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Artin-Schreier Root Stacks and lifts of group actions

This paper proves that the action of a connected affine algebraic group with no non-trivial characters on a smooth projective variety over a field of positive characteristic can be lifted to its associated Artin-Schreier root stacks, while also establishing the existence of a $G$ -linearization on a specific tautological invertible sheaf.

Original authors: Sujoy Chakraborty, Souradeep Majumder

Published 2026-06-15

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Sujoy Chakraborty, Souradeep Majumder

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 designing a building (a mathematical "variety") in a very specific, quirky world called Positive Characteristic. In this world, the usual rules of geometry behave a bit differently than in our standard "Characteristic 0" world.

In the standard world, if you have a building with some special "twisted" corners (called stacky points), you can describe the whole building perfectly just by knowing the main floor plan and a simple list of numbers telling you how twisted each corner is. It's like describing a cake just by its shape and the number of candles on it.

But in this quirky Positive Characteristic world, things get messy. The "twisted" corners can be twisted in complicated, non-repeating ways (non-cyclic or even non-abelian). You can't just use a simple list of numbers anymore; you need a much more detailed blueprint. To handle this, mathematicians invented a new tool called an Artin-Schreier root stack. Think of this as a special, high-tech scaffolding system that wraps around your building to capture all those complicated twists and turns.

The Main Problem: Moving the Scaffolding

Now, imagine you have a giant, invisible robot (a Group $G$ ) that can walk around your building and move it around (a Group Action). Maybe the robot rotates the building or slides it sideways.

The big question the authors asked is: If the robot can move the original building, can it also move the new, complicated scaffolding (the Artin-Schreier root stack) that we built around it?

Usually, when you try to move a complex structure, things might fall apart or get misaligned. The authors wanted to know if there was a rule that guarantees the scaffolding moves perfectly in sync with the building.

The Solution: The "No-Identity" Rule

The authors discovered a specific condition that makes this work. They found that if your robot (the Group $G$ ) is "humble"—meaning it has no "non-trivial characters" (a fancy way of saying it doesn't have any simple, independent ways of scaling or stretching things on its own)—then the answer is YES.

If the robot is humble, and the "twisted" part of the building (the divisor $D$ ) stays in place while the robot moves, then the robot can successfully lift its movement up to the scaffolding.

The Analogy:
Think of the building as a dance floor and the scaffolding as a complex set of mirrors hanging above it.

The robot is a choreographer moving the dancers.
The "twisted part" is a specific spotlight that must stay on a certain dancer.
The "humble robot" rule means the choreographer doesn't have a secret superpower to change the size of the room independently.
The Result: If the choreographer follows these rules, they can move the dancers and the mirrors above them perfectly in sync. The mirrors don't get confused or break; they just follow the dance.

The "Tautological" Treasure

The paper also finds something else hidden in the scaffolding. Inside this new structure, there is a special, invisible rope (an invertible sheaf) that acts like a "p-th root" of the original building's main beam.

Usually, taking a "root" of a shape is hard. But the authors proved that because the robot moves the scaffolding so nicely, it also naturally moves this special rope. The rope is "linearized," meaning it has its own set of instructions that tell it how to move along with the robot, keeping everything perfectly aligned.

Summary

In short, this paper says:

The Setup: We have a complex mathematical structure (a stack) built to handle messy geometry in a specific type of math world.
The Challenge: We want to move this structure using a group of symmetries (a robot).
The Discovery: If the robot is "humble" (has no non-trivial characters) and respects the main features of the building, it can move the complex structure perfectly.
The Bonus: The special "root" object inside the structure also moves perfectly with the robot.

This is a foundational result. It doesn't tell you how to build a bridge or cure a disease (the paper doesn't claim those applications); it simply proves that the mathematical machinery for moving these complex shapes works reliably under specific, well-defined conditions.

