Arcara–Miles conjecture on destabilising line bundles on surfaces

Let $X$ be a smooth projective surface with a fixed Kähler class $[\omega_0]$ and a line bundle $L$. The stability condition $\sigma_{[\omega_0]}$ is the stability condition supported on the tilted heart $\operatorname{Coh}^{\sharp}(X)$ described above, and $L$ and $L[1]$ are viewed as objects of the derived category. Arcara–Miles conjecture. (i) If $L$ is unstable with respect to $\sigma_{[\omega_0]}$, then it is destabilised by $L(-C)$, where $C$ is a curve of negative self-intersection. (ii) If $L[1]$ is unstable with respect to $\sigma_{[\omega_0]}$, then it is destabilised by $L(C)|_C$, where $C$ is a curve of negative self-intersection. This conjecture predicts that instability of these line-bundle objects is detected by curves of negative self-intersection, providing a bridge between Bridgeland stability on surfaces and the geometry of negative curves. The supplied text gives no resolution, so the conjecture is recorded as open.