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Foliated Minimal Models and Flops

This paper investigates minimal models and flops for foliations, establishing that rank one foliations with canonical singularities yield unique MMP outputs while proving existence results for co-rank one flops on threefolds, yet simultaneously demonstrating that rank one foliations exhibit unique pathologies—such as the failure of $D$ -flops and canonical models—that are absent in the classical Minimal Model Program.

Original authors: Paolo Cascini, Roktim Mascharak, Calum Spicer

Published 2026-08-11

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CC BY 4.0

Original authors: Paolo Cascini, Roktim Mascharak, Calum Spicer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to build the most efficient, stable version of a house. In the world of mathematics, specifically a field called algebraic geometry, these "houses" are shapes called varieties. For decades, mathematicians have been playing a game called the "Minimal Model Program" (MMP). Think of this as a renovation process where you try to strip a complex, messy building down to its simplest, most elegant form without tearing it apart. You do this by making specific cuts and swaps, hoping to reach a "minimal model"—a version that is as simple as possible but still holds all the essential structural information.

Usually, if you start with the same messy house and follow the rules of the game, you might end up with a few different-looking minimal models. However, there's a comforting rule: these different versions are all connected by a specific type of swap called a "flop." Imagine a flop as a magical renovation where you swap a hallway for a staircase; the house looks different from the outside, but the total space and the number of rooms remain the same. It's like having two different floor plans that are actually just different angles of the same perfect home. This paper dives into a twist on this game. Instead of just looking at the house itself, the authors are studying "foliations." If a variety is a house, a foliation is like a set of invisible, flowing currents or wind patterns moving through the rooms. These currents can get stuck, swirl, or hit walls in complicated ways. The big question is: if you try to simplify a house that has these tricky wind patterns inside, do you still get the same reliable results? Do the different simplified versions still connect nicely, or does the wind make the whole renovation process chaotic and unpredictable?

This paper, written by Paolo Cascini, Roktim Mascharak, and Calum Spicer, investigates exactly that. They explore what happens when you apply the minimal model rules to these "windy" houses, specifically looking at two different types of wind patterns: those that flow in just one direction (rank one) and those that flow in a way that leaves only one dimension of "room" to move sideways (co-rank one).

The authors discover that the answer depends entirely on which type of wind you are dealing with. For the "one-direction" winds (rank one foliations), the results are surprisingly rigid and unique. They prove that if you start with a specific type of house and a one-direction wind, and you follow the renovation rules to get a minimal model, you will always end up with the exact same result, no matter which path you take. There are no different floor plans to swap between; the destination is fixed. However, they also find that these one-direction winds can cause strange glitches. Sometimes, even when the wind seems stable, the renovation process hits a dead end where a "flop" (the magical swap) simply cannot happen. In other cases, the wind is so stubborn that you can't even find a final, perfect "canonical model" for the house, even if you allow yourself to build in the flexible world of algebraic spaces. It's as if the wind refuses to let the house settle into a final, stable shape.

On the flip side, for the "co-rank one" winds (which are more complex and swirl in a specific way), the authors show that the renovation process works much more smoothly, at least in three-dimensional spaces. They prove that if you have a house with these specific swirling winds and you need to perform a flop, you can almost always do it. They provide a set of conditions—like checking if the wind hits a specific wall or if the curve of the wind is smooth—that guarantee the flop exists. This is a big deal because, in the world of standard house renovations (without wind), sometimes these swaps are impossible to construct. The authors show that the presence of these specific wind patterns actually helps force the swap to happen, making the process more predictable than one might expect.

To reach these conclusions, the team used some clever mathematical tools. For the rigid one-direction winds, they developed a way to look at the "flow" of the wind using something called "foliated jets," which is like taking a high-speed, multi-layered photograph of the wind's path to ensure it doesn't get stuck in a corner. For the swirling winds, they used the concept of "separatrices," which are like invisible fences that the wind follows. By carefully analyzing how these fences interact with the parts of the house being renovated, they could prove that the necessary swaps were possible.

The paper also constructs some fascinating counter-examples to show where the rules break down. They built a specific mathematical house with a one-direction wind where the "null locus" (the part of the house where the wind stops or becomes zero) cannot be removed or contracted, even in the most flexible mathematical settings. This is a stark contrast to the classical rules, where such stubborn parts can usually be smoothed out. They also showed a case where a "flopping contraction" (a specific type of renovation step) exists, but the corresponding "flop" (the swap) does not, proving that the wind can sometimes block the path forward entirely.

In short, this paper tells us that when you add the complexity of flowing patterns to the geometry of shapes, the rules of simplification change. Sometimes, the patterns make the outcome strictly unique and rigid. Other times, they create new, stubborn obstacles that prevent the standard renovation tools from working. And in some specific, complex cases, the patterns actually help force the renovation to succeed where it might otherwise fail. It's a reminder that in the mathematical universe, the "wind" inside a shape can dictate whether the shape can be simplified, and if so, exactly how it will look in the end.

