arxiv: 2511.22600 · v3 · submitted 2025-11-27 · 🧮 math.AG · math.AC

b-divisorial valuations and Berkovich positivity functions

Joaquim Ro\'e , Stefano Urbinati This is my paper

Pith reviewed 2026-05-17 04:09 UTC · model grok-4.3

classification 🧮 math.AG math.AC

keywords Seshadri constantb-divisorBerkovich spacesemicontinuitypositivity invariantvaluationbig and nef divisoralgebraic geometry

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The pith

Seshadri constants and asymptotic orders of vanishing extend to all seminorms in the Berkovich space while remaining semicontinuous.

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

This paper proves semicontinuity for local positivity invariants of big and nef divisors on algebraic varieties. It extends the definitions of the Seshadri constant and the asymptotic order of vanishing to apply to every seminorm in the Berkovich space rather than only to subvarieties. By linking each seminorm to a b-divisor, positivity questions become questions about the shape of cones of b-divisors. The construction works especially cleanly for b-divisorial valuations, which include all Abhyankar valuations as a special case. A reader would care because the result supplies a uniform way to track how positivity behaves when the center moves continuously through a larger space of possible centers.

Core claim

We prove semicontinuity properties for local positivity invariants of big and nef divisors. The usual definition of Seshadri constant and asymptotic order of vanishing along a subvariety is extended to include all seminorms in the Berkovich space, and we obtain semicontinuity of such constants as a function of the center seminorm. We use Shokurov's language of b-divisors; to each seminorm there is an associated b-divisor which can be used to translate questions about positivity into questions about the shape of certain cones of b-divisors. The theory works especially well for what we call b-divisorial valuations, a natural extension of the notion of divisorial valuations which encompasses, e

What carries the argument

The b-divisor associated to a seminorm in the Berkovich space, which converts positivity data into membership questions inside cones of b-divisors.

If this is right

The extended constants vary continuously with the center seminorm, allowing passage to limits in families.
Positivity questions for Abhyankar and other b-divisorial valuations reduce to checking membership in b-divisor cones.
A single framework covers both classical divisorial centers and a wider class of analytic centers.
Local positivity can be compared uniformly across different types of valuations on the same variety.

Where Pith is reading between the lines

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

The same b-divisor translation might apply to other local invariants such as the volume or the mobility function.
Berkovich space could become a natural domain for proving global statements by controlling local semicontinuity.
One could test whether the same cones detect positivity when the divisor is only big rather than nef.

Load-bearing premise

That associating a b-divisor to each seminorm faithfully carries over the positivity information needed to define the extended constants.

What would settle it

An explicit big and nef divisor together with a convergent sequence of b-divisorial valuations whose associated Seshadri constants fail to approach the value at the limit valuation.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

They extend Seshadri constants and vanishing orders to all Berkovich seminorms by associating b-divisors, then prove semicontinuity in the center.

read the letter

The main takeaway is that Roé and Urbinati push the usual definitions of Seshadri constants and asymptotic orders of vanishing out to every seminorm on the Berkovich space. They do this by attaching a b-divisor to each seminorm, which turns questions about positivity of big and nef divisors into statements about the shape of certain cones of b-divisors. Semicontinuity then follows as a function of the center seminorm. The setup is presented as working especially cleanly for b-divisorial valuations, a class that includes Abhyankar valuations and generalizes the classical divisorial ones. This is the concrete new content in the abstract. The b-divisor translation is a reasonable move because that language already handles limits and approximations in birational geometry. The abstract gives a coherent high-level picture without obvious circularity or invented objects that fail to connect to existing theory. The claims line up with standard uses of b-divisors for positivity, so the approach does not look forced. The clearest limitation is that only the abstract is visible here. Without the proofs it is impossible to check how much technical work is needed to establish the semicontinuity or whether the b-divisor association holds for all claimed seminorms without extra restrictions. If hidden conditions appear in the details, the scope could be narrower than the abstract suggests. Still, nothing in the stated program looks internally inconsistent or tautological. This paper is aimed at algebraic geometers already working with Berkovich spaces, non-archimedean valuations, or local positivity invariants. A reader comfortable with b-divisors and valuation theory would find the extension useful. The claims are specific enough and the framework grounded enough that the paper deserves a serious referee rather than a desk rejection. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The paper proves semicontinuity properties for local positivity invariants of big and nef divisors. The usual definitions of Seshadri constants and asymptotic orders of vanishing along subvarieties are extended to all seminorms in the Berkovich space by associating a b-divisor to each seminorm; positivity questions are then translated into statements about the shape of cones of b-divisors. Semicontinuity holds as a function of the center seminorm, and the construction is shown to apply especially cleanly to b-divisorial valuations (a generalization of divisorial valuations that includes Abhyankar valuations).

