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

Let $X$ be a smooth complex projective variety. In 2002, Bridgeland defined a notion of stability for the objects in $D^b(X)$, the bounded derived category of coherent sheaves on $X$, which generalized the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on $X$ and the geometry of the variety. We construct new stability conditions for surfaces containing a curve $C$ whose self-intersection is negative. We show that these stability conditions lie on a wall of the geometric chamber of $\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}(X)$, the stability manifold of $X$. We then construct the moduli space $M_{\sigma }({\mathcal{O}}_X)$ of $\sigma$-semistable objects of class $[{\mathcal{O}}_X]$ in $K_0(X)$ after wall-crossing.