formula for the log canonical threshold of a complex plane branch (with integer coefficients), prove the conjecture of Mustaţă, Takagi and Watanabe for such branches, and bound the log canonical threshold of complex isolated hypersurface singularities.

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BibTeX RIS

BibTeX

@misc{svoray2026bounds,
  title = {{Bounds on the F-Pure Threshold of Isolated Hypersurface Singularities}},
  author = {Yotam Svoray},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.04014},
  eprint = {2606.04014},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.04014},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Bounds on the F-Pure Threshold of Isolated Hypersurface Singularities
AU  - Yotam Svoray
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2606.04014
UR  - https://arxiv.org/abs/2606.04014
AB  - We obtain bounds for the F-pure threshold of isolated hypersurface singularities over an algebraically closed field of positive characteristic p in terms of classical singularity invariants. For an irreducible plane curve whose value semigroup has first two generators β_0,β_1, we show that the F-pure threshold is at most 1/β_0+1/β_1 and differs from it by less than 1/gcd(β_0,β_1)p, with equality when certain base-p digits add without carrying, and we give an example showing that higher weighted-order terms can change the F-pure threshold. We also conjecture a lower bound in terms of the Milnor number, and prove it for semi-quasi-homogeneous singularities, for all plane curves, and whenever the number of variables exceeds p. In particular, it holds in characteristic $2$. As applications, we reprove Igusa's formula for the log canonical threshold of a complex plane branch (with integer coefficients), prove the conjecture of Mustaţă, Takagi and Watanabe for such branches, and bound the log canonical threshold of complex isolated hypersurface singularities.
ER  -

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