Groupoid Action on a DM Stack

Oct 13, 2025 by magicalceylontours.com 70 views

Introduction

In the realm of algebraic geometry, the concept of Deligne-Mumford (DM) stacks plays a crucial role in the study of moduli spaces. A DM stack is a stack that is locally isomorphic to a scheme, and it is a fundamental object of study in algebraic geometry. When a groupoid acts on a DM stack, it is natural to ask whether the quotient of the stack by the groupoid action exists. In this article, we will explore the concept of quotient of a DM stack by a groupoid action and discuss the conditions under which it exists.

Groupoid Action on a DM Stack

Let $\mathcal{X}$ be a Deligne-Mumford stack over a scheme $S$ of characteristic zero. A groupoid $\mathcal{G}$ over $\mathcal{X}$ is a stack with a pair of étale morphisms $s, t: \mathcal{G} \to \mathcal{X}$ , where $s$ is the source map and $t$ is the target map. The groupoid $\mathcal{G}$ acts on $\mathcal{X}$ if there exists a morphism $\mathcal{G} \times_{\mathcal{X}} \mathcal{G} \to \mathcal{G}$ , where $\mathcal{G} \times_{\mathcal{X}} \mathcal{G}$ is the fiber product of $\mathcal{G}$ with itself over $\mathcal{X}$ .

Quotient of a DM Stack by a Groupoid Action

Given a groupoid $\mathcal{G}$ acting on a DM stack $\mathcal{X}$ , we would like to define the quotient stack $\mathcal{X}/\mathcal{G}$ . The quotient stack is a stack that is locally isomorphic to the quotient of the scheme underlying $\mathcal{X}$ by the groupoid $\mathcal{G}$ . To define the quotient stack, we need to specify a morphism $\mathcal{X} \to \mathcal{X}/\mathcal{G}$ and a groupoid $\mathcal{G}/\mathcal{X}$ acting on $\mathcal{X}/\mathcal{G}$ .

Conditions for the Existence of the Quotient Stack

The quotient stack $\mathcal{X}/\mathcal{G}$ exists if and only if the groupoid $\mathcal{G}$ acts freely on $\mathcal{X}$ . This means that for any $x \in \mathcal{X}$ , the stabilizer group $\mathcal{G}_x$ is trivial. In other words, the groupoid $\mathcal{G}$ acts on $\mathcal{X}$ without fixed points.

Construction of the Quotient Stack

Assuming that the groupoid $\mathcal{G}$ acts freely on $\mathcal{X}$ , we can construct the quotient stack $\mathcal{X}/\mathcal{G}$ as follows. Let $\mathcal{X}^{\mathcal{G}}$ be the quotient of the scheme underlying $\mathcal{X}$ by the groupoid $\mathcal{G}$ . Then $\mathcal{X}^{\mathcal{G}}$ is a scheme, and we can define a morphism $\mathcal{X} \to \mathcal{X}^{\mathcal{G}}$ by sending each point $x \in \mathcal{X}$ to its orbit $\mathcal{G}x \subset \mathcal{X}^{\mathcal{G}}$ . The groupoid $\mathcal{G}/\mathcal{X}$ acts on $\mathcal{X}^{\mathcal{G}}$ by sending each point $\mathcal{G}x \in \mathcal{X}^{\mathcal{G}}$ to its orbit $\mathcal{G}(\mathcal{G}x) \subset \mathcal{X}^{\mathcal{G}}$ .

Properties of the Quotient Stack

The quotient stack $\mathcal{X}/\mathcal{G}$ has several important properties. First, it is a DM stack, and it is locally isomorphic to the quotient of the scheme underlying $\mathcal{X}$ by the groupoid $\mathcal{G}$ . Second, the morphism $\mathcal{X} \to \mathcal{X}/\mathcal{G}$ is a quotient map, meaning that it is surjective and the fibers are the orbits of the groupoid $\mathcal{G}$ on $\mathcal{X}$ . Finally, the groupoid $\mathcal{G}/\mathcal{X}$ acts freely on $\mathcal{X}/\mathcal{G}$ , meaning that the stabilizer group $(\mathcal{G}/\mathcal{X})_{\mathcal{X}/\mathcal{G}}$ is trivial.

Examples

There are several examples of DM stacks and groupoids that can be used to illustrate the concept of quotient of a DM stack by a groupoid action. For instance, consider the DM stack $\mathcal{X} = \mathbb{P}^1$ and the groupoid $\mathcal{G} = \mathbb{G}_m$ acting on $\mathcal{X}$ by sending each point $x \in \mathcal{X}$ to its orbit $\mathbb{G}_mx \subset \mathcal{X}$ . Then the quotient stack $\mathcal{X}/\mathcal{G}$ is isomorphic to the DM stack $\mathbb{P}^1/\mathbb{G}_m$ , which is a scheme.

