On Relative Moduli Spaces of Two-Dimensional Sheaves on Threefolds

Abstract

In the study of degeneration methods for moduli problems, Li introduced expanded pairs and expanded degenerations in order to construct proper relative moduli spaces. These notions play a crucial role in the work of Li on relative Gromov-Witten theory, and later on, in the work of Li and Wu on the construction of good degenerations of Quot schemes and the development of degeneration formulas for Donaldson-Thomas invariants of ideal sheaves. Using these constructions, as well as the properties of Quot schemes and the moduli space of semistable sheaves, we construct the relative moduli space of two-dimensional Gieseker-stable sheaves on threefolds and show that it satisfies both the existence and uniqueness parts of the valuative criterion for properness.