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On the singularities of quotients by 1-foliations

This paper investigates the singularities of varieties formed as infinitesimal quotients by 1-foliations in positive characteristic, demonstrating that quotients by log canonical foliations preserve MMP singularities, quotients by multiplicative derivations preserve most F-singularities, and introducing a framework for families of such foliations and their quotients.

Original authors: Quentin Posva

Published 2026-08-26

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CC BY 4.0

Original authors: Quentin Posva

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

In the world of modern geometry, mathematicians study shapes that exist in many dimensions, often trying to understand how these shapes can be broken down, rearranged, or simplified without losing their essential nature. A central tool in this endeavor is the idea of a "foliation," which can be thought of as a way of slicing a shape into layers, much like the pages of a book or the strata of a geological formation. In standard geometry, these layers flow smoothly. However, in a specific branch of mathematics dealing with positive characteristic—a field of study where numbers behave differently than in our everyday arithmetic, often used to solve deep problems in algebra—these layers can develop sharp kinks or singularities. When a shape is sliced by such a foliation, the resulting "quotient" is a new shape formed by collapsing the layers together. The question that has long puzzled researchers is: when we perform this collapse, does the new shape remain well-behaved, or does it become hopelessly broken?

A recent paper by Quentin Posva tackles this question by investigating the specific types of singularities that appear when shapes are quotiented by what are called "one-foliations." These are special, rigid structures that follow strict rules of movement and repetition. Posva's work focuses on two main types of these structures: those that behave in a "multiplicative" way, meaning they repeat in a predictable, scaling pattern, and those that are "canonical," a term describing a specific, mild level of imperfection. The author's primary goal is to determine whether the mathematical properties of the original shape are preserved, improved, or destroyed when it is transformed into its quotient.

The study reveals a surprising resilience in the mathematical structure. When a shape is divided by a multiplicative derivation—a specific kind of rule that generates the slicing layers—the resulting quotient retains many of the original shape's most important health markers. If the original shape was "F-pure" or "F-regular," terms that describe a high degree of structural integrity and resistance to collapse, the new shape remains just as robust. This happens because the process of dividing by these multiplicative rules is mathematically equivalent to a very specific, gentle type of symmetry operation. The author proves that this operation is so well-behaved that it cannot introduce new, severe defects into the geometry. In fact, for surfaces, the paper establishes a precise link: a quotient is perfectly healthy if and only if the slicing rule itself is of the multiplicative type.

The findings are even more striking when looking at the "birational" properties of these shapes, which concern how they can be transformed into one another by stretching or shrinking without tearing. Posva demonstrates that if a shape is sliced by a "canonical" or "log canonical" foliation—rules that are already quite mild—the resulting quotient is guaranteed to have birational singularities that are at least as mild as, or better than, those of the original variety. It is as if the act of slicing, under these specific conditions, acts as a filter that removes the worst imperfections or preserves the existing quality. This is a powerful result because it means that mathematicians can use these quotients to construct new, cleaner shapes from older, more complex ones. However, the paper also draws a firm line in the sand: this smoothing effect is not universal. If the slicing rule is too irregular or "additive" in nature, the resulting shape can become severely damaged, losing properties like being "Cohen-Macaulay," a condition that ensures the shape has no hidden holes or structural gaps.

Beyond the static analysis of single shapes, the paper also explores what happens when these slicing rules vary across a family of shapes, moving from one to another like a sequence of frames in a film. The author introduces a formal definition for these "families" of foliations and asks a critical question: if we take a slice of the family at a specific moment, does it match the slice of the final quotient shape at that same moment? The answer is generally yes, but with a caveat. The paper proves that the two operations—taking a slice and taking a quotient—only perfectly align if the underlying structure of the slicing rule remains smooth and free of sudden jumps. If the rule develops a singularity at a specific point in the sequence, the order of operations matters, and the results can diverge. This distinction is crucial for understanding how these geometric objects evolve and degenerate over time.

Ultimately, this work provides a clear map of the terrain where geometry meets algebra in positive characteristic. It confirms that while the landscape is full of potential pitfalls, there are specific, well-defined paths—those governed by multiplicative and canonical rules—where the journey leads to shapes that are not only intact but often improved. By characterizing exactly when these improvements occur and when they fail, the paper offers a reliable framework for future explorations in this complex field, ensuring that mathematicians know exactly which tools will build a stable structure and which will lead to a collapse.

Technical Summary: On the Singularities of Quotients by 1-Foliations

Problem Statement
In positive characteristic $p > 0$ , a 1-foliation on a normal variety $X$ is defined as a saturated subsheaf of the tangent bundle $T_X$ that is closed under Lie brackets and $p$ -th powers. A fundamental construction associated with such objects is the infinitesimal quotient $X/\mathcal{F}$ , which yields a purely inseparable morphism. While these quotients have been instrumental in constructing pathological examples in algebraic geometry (e.g., [MR83, Hir99a]), the behavior of their singularities remains less understood compared to the characteristic zero case. The paper addresses the qualitative properties of the singularities of $X/\mathcal{F}$ , specifically investigating how the singularity type of the base variety $X$ and the foliation $\mathcal{F}$ influence the resulting quotient.

Methodology
The author employs a combination of local algebraic techniques, the theory of derivations, and birational geometry within the Minimal Model Program (MMP) framework.