Conclusion

In conclusion, the quotient of a DM stack by a groupoid action is a fundamental concept in algebraic geometry. It provides a way to define the quotient of a DM stack by a groupoid action, and it has several important properties. The quotient stack is a DM stack, and it is locally isomorphic to the quotient of the scheme underlying the DM stack by the groupoid action. The morphism from the DM stack to the quotient stack is a quotient map, and the groupoid acting on the quotient stack acts freely. The concept of quotient of a DM stack by a groupoid action has several applications in algebraic geometry, and it is an active area of research.

References

[1] Deligne, P., and Mumford, D. "The irreducibility of the space of curves of given genus." Publications Mathématiques de l'IHÉS, vol. 36, 1969, pp. 75-110.
[2] Mumford, D. "Geometric Invariant Theory." Springer-Verlag, 1965.
[3] Artin, M. "Algebraic stacks." Proceedings of the International Congress of Mathematicians, 1970, pp. 35-55.

Future Directions

Introduction

In our previous article, we discussed the concept of quotient of a DM stack by a groupoid action. We defined the quotient stack and its properties, and we provided several examples to illustrate the concept. In this article, we will answer some frequently asked questions about the quotient of a DM stack by a groupoid action.

Q: What is the difference between a DM stack and a scheme?

A DM stack is a stack that is locally isomorphic to a scheme, whereas a scheme is a locally ringed space that is locally isomorphic to an affine space. In other words, a DM stack is a stack that has a scheme as its underlying space, whereas a scheme is a space that has a ringed space as its underlying space.

A: What is the significance of the groupoid action on a DM stack?

The groupoid action on a DM stack is a way to define the quotient of the DM stack by the groupoid action. The groupoid action provides a way to define the orbits of the groupoid on the DM stack, and it provides a way to define the quotient stack.

Q: What are the conditions for the existence of the quotient stack?

The quotient stack exists if and only if the groupoid acts freely on the DM stack. This means that for any point in the DM stack, the stabilizer group of the point is trivial.

A: What are the properties of the quotient stack?

The quotient stack is a DM stack, and it is locally isomorphic to the quotient of the scheme underlying the DM stack by the groupoid action. The morphism from the DM stack to the quotient stack is a quotient map, and the groupoid acting on the quotient stack acts freely.

Q: Can you provide an example of a DM stack and a groupoid that can be used to illustrate the concept of quotient of a DM stack by a groupoid action?

Consider the DM stack $\mathcal{X} = \mathbb{P}^1$ and the groupoid $\mathcal{G} = \mathbb{G}_m$ acting on $\mathcal{X}$ by sending each point $x \in \mathcal{X}$ to its orbit $\mathbb{G}_mx \subset \mathcal{X}$ . Then the quotient stack $\mathcal{X}/\mathcal{G}$ is isomorphic to the DM stack $\mathbb{P}^1/\mathbb{G}_m$ , which is a scheme.

Q: What are the applications of the quotient of a DM stack by a groupoid action?

The quotient of a DM stack by a groupoid action has several applications in algebraic geometry. It provides a way to define the quotient of a DM stack by a groupoid action, and it provides a way to study the properties of the quotient stack.

A: What are the future directions for research on the quotient of a DM stack by a groupoid action?

There are several future directions for research on the quotient of a DM stack by a groupoid action. One direction is to study the properties of the quotient stack in more detail, such as its cohomology and its moduli space. Another direction is to apply the concept of quotient of a DM stack by a groupoid action to other areas of mathematics, such as representation theory and number theory.

Q: Can you provide a list of references for further reading on the quotient of a DM stack by a groupoid action?

Here are some references for further reading on the quotient of a DM stack by a groupoid action:

[1] Deligne, P., and Mumford, D. "The irreducibility of the space of curves of given genus." Publications Mathématiques de l'IHÉS, vol. 36, 1969, pp. 75-110.
[2] Mumford, D. "Geometric Invariant Theory." Springer-Verlag, 1965.
[3] Artin, M. "Algebraic stacks." Proceedings of the International Congress of Mathematicians, 1970, pp. 35-55.

Conclusion

In conclusion, the quotient of a DM stack by a groupoid action is a fundamental concept in algebraic geometry. It provides a way to define the quotient of a DM stack by a groupoid action, and it has several important properties. The quotient stack is a DM stack, and it is locally isomorphic to the quotient of the scheme underlying the DM stack by the groupoid action. The morphism from the DM stack to the quotient stack is a quotient map, and the groupoid acting on the quotient stack acts freely. We hope that this article has provided a helpful introduction to the concept of quotient of a DM stack by a groupoid action.