---
date: 2022-04-07 19:25
modification date: Thursday 7th April 2022 19:25:17
title: "Artin stack"
aliases: [Artin stack, differentiable stack, geometric stack]
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- Tags: 
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- Refs:
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# Artin stack

- Example: the moduli stack of stable curves. Proving it is Artin uses a method of *tricanonical embeddings*, constructed using [Hilbert schemes](Unsorted/Hilbert%20scheme.md).
- [geometric points](Unsorted/geometric%20fiber.md) do not usually admit universal deformation rings.
- Can prove Artin stacks are [Deligne-Mumford stacks](Unsorted/Deligne-Mumford%20stack.md) using an *unramified diagonal* argument.
- Common requirement: diagonals that are represented by separated [algebraic space](Unsorted/algebraic%20space.md) [of finite type](of%20finite%20type).

![](attachments/Pasted%20image%2020220413092951.png)
![](attachments/Pasted%20image%2020220413093009.png)
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