Local Analysis of Derivations: The paper distinguishes between additive derivations ( $D^{[p]} = 0$ ) and multiplicative derivations ( $D^{[p]} = uD$ for a unit $u$ ). It utilizes the correspondence between multiplicative derivations and actions of the group scheme $\mu_p$ (or $\alpha_p$ for additive cases) to analyze the structure of invariant subrings.
Splitting Arguments: A core technical tool is the observation that, after an appropriate étale cover, quotients by multiplicative derivations correspond to invariants under a $\mu_p$ -action. This allows the author to utilize the fact that the inclusion of the invariant subring splits, preserving various cohomological properties (such as depth and F-singularities).
Birational Comparison: For the study of birational singularities (terminal, canonical, klt, log canonical), the paper establishes pull-back formulas relating the canonical divisors of $X$ , the foliation $\mathcal{F}$ , and the quotient $X/\mathcal{F}$ . This involves a detailed comparison of discrepancy numbers via the adjunction formula for quotients.
Families of Foliations: The paper introduces a formal definition of families of 1-foliations over a base $S$ and investigates the commutativity of the operations of taking fibers and taking quotients.

Key Contributions and Results

1. Preservation of F-Singularities by Multiplicative Derivations
The paper establishes that quotients by 1-foliations with "at worst multiplicative singularities" (those formally generated by commuting multiplicative derivations) preserve a wide range of cohomological and F-singularities.

Theorem 1: If $X$ $X$ is a normal variety over a perfect field and $\mathcal{F}$ $F$ is a 1-foliation with at worst multiplicative singularities, then:
- If $X$ is Cohen–Macaulay, so is $X/\mathcal{F}$ .
- If $X$ is F-pure, F-rational, F-injective, or F-regular, then $X/\mathcal{F}$ retains the same property.
The proof relies on the splitting of the inclusion $A^\mathcal{F} \hookrightarrow A$ after an étale base change, which ensures that depth and F-singularity properties descend to the subring of invariants.
Note: The paper clarifies that while F-singularities are preserved, the Gorenstein property is generally not preserved, whereas the Q-Gorenstein property is.

2. Birational Singularities and the MMP
The paper investigates how the "MMP singularities" of the quotient relate to those of the base and the foliation.

Theorem 2: If $\mathcal{F}$ $F$ is a canonical (resp. log canonical) 1-foliation on a normal variety $X$ $X$ , the quotient $X/\mathcal{F}$ $X / F$ improves or preserves the birational singularities of $X$ $X$ . Specifically:
- If $\mathcal{F}$ is canonical and $X$ is terminal (resp. canonical, klt, log canonical), then $X/\mathcal{F}$ is terminal (resp. canonical, klt, log canonical).
- If $\mathcal{F}$ is log canonical and $X$ is klt (resp. log canonical), then $X/\mathcal{F}$ is klt (resp. log canonical).
The proof utilizes the equality $q^*(K_{X/\mathcal{F}} + \Delta_{X/\mathcal{F}}) = K_X + \Delta + (p-1)K_\mathcal{F}$ to compare discrepancies. The paper notes that the converse implications generally do not hold.

3. Characterization on Regular Surfaces and Higher Dimensions
The paper provides a complete characterization of log canonical 1-foliations on regular surfaces and partial results for higher dimensions.

Theorem 3: Let $S$ $S$ be a regular surface and $\mathcal{F}$ $F$ a rank 1 1-foliation.
- $\mathcal{F}$ is canonical if and only if it is regular (which implies $S/\mathcal{F}$ is regular).
- $\mathcal{F}$ is log canonical if and only if $S/\mathcal{F}$ is F-regular (equivalently F-pure).
Corollary 1: On a regular variety $X$ , a non-trivial 1-foliation cannot be terminal. If it is canonical, it must be regular outside a closed subset of codimension $\ge 3$ . If it has rank 1, it is canonical if and only if it is regular, and log canonical if and only if it has at worst multiplicative singularities.
These results highlight that the log canonical condition is quite restrictive in dimension two.

4. Families of Foliations and Fiber Commutativity
The paper defines families of 1-foliations and investigates whether the formation of quotients commutes with taking fibers.

Theorem 4: Let $(X \to S, \mathcal{F})$ be a family of 1-foliations with $X \to S$ smooth and $S$ regular. Let $Q = T_{X/S}/\mathcal{F}$ . If the fiber $Q_s$ is locally free for some $s \in S$ , then the fiber of the quotient is isomorphic to the quotient of the fiber: $X_s/\mathcal{F}_s \cong (X/\mathcal{F})_s$ .
The paper provides examples (Examples 5.2.10 and 5.2.11) showing that if $Q_s$ is not sufficiently regular (e.g., not Cohen–Macaulay), commutativity may fail. However, the natural morphism $X_s/\mathcal{F}_s \to (X/\mathcal{F})_s$ is always an isomorphism in codimension one.

Significance and Scope
The paper claims to provide a systematic study of the singularities of infinitesimal quotients, filling a gap in the literature where such constructions were often used to produce counterexamples without a detailed analysis of their singularity types.

It generalizes known results on quotients by multiplicative derivations (previously known for regular varieties) to the case where the underlying variety may be singular.
It establishes a direct link between the log canonicity of a foliation and the F-singularity type of its quotient, particularly on surfaces.
The work is modest in scope regarding non-p-closed vector fields, explicitly excluding them from the discussion (Remark 1.0.1).
The results are presented as foundational for understanding the geometry of purely inseparable morphisms in positive characteristic, with specific applications to normalized $p$ -cyclic coverings (Corollary 4.2.10) and the construction of stable families of surfaces (Examples 5.2.8–5.2.9).

The paper does not propose new experimental applications or future research directions beyond the immediate context of the MMP and the theory of foliations in positive characteristic. It positions itself as a consolidation and extension of existing scattered results (referencing [MP97], [PW22], [Lie08], etc.) into a coherent framework for analyzing singularities.

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