pith. sign in

arxiv: 2412.04019 · v2 · submitted 2024-12-05 · 🧮 math.AG

On the coupled stability thresholds of graded linear series

Kento Fujita This is my paper

Pith reviewed 2026-05-23 08:25 UTC · model grok-4.3

classification 🧮 math.AG
keywords graded linear seriescoupled stability thresholdsS-invariantbirational morphismsample seriesstability thresholds
0
0 comments X

The pith

Coupled stability threshold functions for graded linear series extend continuously over the interior of their support when the series contain an ample series.

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

Graded linear series containing an ample series remain graded linear series after pullback under birational morphisms. This property permits the definition of refinements with respect to primitive flags together with explicit formulas for their S-invariants. The paper defines coupled stability thresholds for such series as a generalization of earlier notions and proves that the resulting function extends uniquely and continuously to the interior of the support for any finite collection of the series. It also derives a product-type formula and Abban-Zhuang-style local estimates for these thresholds.

Core claim

For any finite collection of graded linear series each containing an ample series, the coupled stability threshold function admits a unique continuous extension over the interior of the support; a product-type formula holds for the thresholds, and Abban-Zhuang-type formulas estimate the local thresholds.

What carries the argument

The coupled stability threshold function for graded linear series containing an ample series, which encodes the continuous extension property, the product formula, and the local estimates.

If this is right

  • The threshold function is continuous on the interior of the support.
  • A product formula computes the coupled thresholds from the individual series.
  • Abban-Zhuang-type estimates bound the local coupled thresholds.

Where Pith is reading between the lines

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

  • The refinement construction with primitive flags may simplify computation of S-invariants on resolutions of singularities.
  • Continuous extension could allow stability thresholds to be tracked in flat families without jumping at the boundary of the support.
  • The product formula might reduce questions about multi-series stability to single-series calculations in higher-dimensional settings.

Load-bearing premise

The graded linear series under consideration contain an ample series.

What would settle it

An explicit graded linear series containing an ample series for which the coupled stability threshold function exhibits a discontinuity at an interior point of the support would falsify the continuous-extension claim.

read the original abstract

In this paper, we see several basic properties of graded linear series. We firstly see that, if a graded linear series contains an ample series, then so are the pullbacks of the system under birational morphisms. Using this proposition, we define the refinements of graded linear series with respects to primitive flags. Moreover, we give several formulas to compute the $S$-invariant of those refinements. Secondly, we introduce the notion of coupled stability thresholds for graded linear series, which is a generalization of the notion introduced by Rubinstein--Tian--Zhang. We see that, over the interior of the support for finite numbers of graded linear series containing an ample series, the coupled stability threshold function can be uniquely extended continuously, which generalizes the work by Kewei Zhang. Thirdly, we get a product-type formula for coupled stability thresholds, which generalizes the work of Zhuang. Fourthly, we see Abban--Zhuang's type formulas for estimating local coupled stability thresholds.

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

1 major / 2 minor

Summary. The paper establishes several properties of graded linear series on algebraic varieties. It first proves that if a graded linear series contains an ample series, then its pullbacks under birational morphisms remain graded linear series; this is used to define refinements with respect to primitive flags and to derive explicit formulas for the S-invariants of those refinements. It then introduces coupled stability thresholds for graded linear series as a generalization of the Rubinstein-Tian-Zhang notion. The central results are a unique continuous extension of the coupled stability threshold function over the interior of the support for any finite collection of such series containing an ample series (generalizing Kewei Zhang), a product-type formula (generalizing Zhuang), and Abban-Zhuang-type local estimates.

