The intrinsic approach to moduli theory

TL;DR

This paper reviews the shift from extrinsic to intrinsic methods in moduli theory, emphasizing the role of algebraic stacks and recent advances in structure theory for decomposing stacks into simpler components.

Contribution

It surveys recent developments in intrinsic moduli theory using algebraic stacks and geometric invariant theory, highlighting new structure results and future research directions.

Findings

Development of a structure theory for algebraic stacks

Decomposition of stacks into simpler strata

Construction of moduli spaces for each stratum

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.