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

Abstract

We show that the wall-crossing in Bridgeland stability fails to be detected by the birational geometry of stable sheaves, and vice versa. There is a wall in the stability space of canonical genus four curves which does not induce a step in the Minimal Model Program. More precisely, we give an example of a wall-crossing in ${\mathrm{D}}^b({\mathbb{P}}^3)$ 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.

30 pages, 2 figures. Comments welcome