Technical Summary: Artin-Schreier Root Stacks and Lifts of Group Actions

Problem Statement
In characteristic zero, smooth orbifold curves are well-understood as root stacks over their coarse moduli spaces, uniquely determined by their coarse curves and cyclic stabilizer groups. However, in positive characteristic $p > 0$ , this classification fails because stabilizer groups can be non-cyclic or non-abelian. While Artin-Schreier root stacks, introduced by Kobin, provide a framework to classify stacky curves with wild ramification of order $p$ (paralleling the characteristic zero case), the behavior of group actions on these stacks remains an open question. Specifically, given a connected affine algebraic group $G$ acting on a smooth projective variety $X$ in characteristic $p$ , it is not immediately clear under what conditions this action lifts to the associated Artin-Schreier root stack. Furthermore, the existence of compatible linearizations for invertible sheaves on these stacks under such lifted actions requires investigation.

Methodology
The authors employ the theory of algebraic stacks and normalization to address the lifting problem. The methodology proceeds in three main stages:

Construction of the Stack: The paper utilizes the definition of the Artin-Schreier root stack $\wp^{-1}_m((L, s, f)/X)$ as the normalization of a pullback diagram involving the universal Artin-Schreier cover $\wp_m: [\mathbb{P}(1, m)/\mathbb{G}_a] \to [\mathbb{P}(1, m)/\mathbb{G}_a]$ . The stack is defined by a line bundle $L$ , a section $s$ (defining a divisor $D$ ), and a section $f$ of $L^{\otimes m}$ , where $m$ is coprime to $p$ .
Lifting via Normalization: To lift the $G$ -action from $X$ to the root stack, the authors first construct an action on the unnormalized pullback stack $X_m$ . They define the action on the objects and arrows of the fiber groupoid over a scheme $T$ by utilizing the $G$ -linearization of the line bundle $L$ and the invariance properties of the sections $s$ and $f$ .
Application of Normalization Properties: Once the action is established on the unnormalized stack $X_m$ , the authors invoke a result from [AB] (Proposition A.6) which states that a morphism of locally Noetherian algebraic stacks induces a unique morphism on their normalizations, provided the induced map on points is dominant on irreducible components. Since $G$ is smooth and connected, the normalization of $G \times X_m$ is isomorphic to $G \times X_m^\nu$ , allowing the action to descend to the normalized Artin-Schreier root stack $X_m^\nu$ .

Key Contributions and Results

Theorem 3.10 (Lifting of Group Actions): The primary result establishes a criterion for lifting a $G$ -action on a smooth projective variety $X$ to its Artin-Schreier root stack $X_m^\nu$ . The theorem requires:
- $G$ is a connected affine algebraic group with no non-trivial characters (e.g., connected semisimple groups).
- The effective Cartier divisor $D$ (defined by the zero locus of $s$ ) is $G$ -invariant.
- The section $f \in H^0(X, L^{\otimes m})$ has no common zeros with $s$ , and its restriction to the divisor $mD$ is an invariant section under the induced $G$ -linearization on $L^{\otimes m}|_{mD}$ .
  Under these conditions, the $G$ -action on $X$ admits a lift to a $G$ -action on $X_m^\nu$ .
Theorem 4.6 (Linearization of Tautological Sheaf): The paper proves the existence of a tautological invertible sheaf $N$ on the Artin-Schreier root stack $X_m^\nu$ such that $N^{\otimes p} \cong \pi^*L$ , where $\pi: X_m^\nu \to X$ is the coarse moduli map. Crucially, the authors demonstrate that the lifted $G$ -action on $X_m^\nu$ admits a $G$ -linearization on this sheaf $N$ .

Significance and Claims
The paper claims to provide a necessary framework for studying group actions on Artin-Schreier root stacks, which are essential for understanding stacky curves with wild ramification in positive characteristic. By establishing the lifting criteria and the existence of linearized tautological sheaves, the work opens the door to equivariant studies of these stacks. The authors suggest that, analogous to the characteristic zero case where vector bundles on root stacks relate to parabolic bundles on the coarse moduli space, these results may facilitate the study of sheaves on Artin-Schreier root stacks from an equivariant perspective. The work is presented as a direct extension of techniques used in characteristic zero (specifically referencing [CP]) to the more complex setting of positive characteristic, addressing the specific constraints imposed by wild ramification and the structure of the Artin-Schreier extension.

The Main Problem: Moving the Scaffolding

The Solution: The "No-Identity" Rule

The "Tautological" Treasure

Summary

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