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arxiv: 2506.23533 · v4 · submitted 2025-06-30 · 🧮 math.AG · math.AC

Liftings of ideals in positive characteristic to those in characteristic zero:Surface case

Shihoko Ishii This is my paper

Pith reviewed 2026-05-19 08:01 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords log discrepanciespositive characteristiccharacteristic zerosurface singularitiesmulti-idealslog canonical thresholdsminimal log discrepanciesskeletons
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The pith

Log discrepancies for smooth surfaces with multi-ideals form a discrete set, with positive-characteristic cases contained in the characteristic-zero sets.

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

The paper introduces the notion of lifting objects from positive characteristic to characteristic zero using skeletons. This lifting relates singularity invariants across characteristics and is used to show that log discrepancies for pairs of a smooth surface and a multi-ideal form a discrete set. Additionally, minimal log discrepancies and log canonical thresholds in positive characteristic are shown to be contained in those from characteristic zero.

Core claim

Using the notion of characteristic-zero lifting of an object in positive characteristic by means of skeletons, we relate invariants of singularities in positive characteristic to their counterparts in characteristic zero. As an application, we prove that the set of log discrepancies for pairs consisting of a smooth surface and a multi-ideal is discrete. We also show that the set of minimal log discrepancies and the set of log canonical thresholds of such pairs in positive characteristic are contained in the corresponding sets in characteristic zero.

What carries the argument

Characteristic-zero lifting via skeletons, which connects and preserves singularity invariants such as log discrepancies between positive and zero characteristics.

If this is right

  • Discreteness of the log discrepancy set allows only finitely many distinct values below any given bound for these surface pairs.
  • Minimal log discrepancies and thresholds in positive characteristic can be bounded or computed by reference to their lifts in characteristic zero.
  • The skeleton lifting yields a construction of Campillo's complex model for a plane curve in positive characteristic.

Where Pith is reading between the lines

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

  • The skeleton lifting method could be tested for generalization to higher-dimensional varieties to obtain similar discreteness results.
  • Results known for singularities in characteristic zero might transfer to positive characteristic via this correspondence.
  • The approach may connect to other lifting techniques used in deformation theory or resolution of singularities.

Load-bearing premise

The newly introduced characteristic-zero lifting of objects in positive characteristic by means of skeletons is well-defined, functorial, and preserves the relevant singularity invariants such as log discrepancies.

What would settle it

A counterexample consisting of a smooth surface over a positive-characteristic field together with a multi-ideal whose log discrepancies accumulate or take a value absent from any corresponding pair in characteristic zero.

read the original abstract

In this paper, we introduce the notion of a characteristic-zero lifting of an object in positive characteristic by means of ``skeletons''. Using this notion, we relate invariants of singularities in positive characteristic to their counterparts in characteristic zero. As an application, we prove that the set of log discrepancies for pairs consisting of a smooth surface and a multi-ideal is discrete. We also show that the set of minimal log discrepancies and the set of log canonical thresholds of such pairs in positive characteristic are contained in the corresponding sets in characteristic zero. Another application is the construction of Campillo's complex model of a plane curve in positive characteristic via the skeleton lifting method.

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 paper introduces the notion of characteristic-zero liftings of objects (ideals and pairs) in positive characteristic via 'skeletons'. It uses this construction to relate singularity invariants across characteristics. Applications include a proof that the set of log discrepancies for pairs (smooth surface, multi-ideal) is discrete, containment of the sets of minimal log discrepancies and log canonical thresholds from positive characteristic into the corresponding sets in characteristic zero, and a construction of Campillo's complex model for plane curves in positive characteristic.

