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The intrinsic approach to moduli theory

Jarod Alper, Daniel Halpern-Leistner

Abstract

Moduli theory has captured the imagination of algebraic geometers for at least two centuries. Up until the end of the 20th century, moduli spaces were constructed and studied by rigidifying the moduli problem using extrinsic data and applying geometric invariant theory. Over the last several decades, there has been a paradigm shift toward studying moduli problems intrinsically using the language of algebraic stacks. We highlight recent advances in this direction that have incorporated ideas from geometric invariant theory to develop a structure theory for algebraic stacks. In the ideal situation, it allows one to decompose an algebraic stack into simpler strata and construct moduli spaces corresponding to each stratum. In addition to surveying some previous applications of the theory, we take a forward-looking perspective on the field and identify questions for future research.

The intrinsic approach to moduli theory

Abstract

Moduli theory has captured the imagination of algebraic geometers for at least two centuries. Up until the end of the 20th century, moduli spaces were constructed and studied by rigidifying the moduli problem using extrinsic data and applying geometric invariant theory. Over the last several decades, there has been a paradigm shift toward studying moduli problems intrinsically using the language of algebraic stacks. We highlight recent advances in this direction that have incorporated ideas from geometric invariant theory to develop a structure theory for algebraic stacks. In the ideal situation, it allows one to decompose an algebraic stack into simpler strata and construct moduli spaces corresponding to each stratum. In addition to surveying some previous applications of the theory, we take a forward-looking perspective on the field and identify questions for future research.
Paper Structure (29 sections, 12 theorems, 30 equations, 1 figure)

This paper contains 29 sections, 12 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Local Structure of Algebraic Stacks
  3. Global quotient stacks
  4. Theorems
  5. Applications
  6. Beyond Geometric Invariant Theory
  7. Semistable vector bundles over a curve
  8. Geometric invariant theory
  9. Reinterpreting semistability: $\Theta$-semistability
  10. Good moduli spaces
  11. Beyond GIT: verifying $\Theta$- and ${\textsf{S}}$-completeness
  12. Theta-stratifications
  13. One parameter degenerations in algebraic stacks
  14. The construction
  15. Application to equivariant degenerations
  16. ...and 14 more sections

Key Result

Theorem 2.2

Let $\mathcal{X}$ be a quasi-separated algebraic stack of finite type over an algebraically closed field $\Bbbk$, with affine automorphism groups. For a point $x \in \mathcal{X}(\mathbb{C})$ with linearly reductive stabilizer $G_x$, there exists an étale morphism inducing an isomorphism of automorphism groups at $w$ahr.

Figures (1)

  • Figure 1: The effective and ample cone in the $\lambda\delta$-plane

Theorems & Definitions (34)

  • Theorem 2.2: Local Structure Theorem I
  • Theorem 2.3: Local Structure Theorem II
  • Theorem 2.4: Local Structure Theorem III
  • Lemma 3.1
  • Definition 3.2
  • Remark 3.3
  • Example 3.4: GIT
  • Example 3.5: K-semistability
  • Definition 3.6
  • Definition 3.7
  • ...and 24 more