Significance. If the results hold, the work extends the analytic and algebraic machinery of stability thresholds from line bundles to the broader setting of graded linear series, supplying explicit computational tools (S-invariant formulas for refinements, product formulas, and local estimates) that could facilitate calculations in families or higher-dimensional settings. The continuous-extension statement and the preparatory pullback/refinement results are the load-bearing contributions; when the ample-series hypothesis is satisfied they appear to supply a coherent framework generalizing prior work.

major comments (1)
  1. [Introduction / section on coupled stability thresholds] The continuous-extension claim (abstract and the section introducing coupled stability thresholds) is stated only for finite collections of graded linear series that contain an ample series. The preparatory pullback-preservation result and the definition of refinements with respect to primitive flags both invoke this hypothesis explicitly; without it the domain on which the extension, product formula, and local estimates are asserted may shrink. The manuscript provides no alternative construction or discussion of whether the extension statement survives when the ample-series condition is dropped.
minor comments (2)
  1. Notation for the support of a finite collection of graded linear series and for the interior on which the extension is claimed should be introduced with a precise definition (e.g., via a displayed equation) before the extension theorem is stated.
  2. The abstract lists four main results; the body should contain a numbered theorem statement for each, with clear references back to the preparatory propositions on pullbacks and refinements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Introduction / section on coupled stability thresholds] The continuous-extension claim (abstract and the section introducing coupled stability thresholds) is stated only for finite collections of graded linear series that contain an ample series. The preparatory pullback-preservation result and the definition of refinements with respect to primitive flags both invoke this hypothesis explicitly; without it the domain on which the extension, product formula, and local estimates are asserted may shrink. The manuscript provides no alternative construction or discussion of whether the extension statement survives when the ample-series condition is dropped.

    Authors: The pullback-preservation property (Proposition on pullbacks under birational morphisms) holds precisely when the graded linear series contains an ample series; this is the key preparatory result used to define refinements with respect to primitive flags and to obtain the explicit S-invariant formulas. Consequently the continuous-extension theorem, the product formula, and the local estimates are all stated for finite collections satisfying the same hypothesis, as this is the setting in which the refinements are well-defined. The manuscript does not assert that these statements survive without the ample-series condition, nor does it claim an alternative construction in that case. To address the referee's observation we will add a short clarifying remark in the introduction noting that the ample-series hypothesis is required for the pullback property and is therefore retained throughout, while extension beyond this setting is left for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations rely on explicit assumptions and external prior results

full rationale

The paper defines coupled stability thresholds as a generalization of Rubinstein-Tian-Zhang and proves continuous extension, product formulas, and local estimates under the explicit hypothesis that the graded linear series contain an ample series. This hypothesis is stated as a prerequisite for pullbacks and refinements (abstract and section on basic properties), not derived from the target quantities. All cited foundational results (Zhang, Zhuang, Abban-Zhuang) are by non-overlapping authors and function as external inputs rather than self-referential chains. No equations reduce a prediction to a fitted parameter by construction, no uniqueness theorem is imported from the same author, and no ansatz is smuggled via self-citation. The derivation chain is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the paper introduces no explicit free parameters, no new invented entities, and relies on standard background results of algebraic geometry (existence of ample series, birational morphisms, primitive flags).

pith-pipeline@v0.9.0 · 5685 in / 1173 out tokens · 21965 ms · 2026-05-23T08:25:31.474384+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    C.\ Araujo, A-M.\ Castravet, I.\ Cheltsov, K.\ Fujita, A-S.\ Kaloghiros, J.\ Martinez-Garcia, C.\ Shramov, H.\ S\"uss and N.\ Viswanathan, The Calabi problem for Fano threefolds, London Math.\ Soc.\ Lecture Note Ser., 485 Cambridge University Press, Cambridge, 2023, vii+441 pp

    work page 2023

  2. [2]

    H.\ Abban and Z.\ Zhuang, K-stability of Fano varieties via admissible flags, Forum Math.\ Pi 10 (2022), no.\ e15, 1--43

    work page 2022

  3. [3]

    172(6) (2023), 1109--1144

    H.\ Abban and Z.\ Zhuang, Seshadri constants and K-stability of Fano manifolds, Duke Math.\ J. 172(6) (2023), 1109--1144

    work page 2023

  4. [4]

