Embedding Deligne-Mumford stacks into GIT quotient stacks of linear representations

TL;DR

This paper explores a method to embed proper Deligne-Mumford stacks into GIT quotient stacks using ample locally free sheaves and linear representations, facilitating their study via geometric invariant theory.

Contribution

It introduces a new approach to embed Deligne-Mumford stacks into GIT quotients through ample sheaves and linear representations, expanding the toolkit for algebraic geometry.

Findings

Established a construction for embedding stacks into GIT quotients.

Provided conditions under which the embedding exists.

Enhanced understanding of the relationship between stacks and GIT quotients.

Abstract

We study how to use a suitably ample locally free sheaf over a proper Deligne-Mumford stack to furnish an embedding of the stack into a geometric invariant theory (GIT) quotient stack constructed from a finite-dimensional linear representation of the general linear group.