Technical Summary: Foliated Minimal Models and Flops

Problem Statement
The paper investigates the birational geometry of foliations, specifically addressing the uniqueness of minimal models and the existence of flops within the context of the Minimal Model Program (MMP) for foliations. While the classical MMP for varieties establishes that minimal models are unique up to a finite sequence of flops (Kawamata), the behavior of foliated MMPs is less understood. The authors aim to determine whether analogous uniqueness and existence results hold for foliations, distinguishing between two primary cases: rank one foliations and co-rank one foliations on threefolds.

Methodology
The authors employ a combination of techniques from the classical MMP, foliation theory, and complex analytic geometry:

Foliated Jets and Bertini-type Theorems: For rank one foliations, the authors utilize the vector bundle of foliated $m$ -jets ( $J^m_F L$ ) to construct sections of ample line bundles that avoid specific invariant subvarieties. This involves analyzing the non-vanishing of foliated jets on the locus where the foliation is regular and Gorenstein.
Separatrices and Formal Geometry: For co-rank one foliations, the methodology relies heavily on the theory of separatrices (invariant surfaces) and their extension to formal completions along curves. The authors use the Camacho-Sad formula and adjunction results to analyze the intersection properties of these separatrices with flopping curves.
Reduction to klt Models: The existence of flops for co-rank one foliations is reduced to the construction of klt (Kawamata log terminal) ample models. This involves perturbing boundaries to create klt pairs where standard MMP results (specifically [BCHM10]) can be applied.
Counter-Example Construction: To demonstrate pathologies, the authors construct specific examples using Hilbert modular surfaces and projective bundles over them. These constructions involve lifting tautological foliations and introducing tangencies to create foliations with specific cohomological and dynamical properties (e.g., non-abundant foliations).

Key Contributions and Results

Uniqueness of Minimal Models for Rank One Foliations:
The paper proves that for a normal projective $\mathbb{Q}$ -factorial variety $X$ with a rank one foliation $\mathcal{F}$ having canonical singularities and a pseudo-effective canonical divisor $K_{\mathcal{F}}$ , any two outputs of the $K_{\mathcal{F}}$ -MMP are isomorphic.
- Theorem 1.1: If $f_1: X \dashrightarrow Y_1$ and $f_2: X \dashrightarrow Y_2$ are two outputs of a $K_{\mathcal{F}}$ -MMP, there exists an isomorphism $\phi: Y_1 \to Y_2$ such that $f_2 = \phi \circ f_1$ .
- Significance: This contrasts with the classical case where minimal models are only connected by flops; here, for rank one foliations with canonical singularities, the minimal model is unique. The proof utilizes a Bertini-type statement ensuring the existence of divisors containing no log canonical centers of the foliation outside its singular locus.
Existence of Flops for Co-Rank One Foliations:
The authors establish existence results for $D$ -flops in the context of co-rank one foliations on threefolds under specific hypotheses.
- Theorem 1.2 (F-dlt case): For a smooth threefold $X$ with an F-dlt co-rank one foliation $\mathcal{F}$ , if $f: X \to Z$ is a $K_{\mathcal{F}}$ -flopping contraction and specific conditions regarding the irreducibility of the exceptional curve $C$ , the normality of a separatrix $S_1$ , and the ampleness of $-S_1$ are met, then the $D$ -flop exists. The proof relies on the existence of separatrices meeting the flopping curve properly and analyzing Camacho-Sad indices.
- Theorem 1.3 (klt case): For a klt quasi-projective threefold with a co-rank one foliation $\mathcal{F}$ and a boundary $\Delta$ such that $(\mathcal{F}, \Delta)$ is klt, $D$ -flops exist for small contractions where $K_{\mathcal{F}} + \Delta$ is numerically trivial.
Pathologies in Rank One Foliations:
The paper demonstrates that rank one foliations exhibit behaviors absent in the classical MMP, specifically regarding the existence of flops and canonical models.
- Non-existence of Flops (Theorem 5.2): The authors construct a smooth threefold with a rank one foliation having canonical singularities and a flopping contraction that does not admit a $D$ -flop. This occurs because the relevant algebra is not finitely generated, a phenomenon linked to the non-torsion nature of the canonical divisor restricted to the flopping curve.
- Failure of the Basepoint-Free Theorem (Theorem 5.3): The paper constructs a smooth projective threefold with a rank one foliation where the canonical divisor $K_{\mathcal{F}}$ is nef and big, yet the null locus cannot be contracted, even in the category of algebraic spaces. This failure is attributed to the non-abundance of the foliation, obstructing the existence of a canonical model.

Significance
The paper clarifies the structural differences between the birational geometry of varieties and foliations. It establishes that while co-rank one foliations on threefolds largely follow the expected MMP patterns (existence of flops under reasonable conditions), rank one foliations display a unique rigidity in their minimal models (uniqueness up to isomorphism) but also exhibit severe pathologies regarding the existence of flops and canonical models. These results highlight that the foliated MMP cannot be treated as a straightforward extension of the classical MMP; specific dynamical and cohomological obstructions, such as non-abundance and the behavior of separatrices, play critical roles. The work provides necessary counter-examples to the universality of the basepoint-free theorem in the foliated setting and refines the understanding of when flops exist for co-rank one foliations.

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