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

Effective generation for foliated surfaces: results and applications

Calum Spicer , Roberto Svaldi This is my paper

Pith reviewed 2026-05-24 13:50 UTC · model grok-4.3

classification 🧮 math.AG math.DS
keywords foliated surfacesadjoint divisorbirational geometryautomorphism groupsbounded familiesgeneral type foliationsinvariant curves
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The pith

For adjoint general type foliated surfaces the automorphism group is finite and ε-adjoint canonical models form a bounded family.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

read the original 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.

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

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 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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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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