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

TL;DR

This paper demonstrates that wall-crossing phenomena in Bridgeland stability conditions can fail to correspond with the birational geometry of moduli spaces, revealing complexities in their relationship.

Contribution

It provides explicit examples where wall-crossing does not align with the Minimal Model Program, challenging previous assumptions about their correspondence.

Findings

A wall in stability space does not always induce a step in the MMP.

An example of a wall causing different types of contractions in moduli spaces.

The failure of the wall-crossing/MMP correspondence complicates applications in algebraic geometry.

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.