An interesting wall-crossing: failure of the wall-crossing/MMP correspondence

We introduce a novel wall-crossing phenomenon in the space of Bridgeland stability conditions: a wall in the stability space of canonical genus 4 curves that induces non-Q-factorial singularities and hence, it cannot be detected as an operation in the Minimal Model Program of the corresponding moduli space, unlike the case for many surfaces. More precisely, we give an example of a wall-crossing in \(\textrm{D}^{b}(\mathbb {P}^{3})\) ${\text{D}}^b({\mathbb{P}}^3)$ such that the wall induces a small contraction of the moduli space of stable objects associated to one of the adjacent chambers, but a divisorial contraction to the other. This significantly complicates the overall picture in this correspondence to applications of stability conditions to algebraic geometry. The full wall-crossing for canonical genus four curves and the geometry are considered in the published paper (Rezaee in Proc LMS 128(1):e12577, 2024); this article is devoted to describe a particularly interesting wall among the walls in Rezaee (Proc LMS 128(1):e12577, 2024) in full details to explain the novel phenomenon.