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On the quasi-monomiality of the $\alpha$- and $\delta$-invariants

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Pith's one-line read The alpha and delta invariants for projective klt pairs with big Q-Cartier divisors are computed by quasi-monomial valuations.

desk verdict This paper removes the uncountability assumption for quasi-monomial computation of the α and δ invariants via a log resolution reduction. read the letter →

arxiv 2604.18465 v3 pith:NM6UI6M2 submitted 2026-04-20 math.AG

Donghyeon Kim , Dae-Won Lee This is my paper · ORCID
classification math.AG
keywords quasi-monomialvaluationskltpairsalphainvariantdeltabirationalgeometryQ-Cartierdivisorsprojectivevarieties
checked against absolute_floor_iff_bare_distinguishabilityalexander_duality_circle_linking
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that the alpha and delta invariants of a projective klt pair with a big Q-Cartier divisor are computed by quasi-monomial valuations. This holds over algebraically closed fields of characteristic zero without needing the field to be uncountable. A sympathetic reader cares because these invariants help measure the positivity and singularity of divisors in birational geometry, and quasi-monomial valuations are a restricted class that is easier to handle explicitly. The result extends prior statements by removing the uncountability assumption on the base field.

What carries the argument

Quasi-monomial valuations that realize the infima defining the alpha and delta invariants.

What would settle it

A counterexample consisting of a projective klt pair and big Q-Cartier divisor where a non-quasi-monomial valuation gives a strictly smaller value for alpha or delta than any quasi-monomial one.

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Extended reading notes

Core claim

For any projective klt pair (X,Δ) over an algebraically closed field of characteristic 0 and any big Q-Cartier Q-divisor L on X, the invariants α(X,Δ,L) and δ(X,Δ,L) are computed by quasi-monomial valuations.

Load-bearing premise

The pair must be klt and the divisor L must be big and Q-Cartier to control the valuations and apply birational geometry results.

Editorial extensions

If this is right

  • The computation of these invariants can focus on quasi-monomial valuations rather than all possible valuations.
  • The result applies without assuming the base field is uncountable.
  • Existing results on these invariants that assumed uncountability now extend to all algebraically closed fields of characteristic zero.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If true, this could simplify explicit computations in examples where valuations are already known to be quasi-monomial.
  • The technique might extend to related invariants defined via infima over all valuations on the variety.
  • Testable by direct calculation on specific toric or toric-like pairs where the full set of valuations is understood.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for any projective klt pair (X, Δ) over an algebraically closed field of characteristic 0 and any big ℚ-Cartier ℚ-divisor L on X, the invariants α(X,Δ,L) and δ(X,Δ,L) are computed by quasi-monomial valuations, without requiring an uncountability assumption on the base field.

Significance. If the result holds, it removes a longstanding uncountability hypothesis from the computation of these birational invariants, enabling direct application over countable algebraically closed fields such as the algebraic closure of ℚ. This is valuable for explicit computations and potential arithmetic applications. The proof strategy—reducing via log resolution to monomial valuations on a toroidal exceptional locus, controlling discrepancies via the klt condition, ensuring finiteness via bigness of L, and working algebraically with graded pieces of associated graded rings—avoids generic point selection and uncountable extensions, providing a self-contained algebraic argument that strengthens the existing literature on α and δ invariants.

minor comments (3)
  1. The reduction step via log resolution to the toroidal case (described in the proof outline) would benefit from an explicit statement of how the quasi-monomial property is preserved under the chosen resolution and normalization; a short diagram or reference to the precise toroidal coordinates used would improve readability.
  2. In the discussion of the graded pieces of the associated graded rings, the notation for the S-functional and normalized volume could be made uniform with the definitions of α and δ earlier in the text to avoid any ambiguity in the infima restrictions.
  3. A brief remark on why the algebraic closure and characteristic zero suffice for the existence of suitable test configurations (without further base change) would help readers unfamiliar with the precise algebraic closure arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly captures our main result: that α(X,Δ,L) and δ(X,Δ,L) are computed by quasi-monomial valuations for projective klt pairs over algebraically closed fields of characteristic zero, without any uncountability assumption. We appreciate the recognition that the algebraic proof strategy strengthens the existing literature.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained via standard birational reduction

full rationale

The manuscript derives quasi-monomiality of α(X,Δ,L) and δ(X,Δ,L) by applying a log resolution to reduce to a toroidal exceptional locus where candidate valuations become monomial, then invoking the klt hypothesis to ensure positive log discrepancies and bigness of L to guarantee finiteness of the normalized volume and S-functional. The infima are shown to coincide with their restrictions to the quasi-monomial locus by working directly with graded pieces of associated graded rings, using only algebraic closure and characteristic zero. No step equates the target invariants to a fitted parameter or self-citation by construction; the argument relies on external birational geometry results that are independent of the present claim and do not presuppose the quasi-monomial conclusion.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper is a proof in birational geometry that relies on standard background results about klt pairs, valuations, and big divisors rather than introducing new free parameters or entities.

assumptions (1)
  • standard math Standard properties of klt pairs, quasi-monomial valuations, and big ℚ-Cartier divisors in characteristic zero algebraic geometry
    The statement invokes these as given background to establish the computation by quasi-monomial valuations.

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Pith. "Pith review of On the quasi-monomiality of the $\alpha$- and $\delta$-invariants." pith.science (2026). https://pith.science/paper/NM6UI6M2

@misc{pith2026260418465,
  author       = {Pith},
  title        = {Pith review of: On the quasi-monomiality of the $\alpha$- and $\delta$-invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NM6UI6M2}},
  note         = {Machine review of arXiv:2604.18465}
}
abstract

In this paper, we show that for any projective klt pair $(X,\Delta)$ over an algebraically closed field of characteristic \(0\) and any big $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $L$ on $X$, the invariants $\alpha(X,\Delta,L)$ and $\delta(X,\Delta,L)$ are computed by quasi-monomial valuations, without any uncountability assumption on the base field.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Theorem 1.1: there exist quasi-monomial valuations computing α(X,Δ,L) and δ(X,Δ,L)

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Use of Kollár components and fiberwise log resolutions to control centers

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the normalized local volume of a non-closed point

    math.AG 2026-04 unverdicted novelty 7.0 of 10

    The normalized local volume of a non-closed point equals an expression built from the normalized local volumes of closed points.

  2. On the existence of minimizer on a log Fano cone singularity

    math.AG 2026-08 conditional novelty 6.0 of 10

    For any log Fano cone singularity, the infimum defining the local delta invariant is attained by a torus-invariant quasi-monomial valuation.

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