Bridgeland stability conditions on surfaces with curves of negative self-intersection (Q2157800)

Let \(X\) be a smooth projective surface with nef divisor \(H\). Assume that there is a rational curve \(C \subset X\) such that \((H\cdot C) =0\) and \(C^2 = n <0\), e.g, the zero section in the Hirzeburch surface (as projective bundle over \(\mathbb{P}^1\)) for \(n >0\), or \((-2)\)-curves in (quasi)-polarized K3 surfaces. Under this assumption, the authors construct Bridgeland stability conditions \[ \sigma_{H,\beta,s} =(Z_{H,\beta}, \mathcal{B}_{H,k}^{-\Im(z)}) \] on the bounded derived category of coherent sheaves \(D^b(X)\), which satisfy the support property and lie on a wall of the geometric chamber. Via wall-crossing, they construct a morphism to the moduli of \(\sigma\)-stable object in \(D^b(X)\) for some stability condition \(\sigma\) in the geometric chamber, with class \([\mathcal{O}_x] \in K_0(X)\) for some point \(x \in X\): \[ X \sqcup_C \mathbb{P}^{n-1} \to M_{\sigma}(X,[\mathcal{O}_{x}]), \] which is actually an isomorphism.