    R.\ Berman, S.\ Boucksom and M.\ Jonsson, A variational approach to the Yau-Tian-Donaldson conjecture, J.\ Amer.\ Math.\ Soc.\ 34 (2021), no.\ 3, 605--652

    work page 2021

  5. [5]

    S.\ Boucksom and H.\ Chen, Okounkov bodies of filtered linear series, Compos.\ Math.\ 147 (2011), no.\ 4, 1205--1229

    work page 2011

  6. [6]

    S.\ Boucksom, S.\ Cacciola and A.\ Lopez, Augmented base loci and restricted volumes on normal varieties, Math.\ Z.\ 278 (2014), no.\ 3–-4, 979–-985

    work page 2014

  7. [7]

    S.\ Boucksom, C.\ Favre and M.\ Jonsson, Differentiability of volumes of divisors and a problem of Teissier, J.\ Algebraic Geom.\ 18 (2009), no.\ 2, 279--308

    work page 2009

  8. [8]

    C.\ Birkar, The augmented base locus of real divisors over arbitrary fields, Math.\ Ann.\ 368 (2017), 905--921

    work page 2017

  9. [9]

    H.\ Blum and M.\ Jonsson, Thresholds, valuations, and K-stability, Adv.\ Math.\ 365 (2020), 107062, 57 pp

    work page 2020

  10. [10]

    S.\ Boucksom, Corps d'Okounkov (d'apr\`es Okounkov, Lazarsfeld--Musta t a et Kaveh--Khovanskii) , Ast\'erisque No.\ 361 (2014), Exp.\ No.\ 1059, vii, 1--41

    work page 2014

  11. [11]

    I.\ Cheltsov, K.\ Fujita, T.\ Kishimoto and J.\ Park, K-stable Fano 3 -folds in the families 2.18 and 3.4, arXiv:2304.11334v1

    work page arXiv

  12. [12]

    American Mathematical Society, Providence, RI, 2011

    D.\ Cox, J.\ Little and H.\ Schenck, Toric varieties, Graduate Studies in Mathematics, 124. American Mathematical Society, Providence, RI, 2011

    work page 2011

  13. [13]

    K.\ Cho, Y.\ Miyaoka and N.\ I.\ Shepherd-Barron, Characterizations of projective spaces and applications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto, 1997), 1--88, Adv.\ Stud.\ Pure Math.\ 35, Math.\ Soc.\ Japan, Tokyo, 2002

    work page 1997

  14. [14]

    ``Log canonical thresholds of smooth Fano threefolds” : On Tian's invariant and log canonical thresholds, Russian Math.\ Surveys 63 (2008), no.\ 5, 945--950

    J.-P.\ Demailly, Appendix to I.\ Cheltsov and C.\ Shramov's article. ``Log canonical thresholds of smooth Fano threefolds” : On Tian's invariant and log canonical thresholds, Russian Math.\ Surveys 63 (2008), no.\ 5, 945--950

    work page 2008

  15. [15]

    R.\ Dervan, Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds, Ann.\ Fac.\ Sci.\ Toulouse Math.\ 25 (2016), no.\ 4, 919--934

    work page 2016

  16. [16]

    L.\ Ein, R.\ Lazarsfeld, M.\ Musta t a , M.\ Nakamaye and M.\ Popa, Asymptotic invariants of base loci, Ann.\ Inst.\ Fourier (Grenoble) 56 (2006), 1701–-1734

    work page 2006

  17. [17]

    L.\ Ein, R.\ Lazarsfeld, M.\ Musta t a , M.\ Nakamaye and M.\ Popa, Restricted volumes and base loci of linear series, Amer.\ J.\ Math.\ 131 (2009), no.\ 3, 607--651

    work page 2009

  18. [18]

    K.\ Fujita and Y.\ Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math.\ J.\ 70 (2018), no.\ 4, 511--521

    work page 2018

  19. [19]

