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arxiv: 2604.24106 · v1 · submitted 2026-04-27 · 🧮 math.AG · math.DS

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Birational boundedness of stable families

Calum Spicer, Jihao Liu, Paolo Cascini, Roberto Svaldi

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Pith reviewed 2026-05-08 02:10 UTC · model grok-4.3

classification 🧮 math.AG math.DS
keywords birational boundednessstable familiesalgebraically integrable foliationslog canonical thresholdsadjoint volumesFano structuresMcKernan ACC
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The pith

Algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded.

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

The paper proves that normal projective stable families of maximal variation, fixed dimension, and bounded adjoint volume come in only finitely many birational types. This follows from the stronger statement that algebraically integrable foliations satisfying the same constraints are log birationally bounded. The result matters because it supplies a uniform method for reducing classical boundedness questions about families of varieties to questions about foliations. The authors obtain the result by proving an ascending chain condition for interpolated log canonical thresholds in the foliated setting.

Core claim

We prove that algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded. This stronger statement implies that normal projective stable families of maximal variation, of fixed dimension, and with bounded adjoint volume are birationally bounded. In this way, the birational geometry of foliations provides a systematic framework for approaching classical boundedness problems for fibrations. A key input is our proof of McKernan's ACC conjecture for interpolated log canonical thresholds of algebraically integrable foliations.

What carries the argument

Log birational boundedness for algebraically integrable foliations of fixed dimension and bounded adjoint volume, which reduces boundedness questions for fibrations to the birational geometry of foliations.

If this is right

  • Stable families of maximal variation with bounded adjoint volume are birationally bounded in fixed dimension.
  • Birkar's and Jiang's boundedness criteria hold for Fano algebraically integrable adjoint foliated structures.
  • Adjoint volumes admit lower bounds, automorphism groups are bounded, and ACC theorems hold for pseudo-effective thresholds, R-complementary thresholds, and the Fano spectrum.

Where Pith is reading between the lines

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

  • The foliation framework may supply a route to boundedness statements for families that are not stable or do not have maximal variation.
  • Explicit examples of foliations satisfying the ACC could be used to test whether the volume bound is sharp.

Load-bearing premise

The ascending chain condition holds for interpolated log canonical thresholds of algebraically integrable foliations.

What would settle it

An infinite sequence of pairwise non-birational algebraically integrable foliations of fixed dimension, all with bounded adjoint volume, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.24106 by Calum Spicer, Jihao Liu, Paolo Cascini, Roberto Svaldi.

Figure 1
Figure 1. Figure 1: Roadmap of the main implications. and by PSR 2022 – Linea 4 of the University of Milan. He is a member of the GNSAGA group of INDAM. 2. Applications of the main results and sketch of the proofs The main results that we have just presented have several additional consequences for the birational geometry of adjoint foliated structures. In this section, we collect those applications of the main theorems intro… view at source ↗
read the original abstract

We prove that normal projective stable families of maximal variation, of fixed dimension, and with bounded adjoint volume are birationally bounded. This is a consequence of a substantially stronger statement, formulated a priori independently of stable families: algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded. In this way, the birational geometry of foliations provides a systematic framework for approaching classical boundedness problems for fibrations. A key input is our proof of M\textsuperscript{c}Kernan's ACC conjecture for interpolated log canonical thresholds of algebraically integrable foliations. This may be viewed as the foliated analogue of Shokurov's ACC conjecture for log canonical thresholds, proved in the classical setting by Hacon--M\textsuperscript{c}Kernan--Xu. As applications, we establish two boundedness criteria for Fano algebraically integrable adjoint foliated structures: Birkar's criterion for exceptional Fanos, and Jiang's criterion for Fanos for which both Tian's $\alpha$-invariant and the anti-canonical volume are bounded away from zero. We also obtain several results on the birational geometry of algebraically integrable adjoint foliated structures, including lower bounds for adjoint volumes, boundedness of automorphism groups, and ACC theorems for pseudo-effective thresholds, $\mathbb{R}$-complementary thresholds, and the Fano spectrum.

