Geometry of canonical genus four curves

We apply Bridgeland stability conditions machinery to describe the geometry of some classical moduli spaces associated with canonical genus four curves in $\mathbb{P}^3$ via an effective control over its wall-crossing. These moduli spaces with several irreducible components include the moduli space of Pandharipande-Thomas stable pairs, and the Hilbert scheme, which are not previously studied. We give a full list of irreducible components of the space of stable pairs, along with a birational description of each component, and a partial list for the Hilbert scheme.