Significance. If the central claims are established, the work supplies a coherent non-archimedean extension of classical local positivity invariants. The b-divisor formalism converts questions about seminorm centers into geometric properties of cones, which may prove useful for further study of birational geometry and valuation theory over non-archimedean fields. The explicit treatment of b-divisorial valuations, including Abhyankar examples, strengthens the applicability of the framework.

major comments (1)

The abstract states the semicontinuity results but supplies no outline of the key steps or reduction to properties of b-divisor cones; without such an indication in the main text (e.g., in the section introducing the translation), it is difficult to verify that the argument is not tautological once the b-divisor is attached to the seminorm.

minor comments (2)

The notation for seminorms, centers, and associated b-divisors should be fixed early and used consistently; occasional shifts between “seminorm” and “valuation” language can obscure the precise domain of the semicontinuity statements.
A short table or diagram comparing the classical Seshadri constant, the asymptotic order of vanishing, and their Berkovich extensions would help readers track how the new invariants specialize to the old ones.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the recommendation of minor revision. The single major comment is addressed below; we agree that an explicit outline will improve clarity and will incorporate the suggested change.

read point-by-point responses

Referee: The abstract states the semicontinuity results but supplies no outline of the key steps or reduction to properties of b-divisor cones; without such an indication in the main text (e.g., in the section introducing the translation), it is difficult to verify that the argument is not tautological once the b-divisor is attached to the seminorm.

Authors: We appreciate this observation. While the abstract indicates that each seminorm is associated to a b-divisor and that positivity questions are thereby translated into statements about cones of b-divisors, we agree that the main text would benefit from a concise, explicit outline of the logical steps. In the revised manuscript we will insert a short paragraph immediately after the definition of the b-divisor associated to a seminorm (in the section introducing the translation). This paragraph will list the three main reductions: (1) the positivity invariant is expressed as a numerical function of the b-divisor, (2) the relevant cone of b-divisors is shown to be closed under certain operations that encode semicontinuity, and (3) the semicontinuity of the invariant then follows from the continuity of the cone membership function. This addition will make the non-tautological character of the argument transparent without altering any proofs. revision: yes

Circularity Check

0 steps flagged

No circularity: definitional extension followed by independent proof

full rationale

The paper introduces an extension of Seshadri constants and asymptotic vanishing orders to all seminorms on the Berkovich space by associating to each seminorm a b-divisor (using Shokurov's standard language). It then claims to prove semicontinuity of these invariants as functions of the center seminorm, with the construction working especially well for b-divisorial valuations. This is a sequence of new definitions plus a theorem, not a reduction of any claimed result to a fitted input, self-citation chain, or tautological renaming. The abstract and approach give no indication that the semicontinuity statements are forced by construction from the inputs; the b-divisor translation is presented as a tool that aligns with existing usage rather than an internal loop. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the new definitions of b-divisorial valuations and the correspondence between seminorms and b-divisors; these are introduced in the paper rather than derived from prior results.

axioms (2)

domain assumption Standard properties of big and nef divisors on projective varieties
The semicontinuity statements are stated for big and nef divisors.
domain assumption Existence and basic properties of Berkovich spaces and their seminorms
The extension is defined using seminorms on the Berkovich space.

invented entities (1)

b-divisorial valuations no independent evidence
purpose: A natural extension of divisorial valuations that includes Abhyankar valuations and allows the b-divisor translation to work cleanly.
The abstract states that the theory works especially well for these valuations.

pith-pipeline@v0.9.0 · 5423 in / 1429 out tokens · 49825 ms · 2026-05-17T04:09:24.585002+00:00 · methodology

discussion (0)

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/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We use Shokurov’s language of b-divisors; to each seminorm there is an associated b-divisor which can be used to translate questions about positivity into questions about the shape of certain cones of b-divisors.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ε(D, vξ) := sup{t|(D+tD ξ)is nef}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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