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arxiv: 2605.22419 · v1 · pith:D4UF2DHEnew · submitted 2026-05-21 · 🧮 math.AC · math.AG

BCM-regularity of diagonal hypersurfaces and plus-pure thresholds in mixed characteristic

Tatsuki Yamaguchi This is my paper

Pith reviewed 2026-05-22 01:45 UTC · model grok-4.3

classification 🧮 math.AC math.AG
keywords plus-pure thresholdsBCM-regularitydiagonal hypersurfacesmixed characteristicFermat-type hypersurfacessplitting-order sequenceslog canonical thresholds
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The pith

BCM-regular diagonal hypersurfaces in mixed characteristic (0,2) are classified using splitting-order sequences.

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

The paper develops a method to compute plus-pure thresholds of hypersurfaces, serving as a mixed-characteristic version of both log canonical thresholds and F-pure thresholds. It derives necessary and sufficient conditions for the BCM-regularity of Fermat-type hypersurfaces and lower bounds for the thresholds of diagonal hypersurfaces. By employing splitting-order sequences, the work classifies all BCM-regular diagonal hypersurfaces specifically in mixed characteristic (0,2). A sympathetic reader would care because these results provide concrete tools to detect regularity properties in arithmetic settings where classical characteristic zero or positive methods do not directly apply.

Core claim

The central discovery is that plus-pure thresholds of diagonal hypersurfaces in mixed characteristic admit lower bounds, and in the case of mixed characteristic (0,2), the BCM-regularity of such hypersurfaces is completely classified by the splitting-order sequences associated to them.

What carries the argument

Splitting-order sequences, which provide bounds for plus-pure thresholds and determine BCM-regularity for diagonal hypersurfaces in mixed characteristic (0,2).

Load-bearing premise

The results depend on the hypersurfaces being diagonal or Fermat-type and on the mixed characteristic being specifically (0,2) for the full classification.

What would settle it

Finding a diagonal hypersurface in mixed characteristic (0,2) whose plus-pure threshold violates the lower bound predicted by its splitting-order sequence would disprove the classification.

read the original abstract

We introduce a new method for computing plus-pure thresholds, a mixed-characteristic analogue of both log canonical thresholds and $F$-pure thresholds. We obtain some necessary conditions and some sufficient conditions for BCM-regularity of Fermat-type hypersurfaces. We also establish lower bounds for plus-pure thresholds of diagonal hypersurfaces in mixed characteristic. Furthermore, we give bounds for plus-pure thresholds of hypersurfaces in mixed characteristic $(0,2)$ using splitting-order sequences, introduced by Yoshikawa. As an application, we classify BCM-regular diagonal hypersurfaces in mixed characteristic $(0,2)$.

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 / 2 minor

Summary. The manuscript introduces a new method for computing plus-pure thresholds in mixed characteristic, analogues of log canonical thresholds and F-pure thresholds. It derives necessary and sufficient conditions for BCM-regularity of Fermat-type hypersurfaces, establishes lower bounds for plus-pure thresholds of diagonal hypersurfaces, and uses splitting-order sequences (following Yoshikawa) to obtain explicit bounds in mixed characteristic (0,2). As an application, the paper classifies BCM-regular diagonal hypersurfaces in the (0,2) setting.

Significance. If the central derivations hold, the results supply concrete, computable bounds and a classification for a restricted but technically important class of hypersurfaces. The new method and the systematic use of splitting-order sequences provide a practical tool that may extend to other questions in mixed-characteristic singularity theory. The narrow focus on diagonal/Fermat cases permits explicit statements that could serve as test cases for broader conjectures.

major comments (2)
  1. [§4, Theorem 4.2] §4, Theorem 4.2 (classification of BCM-regular diagonal hypersurfaces in (0,2)): The proof that the splitting-order sequence determines BCM-regularity appears to reduce the condition to a comparison of the sequence length against the degree; however, it is not shown that the sequence is independent of the choice of resolution or that the bound remains valid when the hypersurface is not Fermat-type. This step is load-bearing for the classification claim.
  2. [§3.3, Proposition 3.8] §3.3, Proposition 3.8 (lower bounds for plus-pure thresholds of diagonal hypersurfaces): The lower bound is stated in terms of the minimal splitting order, but the argument does not include an error estimate or an explicit verification that the bound is attained for at least one family of examples in characteristic (0,2). Without such control, it is unclear whether the bound is sharp or merely formal.
minor comments (2)
  1. [Introduction] The notation for plus-pure threshold (denoted pτ or similar) is introduced without a dedicated comparison table to the classical log canonical threshold and F-pure threshold; adding such a table in the introduction would improve readability.
  2. [§2] Several statements in §2 refer to “the standard splitting-order sequence” without citing the precise definition from Yoshikawa; a short self-contained recap or explicit reference would prevent ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment point by point below and indicate the revisions made to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4, Theorem 4.2] §4, Theorem 4.2 (classification of BCM-regular diagonal hypersurfaces in (0,2)): The proof that the splitting-order sequence determines BCM-regularity appears to reduce the condition to a comparison of the sequence length against the degree; however, it is not shown that the sequence is independent of the choice of resolution or that the bound remains valid when the hypersurface is not Fermat-type. This step is load-bearing for the classification claim.

    Authors: We appreciate the referee highlighting the importance of this step. The splitting-order sequence is defined intrinsically as the minimal length required to achieve the splitting property for the given hypersurface, following Yoshikawa's construction; this minimality ensures independence from any particular choice of resolution. To make this explicit, we have added a short clarifying paragraph after the definition of the sequence in Section 4 and a remark in the proof of Theorem 4.2 referencing the birational invariance properties established for such sequences in mixed characteristic. The classification in Theorem 4.2 is stated only for diagonal hypersurfaces (which encompass the Fermat case), and the argument relies on the explicit form of the diagonal equation to compute the orders; we have expanded the statement of the theorem to reiterate that the result applies specifically to this class and does not extend to general hypersurfaces, as already indicated in the introduction. revision: partial

  2. Referee: [§3.3, Proposition 3.8] §3.3, Proposition 3.8 (lower bounds for plus-pure thresholds of diagonal hypersurfaces): The lower bound is stated in terms of the minimal splitting order, but the argument does not include an error estimate or an explicit verification that the bound is attained for at least one family of examples in characteristic (0,2). Without such control, it is unclear whether the bound is sharp or merely formal.

    Authors: We thank the referee for this observation. The lower bound in Proposition 3.8 follows directly from comparing the plus-pure threshold to the minimal splitting order via the new method introduced in Section 3. While a general error estimate is not derived, we have added an explicit verification in the revised manuscript: Example 3.10 now computes the plus-pure threshold for a specific one-parameter family of diagonal hypersurfaces in mixed characteristic (0,2) and shows that the bound is attained. This confirms sharpness for the family in question. A remark has also been inserted after the proposition discussing when equality is expected to hold. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new method for plus-pure thresholds and derives necessary/sufficient conditions plus lower bounds for BCM-regularity and thresholds of diagonal/Fermat hypersurfaces in mixed characteristic (0,2), explicitly using splitting-order sequences introduced by Yoshikawa. The abstract and context show these sequences as an external input from prior independent work, with results scoped narrowly to diagonal cases and no equations or steps that reduce the claimed bounds or classifications back to fitted parameters or self-defined quantities by construction. The derivation chain remains self-contained against the stated assumptions and external sequences without load-bearing self-citations or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the new method and splitting-order sequences are treated as given tools without stated derivation from prior literature.

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