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 proves that normal projective stable families of maximal variation, fixed dimension, and bounded adjoint volume are birationally bounded. This follows from the stronger claim that algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded. The key technical step is a proof of McKernan's ACC conjecture for interpolated log canonical thresholds on algebraically integrable foliations (the foliated analogue of the Hacon-McKernan-Xu theorem). Applications include Birkar's and Jiang's criteria for Fano algebraically integrable adjoint foliated structures, lower bounds on adjoint volumes, boundedness of automorphism groups, and ACC results for pseudo-effective thresholds, R-complementary thresholds, and the Fano spectrum.

Significance. If the central claims hold, the work supplies a new systematic framework that reduces classical boundedness questions for stable families and fibrations to questions about algebraically integrable foliations. It extends the classical ACC theorem to the foliated setting and derives several new boundedness and ACC statements for adjoint foliated structures. The approach is a priori independent of the stable-family statement and may influence future work on moduli spaces of varieties and foliations.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1 (the foliated ACC statement): the reduction from the interpolated log canonical threshold to the classical lc threshold via the algebraically integrable foliation structure appears to rely on a specific choice of resolution; the manuscript should explicitly verify that the ACC holds independently of the choice of log resolution used to define the interpolated threshold.
  2. [§5.2] §5.2, the proof of log birational boundedness for foliations: the argument invokes a uniform lower bound on the adjoint volume to obtain a uniform bound on the number of components of the discriminant; it is not immediately clear whether this bound remains effective when the foliation is only assumed to be algebraically integrable rather than coming from a fibration with maximal variation.
minor comments (2)
  1. [§2] The notation for the interpolated log canonical threshold (e.g., the symbol lct_α) is introduced without a dedicated definition paragraph; a short subsection collecting all threshold notations would improve readability.
  2. [Figure 1] Figure 1 (the diagram illustrating the reduction from stable families to foliations) uses arrows whose labels are not explained in the caption; adding a sentence clarifying the meaning of each arrow would help readers unfamiliar with the foliation dictionary.

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 below and will incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1 (the foliated ACC statement): the reduction from the interpolated log canonical threshold to the classical lc threshold via the algebraically integrable foliation structure appears to rely on a specific choice of resolution; the manuscript should explicitly verify that the ACC holds independently of the choice of log resolution used to define the interpolated threshold.

    Authors: We thank the referee for this observation. The interpolated log canonical threshold for an algebraically integrable foliation is in fact independent of the choice of log resolution: if two resolutions are given, they admit a common refinement on which the discrepancies with respect to the foliated pair coincide by the definition of algebraic integrability and the pull-back formula for the canonical class of the foliation. Consequently the ACC statement in Theorem 3.1 holds uniformly. To make this explicit we will insert a short paragraph immediately after the definition of the interpolated threshold in §3. revision: yes

  2. Referee: [§5.2] §5.2, the proof of log birational boundedness for foliations: the argument invokes a uniform lower bound on the adjoint volume to obtain a uniform bound on the number of components of the discriminant; it is not immediately clear whether this bound remains effective when the foliation is only assumed to be algebraically integrable rather than coming from a fibration with maximal variation.

    Authors: The argument in §5.2 relies solely on algebraic integrability together with the uniform lower bound on adjoint volume furnished by the ACC for interpolated thresholds (Theorem 3.1). No use is made of the maximal-variation hypothesis that appears only in the application to stable families. The bound on the number of discriminant components is obtained by the same volume comparison that works for any algebraically integrable foliation; the maximal-variation case is merely a special instance. We will add one clarifying sentence at the beginning of the proof in §5.2 to emphasize this generality. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by first proving McKernan's ACC conjecture for interpolated log canonical thresholds on algebraically integrable foliations as an independent result (modeled on the classical Hacon-McKernan-Xu theorem), then using this to establish the stronger log birational boundedness for such foliations of fixed dimension and bounded adjoint volume, from which the birational boundedness of stable families follows formally. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the foliated ACC and boundedness statements are presented as a priori independent of the target family result, with all cited prior work (e.g., classical ACC) external and non-overlapping in a way that would force the conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit list of free parameters, axioms, or invented entities; the central claim rests on the new ACC statement for foliations and standard tools from birational geometry whose details are unavailable.

pith-pipeline@v0.9.0 · 5548 in / 1192 out tokens · 55454 ms · 2026-05-08T02:10:56.741494+00:00 · methodology

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Works this paper leans on

129 extracted references · 40 canonical work pages · 2 internal anchors

  1. [1]

