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Effective generation for foliated surfaces: results and applications

T0 review · 0 major / 1 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read For adjoint general type foliated surfaces the automorphism group is finite and ε-adjoint canonical models form a bounded family.

desk verdict This paper gives concrete bounds on automorphism groups of adjoint general type foliated surfaces, degrees of invariant curves, and boundedness of their ε-adjoint canonical models. read the letter →

arxiv 2104.11540 v2 submitted 2021-04-23 math.AG math.DS

Calum Spicer , Roberto Svaldi This is my paper · ORCID
classification math.AGmath.DS
keywords foliatedsurfacesadjointdivisorbirationalgeometryautomorphismgroupsboundedfamiliesgeneraltypefoliationsinvariantcurves
checked against alexander_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 authors study the birational geometry of foliated surfaces (X, F) through the adjoint divisor K_F + ε K_X for small ε > 0. They prove that the automorphism group of an adjoint general type foliated surface is finite. They bound the degree of a general curve invariant by an algebraically integrable foliation. They also show that the set of ε-adjoint canonical models of foliations of general type with fixed volume is a bounded family. These results matter because they give effective control over the geometry and classification of such foliated surfaces, extending classical results from algebraic surfaces to the foliated case.

What carries the argument

The adjoint divisor K_F + ε K_X, which is big and nef for small ε on adjoint general type foliated surfaces and carries the birational structure and invariants.

What would settle it

An explicit example of an adjoint general type foliated surface with infinite automorphism group, or an unbounded collection of ε-adjoint canonical models with the same volume, would disprove the main results.

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Extended reading notes

Core claim

The central claim is that the adjoint divisor K_F + ε K_X allows the application of standard techniques from birational geometry to foliated surfaces of adjoint general type, yielding a bound on the automorphism group, a bound on the degree of invariant curves for algebraically integrable foliations, and boundedness of the family of ε-adjoint canonical models with fixed volume.

Load-bearing premise

The foliation must be of adjoint general type so that the adjoint divisor K_F + ε K_X is big and nef for sufficiently small ε.

Editorial extensions

If this is right

  • The automorphism group of an adjoint general type foliated surface is finite.
  • The degree of a general invariant curve under an algebraically integrable foliation on a surface is bounded.
  • The ε-adjoint canonical models of general type foliations with fixed volume form a bounded family.
  • These bounds provide effective generation results for the birational geometry of foliated surfaces.

Reading between the lines

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

  • This framework may allow similar bounds in higher dimensions for foliated varieties.
  • The results could imply effective finiteness theorems for moduli spaces of such foliations.
  • Applications might include classification of foliations with given invariants.
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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 / 1 minor

Summary. The paper explores the birational structure and invariants of a foliated surface (X, F) in terms of the adjoint divisor K_F + ε K_X for 0 < ε ≪ 1. It claims to establish a bound on the automorphism group of an adjoint general type foliated surface (X, F), a bound on the degree of a general curve invariant by an algebraically integrable foliation on a surface, and that the set of ε-adjoint canonical models of foliations of general type with fixed volume form a bounded family.

Significance. If the results hold, they would contribute to effective birational geometry for foliations on surfaces by supplying explicit bounds and a boundedness statement, extending standard techniques from the minimal model program to the foliated setting. This could aid in classification problems, though the significance remains potential given the absence of verifiable derivations.

minor comments (1)
  1. The abstract provides a high-level overview but does not state the explicit bounds or reference specific theorems/sections, which limits immediate assessment of the strength of the claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for acknowledging the potential significance of our results on effective birational geometry for foliated surfaces. The recommendation is uncertain due to concerns over verifiable derivations; we address this below. We note that no specific major comments were listed in the report.

read point-by-point responses

  1. Referee: absence of verifiable derivations

    Authors: The proofs of the main results (bounds on automorphism groups, invariant curve degrees, and boundedness of ε-adjoint canonical models) are given in full detail in Sections 3–6 of the manuscript, adapting standard MMP techniques to the foliated setting with explicit references to the relevant lemmas on adjoint divisors. We followed the same level of detail as in prior works on foliations (e.g., the cited papers on canonical models). If particular steps remain unclear, we are prepared to add further explanations or diagrams in a revision. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper applies standard birational geometry techniques to the adjoint divisor K_F + ε K_X (assumed big and nef for small ε > 0 under the general type hypothesis) to derive bounds on automorphism groups, invariant curve degrees, and boundedness of canonical models. No load-bearing steps reduce by construction to self-definitions, fitted inputs renamed as predictions, or self-citation chains; the central claims rest on external results in birational geometry that are independent of the present derivations. The derivation chain is self-contained against external benchmarks.

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

Abstract supplies no information on free parameters, background axioms, or newly postulated entities; all such items remain unknown.