    K.\ Fujita, Uniform K-stability and plt blowups of log Fano pairs, Kyoto J.\ Math.\ 59 (2019), no.\ 2, 399--418

    work page 2019

  20. [20]

    K.\ Fujita, On K-stability for Fano threefolds of rank 3 and degree 28 , Int.\ Math.\ Res.\ Not.\ IMRN (2023), no.\ 15, 12601--12784

    work page 2023

  21. [21]

    Y.\ Hashimoto, Anticanonically balanced metrics and Hilbert--Mumford criterion for the _m -invariant of Fujita--Odaka, Ann.\ Global Anal.\ Geom.\ 64, (2023) no.\ 8, 40 pp

    work page 2023

  22. [22]

    om, Coupled K\

    J.\ Hultgren and D.\ W.\ Nystr\"om, Coupled K\"ahler--Einstein metrics, Int.\ Math.\ Res.\ Not.\ IMRN 21 (2019), 6765–-6796

    work page 2019

  23. [23]

    S.\ Ishii, Extremal functions and prime blow-ups, Comm.\ Algebra 32 (2004), no.\ 3, 819--827

    work page 2004

  24. [24]

    Y.\ Hu and S.\ Keel, Mori dream spaces and GIT, Michigan Math.\ J.\ 48 (2000), 331--348

    work page 2000

  25. [25]

    S.\ Kebekus, Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, Complex geometry (G\"ottingen, 2000), 147--155, Springer, Berlin, 2002

    work page 2000

  26. [26]

    Cambridge Tracts in Math., 134, Cambridge University Press, Cambridge, 1998

    J.\ Koll \'a r and S.\ Mori, Birational geometry of algebraic varieties, With the collaboration of C.\ H.\ Clemens and A.\ Corti. Cambridge Tracts in Math., 134, Cambridge University Press, Cambridge, 1998

    work page 1998

  27. [27]

    R.\ Lazarsfeld, Positivity in algebraic geometry, I: Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete.\ (3) 48, Springer, Berlin, 2004

    work page 2004

  28. [28]

    R.\ Lazarsfeld and M.\ Musta t a , Convex bodies associated to linear series, Ann.\ Sci.\ \'Ec.\ Norm.\ Sup\'er.\ 42 (2009), no.\ 5, 783--835

    work page 2009

  29. [29]

    A.\ Lopez, Augmented base loci and restricted volumes on normal varieties, II: The case of real divisors, Math.\ Proc.\ Cambridge Philos.\ Soc.\ 159 (2015), no.\ 3, 517-–527

    work page 2015

  30. [30]

    Mathematical Society of Japan, Tokyo, 2004

    N.\ Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, 14. Mathematical Society of Japan, Tokyo, 2004

    work page 2004

  31. [31]

    S.\ Okawa, On images of Mori dream spaces, Math.\ Ann.\ 364 (2016), no.\ 3-4, 1315--1342

    work page 2016

  32. [32]

    Y.\ Rubinstein, G.\ Tian and K.\ Zhang, Basis divisors and balanced metrics, J.\ Reine Angew.\ Math.\ 778 (2021), 171--218

    work page 2021

  33. [33]

    C.\ Xu, K-stability of Fano varieties (version 2024/04/30), book available at https://web.math.princeton.edu/\ chenyang/Kstabilitybook.pdf

    work page 2024

  34. [34]

    K.\ Zhang, Continuity of delta invariants and twisted K\"ahler-Einstein metrics, Adv.\ Math.\ 388 (2021), Paper No.\ 107888, 25 pp

    work page 2021

  35. [35]

    K.\ Zhang, A quantization proof of the uniform Yau--Tian--Donaldson conjecture, J.\ Eur.\ Math.\ Soc.\ (2023), DOI 10.4171/JEMS/1373

    work page doi:10.4171/jems/1373 2023

  36. [36]

    Z.\ Zhuang, Product theorem for K-stability, Adv.\ Math.\ 371 (2020), 107250, 18 pp

    work page 2020