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Foliations with small singular set in arbitrary characteristic

This paper investigates the geometry of foliations on smooth algebraic varieties over fields of arbitrary characteristic by classifying regular foliations on specific surfaces, extending the Camacho-Sad index formula, establishing conditions for $p$ -closedness on projective spaces, and proving Bott vanishing theorems for $p$ -closed foliations.

Original authors: Thiago Fassarella, Wodson Mendson, João Pedro dos Santos, Frédéric Touzet

Published 2026-07-22

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CC BY 4.0

Original authors: Thiago Fassarella, Wodson Mendson, João Pedro dos Santos, Frédéric Touzet

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 the universe not as a collection of separate objects, but as a giant, flowing river. In mathematics, this river is called a "variety," and the way the water moves is described by something called a "foliation." Think of a foliation like a stack of paper or a loaf of sliced bread: the whole shape is made up of many thin, smooth layers (called "leaves") that fit together perfectly without tearing or overlapping. If you were a tiny ant walking on one of these leaves, you would never be able to step onto a different leaf without jumping; you are stuck on your own path.

For a long time, mathematicians studied these rivers in a world where the rules of arithmetic were the same as the ones we use in school (like $1+1=2$ ). This is called "characteristic zero." But there is another world, a strange and magical place called "positive characteristic," where the rules of counting are different. In this world, if you count up to a certain number $p$ (a prime number like 2, 3, or 5), you don't keep going to $p+1$ ; instead, you wrap around and start over at zero. It's like a clock that only has 5 hours: after 5 comes 0 again. In this weird world, the behavior of our river of leaves changes in surprising ways. Sometimes the leaves stop flowing smoothly and get stuck in loops, or they behave in ways that seem impossible in our normal world. Mathematicians care about this because understanding these strange rules helps them see the deeper, hidden structure of geometry itself, revealing patterns that are invisible when we only look at the "normal" world.

This paper is a detective story set in that magical, wrap-around world. The authors, Thiago Fassarella and his team, are investigating what happens to these "leafy" rivers when they are smooth and regular, but live in a landscape that has very few "rough spots" (singularities). They wanted to know: Do the leaves in this strange world behave like they do in our normal world, or do they do something completely wild?

First, they tackled the simplest landscapes: surfaces that look like spheres or twisted tubes (called rational and del Pezzo surfaces). In our normal world, if a river flows smoothly across such a surface, it usually follows a predictable path. The authors found that in the wrap-around world, this is mostly true too, but with a twist. If the surface is a simple sphere or a specific type of twisted tube, the leaves are always "p-closed." This is a fancy way of saying the leaves are trapped in perfect, repeating loops that fit the rules of the wrap-around clock. However, they discovered a surprising exception: on a slightly more complex surface (a "weak del Pezzo" surface) in a world where the clock only has 2 hours (characteristic 2), the leaves can break free and flow in a way that is not p-closed. It's like finding a river that refuses to loop back on itself, even though the rules of the world say it should.

Next, the team looked at what happens when the river hits a wall (an invariant hypersurface). In the wrap-around world, if a smooth wall exists that the river flows along, and that wall doesn't touch any of the river's rough spots, the river is forced to behave in a very specific, rigid way. The authors proved that the river must be p-closed, and the "degree" of the river (a measure of its complexity) must be a multiple of the clock's number $p$ . It's as if the wall acts like a strict teacher, forcing the river to follow the rules of the wrap-around clock perfectly. They also showed that if the river is not following these rules, it simply cannot have such a smooth wall without hitting a rough spot.

Finally, the authors connected these findings to a famous mathematical tool called the "Bott vanishing theorem." In the normal world, this theorem says that certain complex numbers associated with the river's shape must disappear (become zero) under specific conditions. The team proved that in the wrap-around world, these numbers don't just disappear; they vanish in a very specific way, becoming multiples of the clock number $p$ . This confirms that the strange, wrap-around rules of this world leave a distinct fingerprint on the geometry of the river.

In short, the paper maps out the rules of these leafy rivers in the strange wrap-around world. It confirms that while most rivers behave predictably and get trapped in loops, there are rare, exotic landscapes where they can break free. It also proves that if a river flows smoothly along a wall without hitting a bump, it is forced to obey the strict, repeating rules of the wrap-around clock. These findings don't just solve a puzzle; they provide a new, clearer picture of how geometry behaves when the fundamental rules of counting are turned upside down.

Technical Summary: Foliations with Small Singular Set in Arbitrary Characteristic

Problem Statement
This paper investigates the geometry of foliations on smooth algebraic varieties over an algebraically closed field $k$ of arbitrary characteristic $p \geq 0$ . The central problem is to understand how the behavior of foliations in positive characteristic diverges from or aligns with the classical theory in characteristic zero. Specifically, the authors address the classification of regular foliations on specific rational surfaces, the conditions under which foliations are $p$ -closed (a property unique to positive characteristic where the $p$ -th power of a vector field tangent to the foliation remains tangent), and the constraints imposed by the existence of invariant subvarieties with small singular sets.

