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arxiv: 2604.09027 · v1 · submitted 2026-04-10 · 🧮 math.AG

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Recent progress on the Minimal Model Program for foliations

Paolo Cascini , Calum Spicer
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:17 UTC · model grok-4.3

classification 🧮 math.AG
keywords foliationsminimal model programbirational geometrysingularitiesadjunctionsurfacesthreefolds
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The pith

Recent progress has established the minimal model program for foliations on surfaces and the existence of flips on threefolds.

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

This paper surveys recent advances in the birational geometry of foliations on complex varieties, viewed through the lens of the minimal model program. It covers the development of singularity theory for foliations and adjunction formulas that link foliation invariants to the underlying variety. These ingredients enable the full minimal model program for foliations on surfaces and the construction of flips for foliations on threefolds. Sympathetic readers would care because this organizes the birational classification of foliated varieties, which appear in many contexts from differential geometry to algebraic geometry.

Core claim

By focusing on the minimal model program viewpoint, the survey shows that singularities of foliations can be studied in a way compatible with birational geometry, adjunction holds for foliations, the MMP works for foliations on surfaces, and flips exist for foliations on threefolds.

What carries the argument

The minimal model program adapted to foliated varieties, using foliation-specific definitions of singularities and the canonical class of the foliation.

If this is right

  • The MMP can be run for foliations on any surface, producing minimal models.
  • Flips for foliations on threefolds exist, so the MMP can be initiated in that dimension.
  • Adjunction formulas allow control of singularities when restricting foliations to divisors or curves.

Where Pith is reading between the lines

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

  • These results might be used to study the moduli space of foliations or their deformation theory.
  • Extending the framework to higher dimensions or to foliations in positive characteristic could be a natural next step.

Load-bearing premise

The survey assumes the reader has substantial background in the classical minimal model program and the theory of foliations on complex varieties.

What would settle it

Discovery of a foliated threefold where no flip exists for a given extremal ray would falsify the existence claim for flips.

read the original abstract

We survey recent progress on the birational geometry of foliations on complex varieties. We focus on the MMP viewpoint: singularities, adjunction and applications to the MMP for foliations on surfaces and to the existence of flips on threefolds.

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 manuscript is a survey of recent progress on the birational geometry of foliations on complex varieties. It focuses on the minimal model program viewpoint, covering singularities of foliations, adjunction formulas, applications of the MMP to foliations on surfaces, and the existence of flips for foliations on threefolds.

Significance. As a survey consolidating existing results without advancing new theorems, the paper offers a useful reference for specialists if the cited literature is represented accurately. It highlights key developments in a specialized subfield of algebraic geometry, potentially aiding navigation of the MMP for foliations.

minor comments (1)
  1. [Abstract] The abstract provides no definitions or introductory material, assuming substantial prior knowledge of the classical MMP and foliation theory; this is a presentation choice but may affect broader accessibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our survey manuscript and for recommending acceptance. The report accurately captures the scope of the paper as a review of recent progress on the MMP for foliations.

Circularity Check

0 steps flagged

Survey paper with no original derivations or quantitative claims

full rationale

This manuscript is explicitly a survey of existing results on the birational geometry of foliations, covering singularities, adjunction formulas, the MMP on surfaces, and flips in dimension three. It advances no new theorems, equations, or deductive chains of its own. All technical content is referential to prior literature, with no self-contained derivations that could reduce to fitted parameters, self-definitions, or self-citation chains. The abstract and structure confirm the absence of any load-bearing original claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The survey rests on the standard framework of birational geometry over the complex numbers, including established notions of singularities and adjunction for foliations; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard axioms of algebraic geometry over the complex numbers, including resolution of singularities and properties of the canonical divisor.
    The paper operates within the classical setting of complex varieties and the MMP.
  • domain assumption Existence and basic properties of foliations and their singularities as developed in prior literature.
    The survey focuses on MMP applications to foliations, presupposing the definition and basic theory of foliated varieties.

pith-pipeline@v0.9.0 · 5315 in / 1337 out tokens · 54397 ms · 2026-05-10T17:17:42.181866+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 7 canonical work pages

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