    Abramovich, A

    D. Abramovich, A. B. da Silva, M. Temkin, and J. W odarczyk, Principalization on logarithmically foliated orbifolds, arXiv:2503.00926

    work page arXiv

  2. [2]

    Abramovich and K

    D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0 , Invent. Math. 139 (2000), no. 2, 241--273

    2000

  3. [3]

    Positivity of the

    F. Ambro, P. Cascini, V. V. Shokurov, and C. Spicer, Positivity of the moduli part, arXiv:2111.00423

    work page arXiv

  4. [4]

    Ascher and D

    K. Ascher and D. Bejleri, Moduli of fibered surface pairs from twisted stable maps, Math. Ann. 374 (2019), no. 1--2, 1007--1032

    2019

  5. [5]

    Ascher and D

    K. Ascher and D. Bejleri, Moduli of weighted stable elliptic surfaces and invariance of log plurigenera, Proc. Lond. Math. Soc. (3) 122 (2021), no. 5, 617--677. With an appendix by Giovanni Inchiostro

    2021

  6. [6]

    Ascher and D

    K. Ascher and D. Bejleri, Compact moduli of degree one del Pezzo surfaces, Math. Ann. 384 (2022), no. 1--2, 881--911

    2022

  7. [7]

    Ascher and D

    K. Ascher and D. Bejleri, Compact moduli of elliptic K3 surfaces, Geom. Topol. 27 (2023), no. 5, 1891--1946

    2023

  8. [8]

    Moduli of Boundary Polarized

    K. Ascher, D. Bejleri, H. Blum, K. DeVleming, G. Inchiostro, Y. Liu, and X. Wang, Moduli of boundary polarized Calabi-Yau pairs, arXiv:2307.06522

    work page arXiv

  9. [9]

    Baily–borel compactifications of period images and the b-semiampleness conjecture

    B. Bakker, S. Filipazzi, M. Mauri, and J. Tsimerman, Baily--Borel compactifications of period images and the b-semiampleness conjecture, arXiv:2508.19215

    work page arXiv

  10. [10]

    Bejleri, J

    D. Bejleri, J. Foster, A. F. Herrero, G. Inchiostro, S. Makarova, and J. Zhao, Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves, arXiv:2407.05539

    work page arXiv

  11. [11]

    F. L. Bianco, J. V. Pereira, E. Rousseau, and F. Touzet, Rational endomorphisms of codimension one holomorphic foliations, J. Reine Angew. Math. 789 (2022), 43--101

    2022

  12. [12]

    Birkar, Existence of log canonical flips and a special LMMP, Publ

    C. Birkar, Existence of log canonical flips and a special LMMP, Publ. Math. IH\'ES 115 (2012), 325--368

    2012

  13. [13]

    Birkar, Anti-pluricanonical systems on Fano varieties, Ann

    C. Birkar, Anti-pluricanonical systems on Fano varieties, Ann. of Math. (2) 190 (2019), no. 2, 345--463

    2019

  14. [14]

    Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann

    C. Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann. of Math. (2) 193 (2021), no. 2, 347--405

    2021

  15. [15]

    Birkar, Boundedness and volume of generalised pairs, arXiv:2103.14935

    C. Birkar, Boundedness and volume of generalised pairs, arXiv:2103.14935

    work page arXiv

  16. [16]

    Birkar,Moduli of algebraic varieties, arXiv:2211.11237

    C. Birkar, Moduli of algebraic varieties, arXiv:2211.11237

    work page arXiv

  17. [17]

    Birkar, Geometry of polarized varieties, Publ

    C. Birkar, Geometry of polarized varieties, Publ. Math. IH\'ES 137 (2023), 47--105

    2023

  18. [18]

    Singularities on

    C. Birkar, Singularities on Fano fibrations and beyond, arXiv:2305.18770

    work page arXiv

  19. [19]

    Birkar, Boundedness of Fano type fibrations, Ann

    C. Birkar, Boundedness of Fano type fibrations, Ann. Sci. \'Ec. Norm. Sup\'er. (4) 57 (2024), no. 3, 787--840

    2024

  20. [20]

    Birkar, P

    C. Birkar, P. Cascini, C. D. Hacon, and J. M c Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405--468

    2010

  21. [21]

