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arxiv: 2509.07217 · v2 · submitted 2025-09-08 · 🧮 math.AC · math.AG

Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings

Marta Benozzo , Vignesh Jagathese , Vaibhav Pandey , Pedro Ram\'irez-Moreno , Karl Schwede , Prashanth Sridhar This is my paper

Pith reviewed 2026-05-18 18:36 UTC · model grok-4.3

classification 🧮 math.AC math.AG
keywords plus-pure thresholdF-pure thresholdmixed characteristichypersurfacesperfectoid pure singularityramificationelliptic curvestest ideals
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The pith

The plus-pure threshold of hypersurfaces in mixed characteristic limits to the F-pure threshold as the base DVR ramifies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies the plus-pure thresholds of hypersurfaces over regular rings in mixed characteristic. It establishes that these thresholds converge to the F-pure thresholds when the base discrete valuation ring is ramified to higher degrees. The authors prove that certain extremal singularities known from positive characteristic cannot reach the same threshold values in the unramified mixed characteristic setting. They also derive bounds for hypersurfaces tied to elliptic curves and show that these thresholds differ from both the F-pure threshold in characteristic p and the log canonical threshold in characteristic zero. The work further demonstrates that p times the plus-pure threshold is not always a jumping number.

Core claim

The paper shows that the plus-pure threshold limits to the F-pure threshold as the base DVR ramifies. In particular, given a complete unramified regular local ring of mixed characteristic p>0, the equation f^p + p^2 g does not define a perfectoid pure singularity for any f and g. It also computes explicit bounds on the plus-pure thresholds of hypersurfaces related to elliptic curves, yielding examples where the threshold is neither the corresponding F-pure threshold nor the log canonical threshold and where p times the threshold fails to be a jumping number.

What carries the argument

The plus-pure threshold, a numerical invariant for singularities in mixed characteristic obtained via test ideal constructions or limits over ramified extensions of the base DVR.

If this is right

  • Analogs of positive characteristic extremal singularities cannot attain the same extremal plus-pure threshold values in the unramified setting.
  • Equations that admit p-th roots modulo p^2 have their plus-pure thresholds bounded in a controlled way.
  • Hypersurfaces associated to elliptic curves have plus-pure thresholds distinct from both the F-pure threshold and the log canonical threshold.
  • Multiples of the plus-pure threshold by p are not necessarily jumping numbers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Ramification of the base ring appears necessary to recover the full range of purity thresholds known in positive characteristic.
  • Separate classification tools may be needed for singularities in unramified versus ramified mixed characteristic rings.
  • The elliptic curve examples could be generalized to other curves to produce further thresholds that interpolate between characteristic p and zero.

Load-bearing premise

The plus-pure threshold is defined so that it varies continuously under ramification of the base DVR while the rings stay regular after adjoining p-th roots or reducing modulo p squared.

What would settle it

An explicit pair of polynomials f and g such that f^p + p^2 g defines a perfectoid pure singularity in a complete unramified regular local ring of mixed characteristic p>0 would disprove the non-purity statement.

read the original abstract

We study the plus-pure threshold (ppt) of hypersurfaces in mixed characteristic. We show that the ppt limits to the $F$-pure threshold (fpt) as we ramify the base DVR. Additionally, we show that analogs of some positive characteristic extremal singularities cannot attain the same `extremal' ppt values in the unramified setting. We also study equations which have controlled ramification when we adjoin their $p$-th roots as well as equations which admit $p$-th roots modulo $p^2$ (or modulo other values), bounding their ppts. In particular, given a complete unramified regular local ring of mixed characteristic $p>0$, $f^p + p^2 g$ does not define a perfectoid pure singularity for any $f$ and $g$. Finally, we compute bounds on the ppt of hypersurfaces related to elliptic curves. This gives examples where the ppt is neither the corresponding fpt in characteristic $p > 0$ nor the lct in characteristic zero. This also provides examples where $p$ times the ppt is not a jumping number, in stark contrast with the characteristic $p > 0$ picture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the plus-pure threshold (ppt) of hypersurfaces in mixed characteristic regular rings. It proves that the ppt converges to the F-pure threshold (fpt) as the base DVR is ramified. It provides bounds on the ppt for hypersurfaces with controlled ramification when adjoining p-th roots and for those admitting p-th roots modulo p^2. In particular, it shows that f^p + p^2 g does not define a perfectoid pure singularity in complete unramified regular local rings of mixed characteristic. It also gives bounds for hypersurfaces related to elliptic curves, yielding examples where the ppt is distinct from both the positive characteristic fpt and the characteristic zero lct, and where p times the ppt is not a jumping number.

