Moduli spaces of semistable pairs on projective Deligne–Mumford stacks

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.