Moduli spaces of semistable pairs on projective Deligne-Mumford stacks

TL;DR

This paper extends the construction of moduli spaces of semistable pairs to projective Deligne-Mumford stacks, analyzing their deformation theories and defining invariants in three dimensions.

Contribution

It generalizes the moduli space construction for semistable pairs to the setting of projective Deligne-Mumford stacks, including deformation, obstruction theories, and virtual classes.

Findings

Constructed moduli spaces of semistable pairs on stacks.

Developed deformation and obstruction theories for stable pairs.

Defined Pandharipande-Thomas invariants for three-dimensional stacks.

Abstract

We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective Deligne-Mumford stacks. We study the deformation and obstruction theories of stable pairs, and then prove the existence of virtual fundamental classes for some cases of dimension two and three. This leads to a definition of Pandharipande-Thomas invariants on three-dimensional smooth projective Deligne-Mumford stacks.