Significance. If the central claims hold, the work introduces and bounds the plus-pure threshold as a mixed-characteristic singularity invariant, establishing its convergence to the fpt under ramification and supplying explicit examples (including elliptic-curve hypersurfaces) that separate it from both the fpt and the lct. The non-attainment result for f^p + p^2 g in the unramified setting and the observation that p·ppt need not be a jumping number are concrete contributions that could guide the development of test-ideal and perfectoid techniques in mixed characteristic.

major comments (2)
  1. [§4] §4, the limit statement: the argument that ppt approaches fpt under ramification of the base DVR assumes continuity of the underlying test-ideal construction, but the manuscript does not verify that regularity is preserved after adjoining p-th roots in the families considered (e.g., the elliptic-curve hypersurfaces).
  2. [Theorem 6.3] Theorem 6.3 (or the statement on f^p + p^2 g): the claim that such an equation never yields a perfectoid pure singularity in the unramified case is load-bearing for the non-attainment results; the proof sketch relies on the unramified hypothesis, yet the precise role of the p^2 coefficient versus higher powers is not compared with the ramified setting.
minor comments (3)
  1. [Introduction] The definition of the plus-pure threshold is introduced via a limit process; a short paragraph recalling the precise construction (test ideals or perfectoid analogs) at the first appearance would aid readers unfamiliar with the mixed-characteristic literature.
  2. [Notation] Notation for ramification index and the base DVR is used consistently but never tabulated; a small notation table or list of standing assumptions would improve readability.
  3. [Final section] The elliptic-curve examples in the final section are computed for specific Weierstrass models; stating the precise equations and the resulting ppt bounds in a table would make the contrast with fpt and lct easier to verify.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§4] §4, the limit statement: the argument that ppt approaches fpt under ramification of the base DVR assumes continuity of the underlying test-ideal construction, but the manuscript does not verify that regularity is preserved after adjoining p-th roots in the families considered (e.g., the elliptic-curve hypersurfaces).

    Authors: We agree that an explicit verification of regularity preservation would strengthen the continuity argument underlying the limit statement in §4. For the elliptic-curve hypersurfaces and similar families, the base ring is regular and the controlled ramification when adjoining p-th roots ensures the resulting ring remains regular (as the extension is finite and the singularity is isolated in a manner compatible with regularity). We will add a clarifying remark or short paragraph in §4 to verify this preservation explicitly for the families considered. revision: yes

  2. Referee: [Theorem 6.3] Theorem 6.3 (or the statement on f^p + p^2 g): the claim that such an equation never yields a perfectoid pure singularity in the unramified case is load-bearing for the non-attainment results; the proof sketch relies on the unramified hypothesis, yet the precise role of the p^2 coefficient versus higher powers is not compared with the ramified setting.

    Authors: The unramified hypothesis is indeed essential to the proof of Theorem 6.3, which relies on specific valuation and test-ideal properties in unramified regular local rings to show that f^p + p^2 g cannot define a perfectoid pure singularity. In the ramified setting the limit results of §4 show that the ppt approaches the fpt, allowing different behavior. We will expand the discussion of Theorem 6.3 to compare the role of the p^2 coefficient with higher powers (e.g., p^3 or above) and to contrast the unramified non-attainment with the ramified case. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's claims center on defining plus-pure thresholds via limit processes and test ideal constructions in mixed-characteristic regular rings, then proving that these limits recover the F-pure threshold under ramification of the base DVR, along with explicit bounds for hypersurfaces such as f^p + p^2 g and those tied to elliptic curves. These steps rely on independent ring-theoretic definitions and continuity properties under ramification, without any reduction of a derived quantity to a fitted parameter or self-referential input by construction. No self-citation is shown to be load-bearing for the central results, and the derivations remain self-contained against external algebraic benchmarks rather than renaming or smuggling ansatzes from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the regularity of the local rings, the existence of a well-defined plus-pure threshold via some test-ideal or perfectoid construction, and the ability to adjoin p-th roots while controlling ramification. No free parameters or invented entities with independent evidence are visible from the abstract.

axioms (2)
  • domain assumption The base ring is a complete regular local ring in mixed characteristic p>0
    Invoked for all statements about unramified and ramified cases.
  • domain assumption The plus-pure threshold is defined so that it varies continuously under ramification of the DVR
    Required for the limit statement to hold.
invented entities (1)
  • plus-pure threshold no independent evidence
    purpose: Singularity measure in mixed characteristic
    Central new or adapted invariant whose properties are being bounded.

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