[Paper Review] Algebraic stacks

Tomás L. Gómez|arXiv (Cornell University)|Nov 25, 1999

Advanced Topics in Algebra75 citations

TL;DR

This expository paper develops the theory of algebraic stacks, focusing on the moduli stack of vector bundles and comparing it in detail with the moduli scheme constructed via geometric invariant theory. It provides a comprehensive, accessible introduction to algebraic stacks with a central example illustrating the advantages and distinctions of the stack-theoretic approach over classical moduli schemes.

ABSTRACT

This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector budles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.

Motivation & Objective

• To provide a clear, accessible introduction to the general theory of algebraic stacks for researchers unfamiliar with the framework.
• To examine the moduli stack of vector bundles as a central example, illustrating how stacks resolve limitations of classical moduli spaces.
• To compare the stack-theoretic moduli space of vector bundles with the moduli scheme obtained via geometric invariant theory (GIT).
• To highlight the advantages of stacks in capturing automorphisms and parametrizing families with non-trivial stabilizers.
• To serve as a foundational reference for researchers working in moduli theory, algebraic geometry, and related areas.

Proposed method

• Introduces algebraic stacks via the language of fibered categories and descent theory, emphasizing their role as generalizations of schemes.
• Uses the category of vector bundles over a base scheme as a motivating example to illustrate stack structure.
• Applies the theory of quotient stacks to describe the moduli stack of vector bundles as a quotient of a parameter space by a group action.
• Compares the stack-theoretic construction with the GIT quotient construction, focusing on the difference in universal properties and stabilizer groups.
• Employs descent data and groupoid presentations to formalize the stack structure and verify representability.
• Analyzes the relationship between the stack and its coarse moduli space, particularly in terms of points and automorphisms.

Experimental results

Research questions

• RQ1 How do algebraic stacks improve the construction of moduli spaces when automorphisms are non-trivial?
• RQ2 What is the precise relationship between the moduli stack of vector bundles and the moduli scheme obtained via geometric invariant theory?
• RQ3 In what ways does the stack-theoretic approach resolve issues present in classical moduli schemes, such as non-separatedness or failure to represent families?
• RQ4 How do stabilizer groups in the stack reflect the geometry of vector bundles with automorphisms?
• RQ5 What are the key structural differences between the stack and its coarse moduli space in the context of vector bundles?

Key findings

• The moduli stack of vector bundles provides a finer and more natural parametrization of families than the GIT moduli scheme, especially when bundles have non-trivial automorphisms.
• The stack captures the full automorphism data of vector bundles, which the GIT scheme often fails to represent correctly.
• The coarse moduli space obtained via GIT is not always a fine moduli space, whereas the stack serves as a fine moduli space in the stack-theoretic sense.
• The comparison reveals that the stack has better functorial properties and is more suitable for universal constructions.
• The stack-theoretic approach resolves issues such as non-separatedness and non-representability that can arise in classical moduli problems.
• The paper demonstrates that the stack of vector bundles is a Deligne-Mumford stack when the base scheme is Noetherian and the bundles are stable, under appropriate conditions.