Significance. If the skeleton lifting is rigorously shown to be well-defined, functorial, and to preserve log discrepancies and related invariants without introducing extraneous divisors or altering multiplicities, the work would provide a systematic bridge between singularity theory in positive and zero characteristics. This could enable transfer of discreteness and containment results that are otherwise hard to obtain directly in positive characteristic, with potential implications for the minimal model program and resolution of singularities in mixed characteristic.

major comments (2)
  1. [Definition of skeletons and main theorems on invariance] The central claims on discreteness and containment (stated in the abstract and presumably proved in the main theorems) rest entirely on the skeleton lifting preserving log discrepancies. The manuscript must explicitly verify in the definition of skeletons (likely §2 or §3) that the lifting commutes with the operations computing log discrepancies, including that no extra divisors are introduced and multiplicities are preserved canonically; without this, neither discreteness nor the containment follows.
  2. [Functoriality and invariance properties] For the functoriality claim needed for pairs on smooth surfaces, the paper should provide a concrete check that the skeleton construction is independent of choices (e.g., of local coordinates or embeddings) and commutes with blow-ups or other birational operations used in computing minimal log discrepancies; any dependence on auxiliary data would undermine the applications to multi-ideals.
minor comments (2)
  1. [Introduction and related work] Clarify the precise relationship between the new skeleton lifting and existing notions such as generic projections or generic hyperplane sections used in characteristic-zero singularity theory.
  2. [Applications section] Ensure all statements about containment of sets of log canonical thresholds explicitly reference the corresponding theorems in characteristic zero for comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which help us clarify the key properties of the skeleton lifting. We address the major comments point by point below, indicating the revisions we plan to incorporate.

read point-by-point responses

  1. Referee: [Definition of skeletons and main theorems on invariance] The central claims on discreteness and containment (stated in the abstract and presumably proved in the main theorems) rest entirely on the skeleton lifting preserving log discrepancies. The manuscript must explicitly verify in the definition of skeletons (likely §2 or §3) that the lifting commutes with the operations computing log discrepancies, including that no extra divisors are introduced and multiplicities are preserved canonically; without this, neither discreteness nor the containment follows.

    Authors: We agree that making the preservation of log discrepancies fully explicit strengthens the foundation of the main theorems. In Section 2 the skeleton is constructed from the data of a resolution of the ideal in positive characteristic, with coefficients lifted directly to characteristic zero; by design the exceptional divisors correspond bijectively and the multiplicity sequence along each divisor is preserved. Proposition 2.5 already records that the log discrepancy of the lifted pair equals that of the original pair because the discrepancy formula depends only on these data. To address the referee’s request for an explicit verification, we will insert a new Lemma 2.7 that isolates the commutation statement, proves the absence of extraneous divisors, and confirms canonical preservation of multiplicities. This lemma will be cited directly in the proofs of the discreteness and containment results. revision: yes

  2. Referee: [Functoriality and invariance properties] For the functoriality claim needed for pairs on smooth surfaces, the paper should provide a concrete check that the skeleton construction is independent of choices (e.g., of local coordinates or embeddings) and commutes with blow-ups or other birational operations used in computing minimal log discrepancies; any dependence on auxiliary data would undermine the applications to multi-ideals.

    Authors: The skeleton is defined intrinsically via the associated graded ring and the valuation determined by the ideal, without reference to a particular choice of local coordinates or embedding. In the proof of Theorem 3.1 we already verify that two different presentations of the same ideal yield isomorphic skeletons after base change, hence the same lifted ideal in characteristic zero. Compatibility with blow-ups follows because the skeleton encodes the multiplicity data along the strict transform, so the lifted ideal on the blow-up is obtained by the same lifting procedure applied to the strict transform. To make this concrete, we will add a short subsection (or appendix example) that carries out the construction explicitly for a sample multi-ideal on a smooth surface, showing independence of coordinates and direct commutation with a single blow-up. This will also serve as a template for the multi-ideal case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new definition

full rationale

The paper introduces a new notion of skeleton-based characteristic-zero lifting for positive-characteristic objects on smooth surfaces. It then establishes that this lifting is well-defined and functorial, and uses the construction to relate log discrepancies, minimal log discrepancies, and log canonical thresholds across characteristics, yielding discreteness and containment results. No quoted step reduces a claimed prediction or invariant to a fitted input, self-citation, or definitional tautology; the central claims rest on independent verification of the lifting's properties rather than on renaming or smuggling prior results. The derivation therefore does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claims rest on the well-definedness of the new skeleton lifting and on standard facts from algebraic geometry about log discrepancies and log canonical thresholds; no explicit free parameters or invented entities are visible in the abstract.

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