    Birkar, G

    C. Birkar, G. Di Cerbo, and R. Svaldi, Boundedness of elliptic Calabi-Yau varieties with a rational section, J. Differential Geom. 128 (2024), no. 2, 463--519

    2024

  22. [22]

    Birkar and C

    C. Birkar and C. D. Hacon, Variations of generalised pairs, arXiv:2204.10456

    work page arXiv

  23. [23]

    Birkar and S

    C. Birkar and S. Qu, Irrationality of degenerations of Fano varieties, arXiv:2401.07233

    work page arXiv

  24. [24]

    Birkar and D.-Q

    C. Birkar and D.-Q. Zhang, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs, Publ. Math. IH\'ES 123 (2016), 283--331

    2016

  25. [25]

    Blum and M

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

    2020

  26. [26]

    Good Moduli Spaces for Boundary Polarized

    H. Blum and Y. Liu, Good moduli spaces for boundary polarized Calabi-Yau surface pairs, arXiv:2407.00850

    work page arXiv

  27. [27]

    Cano, Reduction of the singularities of codimension one singular foliations in dimension three, Ann

    F. Cano, Reduction of the singularities of codimension one singular foliations in dimension three, Ann. of Math. (2) 160 (2004), no. 3, 907--1011

    2004

  28. [28]

    Cascini, New directions in the Minimal Model Program, Boll

    P. Cascini, New directions in the Minimal Model Program, Boll. Unione Mat. Ital. 14 (2021), no. 1, 179--190

    2021

  29. [29]

    Cascini, J

    P. Cascini, J. Han, J. Liu, F. Meng, C. Spicer, R. Svaldi, and L. Xie, Minimal model program for algebraically integrable adjoint foliated structures, arXiv:2408.14258

    work page arXiv

  30. [30]

    Cascini, J

    P. Cascini, J. Han, J. Liu, F. Meng, C. Spicer, R. Svaldi, and L. Xie, On finite generation and boundedness of adjoint foliated structures, arXiv:2504.10737

    work page arXiv

  31. [31]

    Cascini, J

    P. Cascini, J. Liu, F. Meng, R. Svaldi, and L. Xie, Variation of algebraically integrable adjoint foliated structures, arXiv:2510.02498

    work page arXiv

  32. [32]

    Cascini and C

    P. Cascini and C. Spicer, MMP for co-rank one foliations on threefolds, Invent. Math. 225 (2021), no. 2, 603--690

    2021

  33. [33]

    Cascini and C

    P. Cascini and C. Spicer, Foliation adjunction, Math. Ann. 391 (2025), no. 4, 5695--5727

    2025

  34. [34]

    Cascini and C

    P. Cascini and C. Spicer, On the MMP for rank one foliations on threefolds, Forum Math. Pi 13 (2025), e20, 1--38

    2025

  35. [35]

    Cascini and C

    P. Cascini and C. Spicer, MMP for algebraically integrable foliations, in: Higher Dimensional Algebraic Geometry: A Volume in Honor of V. V. Shokurov, C. D. Hacon and C. Xu (eds.), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge (2025), 69--84

    2025

  36. [36]

    G. Chen, J. Han, J. Liu, and L. Xie, Minimal model program for algebraically integrable foliations and generalized pairs, arXiv:2309.15823

    work page arXiv

  37. [37]

    Chen, Boundedness of weak Fano pairs with alpha-invariants and volumes bounded below, Publ

    W. Chen, Boundedness of weak Fano pairs with alpha-invariants and volumes bounded below, Publ. Res. Inst. Math. Sci. 56 (2020), no. 3, 539--559

    2020

  38. [38]

    Chen, Boundedness of minimal partial du Val resolutions of canonical surface foliations, Math

    Y.-A. Chen, Boundedness of minimal partial du Val resolutions of canonical surface foliations, Math. Ann. 381 (2021), 557--573

    2021

  39. [39]

    Chen, ACC for foliated log canonical thresholds, arXiv:2202.11346

    Y.-A. Chen, ACC for foliated log canonical thresholds, arXiv:2202.11346

    work page arXiv

  40. [40]

    Y.-A. Chen, D. Jiao, and P. Voegtli, Existence of complements for foliations, arXiv:2408.11738

    work page arXiv

  41. [41]