Methodology
The authors employ a combination of algebraic geometry techniques tailored to positive characteristic:

Local Analysis and Blow-ups: The study of singularities via local coordinates, blow-ups, and the analysis of strict transforms. The authors utilize the interplay between the vanishing order of defining 1-forms and the geometry of exceptional divisors.
Index Theory: A significant portion of the methodology involves extending the Camacho-Sad index and its associated sum formula to arbitrary characteristic. This allows for the calculation of self-intersection numbers of invariant curves in terms of local indices.
Intersection Theory: The use of intersection formulas involving the normal bundle $N_F$ , the canonical bundle $K_X$ , and the tangent sheaf $T_F$ to derive constraints on the existence of regular foliations.
Frobenius Morphisms and Descent: For $p$ -closed foliations, the authors utilize the geometric Frobenius morphism and the theory of $F$ -quotients (descent to the Frobenius twist) to establish vanishing theorems in the Chow ring.
Cohomological Tools: The paper applies Bott vanishing theorems in Hodge cohomology and the Chow ring, relating the vanishing of Chern classes of the normal bundle to the $p$ -closed nature of the foliation.

Key Contributions and Results

Classification on Rational Surfaces:
- Minimal Rational Surfaces: The authors provide a complete classification of regular foliations on minimal rational surfaces. They prove that such surfaces must be Hirzebruch surfaces $\mathbb{F}_e$ with $e \neq 1$ . In characteristic zero, only the standard $\mathbb{P}^1$ -fibration exists. In positive characteristic $p$ , if $p$ divides $e$ , a new class of regular Riccati foliations arises. Crucially, they establish that all regular foliations on minimal rational surfaces in positive characteristic are $p$ -closed.
- Del Pezzo Surfaces: For smooth projective del Pezzo surfaces, the authors show that any regular foliation leaves one of the standard $\mathbb{P}^1$ -fibrations invariant. Consequently, these foliations are also $p$ -closed.
- Weak Del Pezzo Surfaces: The paper presents a counterexample to the general expectation that regular foliations on rationally connected varieties are $p$ -closed. They construct regular foliations on weak del Pezzo surfaces of degree 4 in characteristic 2 that are not 2-closed. However, they prove that for weak del Pezzo surfaces of degree 6 or 8, regular foliations are $p$ -closed.
Invariant Subvarieties and Singular Sets:
- Small Singular Sets: The authors extend the Camacho-Sad formula to arbitrary characteristic. Using this, they prove that if a reduced curve $C \subset \mathbb{A}^2_k$ is invariant under a foliation $\mathcal{F}$ and satisfies $\deg(C)(\deg(C)-1) < p$ , then the singular locus of the curve is contained in the singular locus of the foliation ( $\text{Sing}(C) \subset \text{Sing}(\mathcal{F})$ ).
- Projective Spaces: For codimension one distributions on $\mathbb{P}^n_k$ , the existence of an invariant hypersurface $Y$ such that $Y \cap \text{Sing}(D)$ has codimension at least 3 forces the distribution to be integrable, $p$ -closed, and defined by the kernel of an exact differential $dF$.
- Degree Constraints: These results imply strict divisibility conditions on the degree of the normal bundle. Specifically, if a codimension one foliation on $\mathbb{P}^n_k$ has a singular set of codimension $\geq 3$ , then $p$ divides the degree of the normal bundle (which is $d+2$ ).
Vanishing Theorems:
- Hodge Cohomology: The paper establishes versions of the Bott vanishing theorem in Hodge cohomology for regular foliations, showing that suitable powers of Chern classes of the normal bundle vanish.
- Chow Ring: A novel contribution is the establishment of a "Bott vanishing" theorem in the Chow ring for $p$ -closed foliations. The authors prove that for a regular $p$ -closed foliation $\mathcal{F}$ of codimension $q$ on a smooth projective variety $X$ , the $\ell$ -th Chern class of the normal bundle satisfies $\text{Chern}_\ell(N_\mathcal{F}) \in p^{\ell-q} \cdot \text{CH}_\ell(X)$ for $\ell > q$ . This result relies on the descent of the normal bundle to the Frobenius twist of the variety.

Significance and Claims
The paper claims to highlight both the similarities and differences between foliation theory in characteristic zero and positive characteristic.

Contrast with Characteristic Zero: The existence of regular, non- $p$ -closed foliations on weak del Pezzo surfaces in characteristic 2 serves as a distinct feature of positive characteristic, contrasting with the algebraic integrability conjectures often made for characteristic zero.
Generalization of Classical Results: The extension of the Camacho-Sad formula and the Bott vanishing theorem to arbitrary characteristic provides a robust framework for studying foliations with small singular sets.
Structural Insights: The results on projective spaces and rational surfaces suggest that while "wild" behavior (non- $p$ -closed foliations) exists, it is constrained to specific geometric configurations (e.g., specific degrees of weak del Pezzo surfaces). For most "standard" rational varieties (projective spaces, minimal rational surfaces, del Pezzo surfaces), the regular foliations exhibit the expected $p$ -closed behavior.

The authors emphasize that their work provides necessary conditions for the existence of regular foliations and establishes that the presence of invariant hypersurfaces with small singular sets imposes strong arithmetic constraints (divisibility by $p$ ) on the geometry of the foliation.

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