Moduli space of pairs over projective stacks

TL;DR

This paper develops a framework for defining and constructing moduli spaces of semistable pairs over projective stacks, extending classical concepts to a stack setting with GIT-based stability conditions.

Contribution

It introduces a new notion of pairs over projective stacks, constructs their moduli stack and space, and defines semistability via a polynomial parameter and GIT methods.

Findings

Constructed the stack and moduli space of semistable pairs.

Defined semistability depending on a polynomial parameter.

Extended classical moduli theory to the setting of projective stacks.

Abstract

Let $\clX$ a projective stack over an algebraically closed field $k$ of characteristic 0. Let $\clE$ be a generating sheaf over $\clX$ and $\clO_X(1)$ a polarization of its coarse moduli space $X$. We define a notion of pair which is the datum of a non vanishing morphism $\Gamma\otimes\clE\to \clF$ where $\Gamma$ is a finite dimensional $k$ vector space and $\clF$ is a coherent sheaf over $\clX$. We construct the stack and the moduli space of semistable pairs. The notion of semistability depends on a polynomial parameter and it is dictated by the GIT construction of the moduli space.