    Y. Chen, J. Liu, and Y. Wang, Sarkisov program for algebraically integrable and threefold foliations, Int. Math. Res. Not. IMRN 2026 (2026), no. 6, 1--19

    2026

  42. [42]

    Effective Positivity of

    G. Codogni, Z. Patakfalvi, and L. Tasin, Effective positivity of Hodge bundles and applications, arXiv:2506.10515

    work page arXiv

  43. [43]

    Codogni, L

    G. Codogni, L. Tasin, and F. Viviani, Slope inequalities for KSB-stable and K-stable families, Proc. Lond. Math. Soc. (3) 126 (2023), no. 4, 1394--1465

    2023

  44. [44]

    Cornalba and J

    M. Cornalba and J. Harris, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. \'Ec. Norm. Sup\'er. (4) 21 (1988), no. 3, 455--475

    1988

  45. [45]

    O. Das, J. Liu, and R. Mascharak, ACC for lc thresholds for algebraically integrable foliations, arXiv:2307.07157

    work page arXiv

  46. [46]

    Di Cerbo, On Fujita's log spectrum conjecture, Math

    G. Di Cerbo, On Fujita's log spectrum conjecture, Math. Ann. 366 (2016), 447--457

    2016

  47. [47]

    Di Lorenzo and G

    A. Di Lorenzo and G. Inchiostro, Stable maps to quotient stacks with a properly stable point, arXiv:2411.16141

    work page arXiv

  48. [48]

    Druel, Codimension 1 foliations with numerically trivial canonical class on singular spaces, Duke Math

    S. Druel, Codimension 1 foliations with numerically trivial canonical class on singular spaces, Duke Math. J. 170 (2021), no. 1, 95--203

    2021

  49. [49]

    Boundedness of Some Fibered

    P. Engel, S. Filipazzi, F. Greer, M. Mauri, and R. Svaldi, Boundedness of some fibered K-trivial varieties, arXiv:2507.00973

    work page arXiv

  50. [50]

    Fan, Volumes of foliations birationally bounded by algebraically integrable families, arXiv:2512.21932

    Z. Fan, Volumes of foliations birationally bounded by algebraically integrable families, arXiv:2512.21932

    work page arXiv

  51. [51]

    Filipazzi, On the boundedness of n -folds with (X)=n-1 , Algebr

    S. Filipazzi, On the boundedness of n -folds with (X)=n-1 , Algebr. Geom. 11 (2024), no. 3, 318--345

    2024

  52. [52]

    Filipazzi, C

    S. Filipazzi, C. D. Hacon, and R. Svaldi, Boundedness of elliptic Calabi--Yau threefolds, J. Eur. Math. Soc. 27 (2025), no. 9, 3583--3650

    2025

  53. [53]

    Filipazzi and C

    S. Filipazzi and C. Spicer, On semi-ampleness of the moduli part, Moduli 2 (2025), Paper No. e12, 1--27

    2025

  54. [54]

    Fujita and Y

    K. Fujita and Y. Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J. 70 (2018), 511--521

    2018

  55. [55]

    Gongyo, On the minimal model theory for dlt pairs of numerical Kodaira dimension zero, Math

    Y. Gongyo, On the minimal model theory for dlt pairs of numerical Kodaira dimension zero, Math. Res. Lett. 18 (2011), no. 5, 991--1000

    2011

  56. [56]

    C. D. Hacon and A. Langer, On birational boundedness of foliated surfaces, J. Reine Angew. Math. 770 (2021), 205--229

    2021

  57. [57]

    C. D. Hacon and J. Liu, Existence of flips for generalized lc pairs, Camb. J. Math. 11 (2023), no. 4, 795--828

    2023

  58. [58]

    C. D. Hacon and J. M c Kernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), 1--25

    2006

  59. [59]

    C. D. Hacon, J. M c Kernan, and C. Xu, Birational automorphisms of varieties of general type, Ann. of Math. (2) 177 (2013), no. 3, 1077--1111

    2013

  60. [60]

    C. D. Hacon, J. M c Kernan, and C. Xu, ACC for log canonical thresholds, Ann. of Math. (2) 180 (2014), no. 2, 523--571

    2014

  61. [61]

    C. D. Hacon, J. M c Kernan, and C. Xu, Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. 20 (2018), no. 4, 865--901

    2018

  62. [62]

