[Paper Review] On the MMP for rank one foliations on threefolds

Paolo Cascini, Calum Spicer|arXiv (Cornell University)|Dec 21, 2020

Algebraic Geometry and Number Theory63 references26 citations

TL;DR

This paper establishes the existence of flips for log canonical rank one foliated pairs on Q-factorial klt threefolds, extending McQuillan's minimal model program for foliations. By constructing a foliated complement and analyzing the formal neighborhood of a flipping curve via a normal form of the induced foliation, the authors prove that flips exist under mild singularities, enabling the running of the foliated MMP and minimal model existence for pseudo-effective canonical foliations.

ABSTRACT

We prove existence of flips for log canonical foliated pairs of rank one on a Q-factorial projective klt threefold. This, in particular, provides a proof of the existence of a minimal model for a rank one foliation on a threefold for a wider range of singularities, after McQuillan.

Motivation & Objective

• To extend McQuillan's minimal model program for foliations to threefolds with log canonical singularities.
• To establish the existence of flips for rank one foliated pairs (F, Δ) on Q-factorial klt threefolds.
• To relate the birational geometry of foliations to classical birational geometry by bounding singularities of the ambient variety.
• To prove that a threefold with a canonical foliation and an isolated singularity must have log canonical singularities.
• To provide a framework for the existence of minimal models for foliations on threefolds under broader singularities than previously known.

Proposed method

• Reduces the KF-flip to a (KX + D)-flip by constructing a highly singular, F-invariant divisor D containing the flipping curve C.
• Uses a foliated complement E on the target Z such that Kf∗F + E is Q-Cartier and (f∗F, E) has mild singularities.
• Analyzes the induced foliation f∗F on Z via a normal form given by a vector field of the form ∑nixi∂/∂xi with non-negative integers ni.
• Performs a detailed analysis of the formal neighborhood of the flipping curve C to construct a surgery resembling a flip via weighted blow-ups and blow-downs.
• Applies the classical MMP techniques to the pair (X, D) while preserving foliation invariance through the choice of D.
• Uses induction and divisorial contraction arguments on auxiliary divisors Gk and Eℓ to show that the MMP over Z terminates and contracts all exceptional divisors.

Experimental results

Research questions

• RQ1 Can flips be constructed for log canonical foliated pairs of rank one on threefolds with klt singularities, extending McQuillan’s result beyond quotient singularities?
• RQ2 What is the relationship between the singularities of the ambient threefold and the singularities of a canonical foliation on it?
• RQ3 Can the existence of minimal models for foliations on threefolds be established under broader singularities than previously known?
• RQ4 Is it possible to bound the singularities of the ambient variety by those of the foliation without assuming a canonical model theorem for foliations on threefolds?
• RQ5 How can the formal neighborhood of a flipping curve be described in terms of foliation invariants to enable explicit construction of the flip?

Key findings

• The existence of flips is proven for log canonical foliated pairs (F, Δ) of rank one on Q-factorial klt projective threefolds, as stated in Theorem 8.8.
• The minimal model program for such foliations can be run under the assumption that KF + Δ is pseudo-effective, leading to the existence of a minimal model, as in Theorem 8.10.
• If a normal threefold admits a canonical foliation with an isolated singularity, then the threefold itself must have log canonical singularities, as shown in Theorem 4.3.
• The formal neighborhood of a flipping curve in a threefold with a rank one foliation admits a normal form described by a vector field with non-negative integer coefficients, enabling explicit construction of the flip.
• The MMP over the target Z of a flipping contraction contracts all exceptional divisors Gk and Eℓ, ensuring termination and the existence of the flip.
• The final flip f+: X+ → Z is small and KF+ is ample over Z, confirming that the flip satisfies the required minimality condition.