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Cite this review

Pith. "Pith review of Effective generation for foliated surfaces: results and applications." pith.science (2026). https://pith.science/paper/2104.11540

@misc{pith2026210411540,
  author       = {Pith},
  title        = {Pith review of: Effective generation for foliated surfaces: results and applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2104.11540}},
  note         = {Machine review of arXiv:2104.11540}
}
abstract

We explore the birational structure and invariants of a foliated surface $(X, \mathcal F)$ in terms of the adjoint divisor $K_{\mathcal F}+\epsilon K_X$, $0< \epsilon \ll 1$. We then establish a bound on the automorphism group of an adjoint general type foliated surface $(X, \mathcal F)$, provide a bound on the degree of a general curve invariant by an algebraically integrable foliation on a surface and prove that the set of $\epsilon$-adjoint canonical models of foliations of general type and with fixed volume form a bounded family.

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Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    C. Birkar. Geometry and moduli of polarised varieties, 2020. Arxiv e-print, arXiv:2006.11238

    work page arXiv 2020

  2. [2]

    Bierstone and P

    E. Bierstone and P. D. Milman. Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. , 128(2):207--302, 1997

    work page 1997

  3. [3]

    Bogomolov and M

    F. Bogomolov and M. McQuillan. Rational curves on foliated varieties. In Foliation theory in algebraic geometry , Simons Symp., pages 21--51. Springer, Cham, 2016

    work page 2016

  4. [4]

    Brunella

    M. Brunella. Birational geometry of foliations . Monograf\' as de Matem\'atica. Instituto de Matem\'atica Pura e Aplicada (IMPA), Rio de Janeiro, 2000

    work page 2000

  5. [5]

    Corr\^ e a, Jr

    M. Corr\^ e a, Jr. and T. Fassarella. On the order of the automorphism group of foliations. Math. Nachr. , 287(16):1795--1803, 2014

    work page 2014

  6. [6]

    Y.-A. Chen. Boundedness of minimal partial du val resolutions of canonical surface foliations, 2019. Arxiv e-print, arXiv:1912.02931

    work page arXiv 2019

  7. [7]

    Y.-A. Chen. Generalized canonical models of foliated surfaces, 2021. Arxiv e-print, arXiv:2103.12669

    work page arXiv 2021

  8. [8]

    Cascini and C

    P. Cascini and C. Spicer. MMP for co-rank one foliations on threefolds, 2018. ArXiv e-print, arXiv:1808.02711

    work page arXiv 2018

Show all 21 references
  1. [9]

    Cascini and C

    P. Cascini and C. Spicer. On the MMP for rank one foliations on threefolds, 2020. ArXiv e-print, arXiv:2012.11433

    2020

  2. [10]

    S. Druel. Codimension 1 foliations with numerically trivial canonical class on singular spaces. Duke Math. J. , 170(1):95--203, 2021

    2021

  3. [11]

    Filipazzi

    S. Filipazzi. Some remarks on the volume of log varieties. Proc. Edinb. Math. Soc. (2) , 63(2):314--322, 2020

    2020

  4. [12]

    C. D. Hacon and A. Langer. On birational boundedness of foliated surfaces. Journal für die reine und angewandte Mathematik (Crelles Journal) , 2021(770):205--229, 2021

    2021

  5. [13]

    Hacon, J

    C. Hacon, J. M c Kernan, and C. Xu. ACC for log canonical thresholds. Ann. of Math. , 180(2):523--571, 2014

    2014

  6. [14]

    Koll\' a r and S

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

    1998

  7. [15]

    Matsusaka

    T. Matsusaka. On polarized normal varieties. I . Nagoya Mathematical Journal , 104(none):175 -- 211, 1986

    1986

  8. [16]

    McQuillan

    M. McQuillan. Canonical models of foliations. Pure Appl. Math. Q. , 4(3, part 2):877--1012, 2008

    2008

  9. [17]

    McQuillan and D

    M. McQuillan and D. Panazzolo. Almost \'etale resolution of foliations. J. Differential Geom. , 95(2):279--319, 2013

    2013

  10. [18]

    Martinelli, S

    D. Martinelli, S. Schreieder, and L. Tasin. On the number and boundedness of log minimal models of general type. Ann. Sci. \' E c. Norm. Sup\' e r. (4) , 53(5):1183--1207, 2020

    2020

  11. [19]

    J. V. Pereira and P. Fernandez S\' a nchez. Transformation groups of holomorphic foliations. Comm. Anal. Geom. , 10(5):1115--1123, 2002

    2002

  12. [20]

    J. V. Pereira and R. Svaldi. Effective algebraic integration in bounded genus. Algebr. Geom. , 6(4):454--485, 2019

    2019

  13. [21]

    C. Spicer. Higher-dimensional foliated M ori theory. Compos. Math. , 156(1):1--38, 2020

    2020

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