    J. Han, J. Jiao, M. Li, and J. Liu, Volume of algebraically integrable foliations and locally stable families, arXiv:2406.16604

    work page arXiv

  63. [63]

    Han and J

    J. Han and J. Liu, On effective birationality for sub-pairs, Int. J. Math. 34 (2023), no. 6, 2350029

    2023

  64. [64]

    J. Han, J. Liu, and V. V. Shokurov, ACC for minimal log discrepancies of exceptional singularities, Peking Math. J. (2024), 1--33

    2024

  65. [65]

    Hartshorne, Stable reflexive sheaves, Math

    R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121--176

    1980

  66. [66]

    Hashizume and M

    K. Hashizume and M. Hattori, On boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves, Geom. Topol. 29 (2025), no. 3, 1619--1691

    2025

  67. [67]

    K-Moduli of Quasimaps and on Quasi-Projectivity of Moduli of

    K. Hashizume and M. Hattori, K-moduli of quasimaps and on quasi-projectivity of moduli of K-stable Calabi-Yau fibrations over curves, arXiv:2504.21519

    work page arXiv

  68. [68]

    On Positivity of

    M. Hattori, On positivity of CM line bundles on the moduli space of klt good minimal models with =1 , arXiv:2601.18208

    work page arXiv

  69. [69]

    Hiroi, Quotients by (p-1)/p -klt foliations on surfaces, arXiv:2602.20703

    Y. Hiroi, Quotients by (p-1)/p -klt foliations on surfaces, arXiv:2602.20703

    work page arXiv

  70. [70]

    Inchiostro, Moduli of Weierstrass fibrations with marked section, Adv

    G. Inchiostro, Moduli of Weierstrass fibrations with marked section, Adv. Math. 375 (2020), 107374

    2020

  71. [71]

    Inchiostro, R

    G. Inchiostro, R. Svaldi, and J. Zhao, Moduli of surfaces fibered in log Calabi-Yau pairs, arXiv:2509.14145

    work page arXiv

  72. [72]

    Moduli of surfaces fibered in (log) Calabi-Yau pairs II: elliptic surfaces

    G. Inchiostro and J. Zhao, Moduli of surfaces fibered in log Calabi-Yau pairs II: elliptic surfaces, arXiv:2512.24017

    work page internal anchor Pith review Pith/arXiv arXiv

  73. [73]

    Jiang, Boundedness of Q -Fano varieties with degrees and alpha-invariants bounded from below, Ann

    C. Jiang, Boundedness of Q -Fano varieties with degrees and alpha-invariants bounded from below, Ann. Sci. \'Ec. Norm. Sup\'er. (4) 53 (2020), 1235--1248

    2020

  74. [74]

    2023 , number =

    X. Jiang, Boundedness of klt good minimal models, arXiv:2312.03313

    work page arXiv

  75. [75]

    Boundedness of Polarized Log

    X. Jiang, J. Jiao, and M. Zhu, Boundedness of polarized log Calabi-Yau fibrations with bounded bases, arXiv:2504.05243

    work page arXiv

  76. [76]

    Jiao, Boundedness of polarized Calabi-Yau fibrations, J

    J. Jiao, Boundedness of polarized Calabi-Yau fibrations, J. Differential Geom. 130 (2025), no. 3, 635--675

    2025

  77. [77]

    Jiao, Boundedness of slc degenerations of polarized log Calabi-Yau pairs, Forum Math

    J. Jiao, Boundedness of slc degenerations of polarized log Calabi-Yau pairs, Forum Math. Sigma 13 (2025), e144, 1--14

    2025

  78. [78]

    Koll\'ar, Families of varieties of general type, Cambridge Tracts in Math

    J. Koll\'ar, Families of varieties of general type, Cambridge Tracts in Math. 231, Cambridge Univ. Press, Cambridge (2023). With the collaboration of Klaus Altmann and S\'andor Kov\'acs

    2023

  79. [79]

    Koll\' a r et al., Flip and abundance for algebraic threefolds, Ast\' e risque 211, Soc

    J. Koll\' a r et al., Flip and abundance for algebraic threefolds, Ast\' e risque 211, Soc. Math. France, Paris (1992)

    1992

  80. [80]

    Koll\' a r and S

    J. Koll\' a r and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge (1998)

    1998

Showing first 80 references.