The birational geometry of moduli spaces via Bridgeland stability

In 2006, Tom Bridgeland introduced the notion of stability conditions for objects in an arbitrary triangulated category. The original motivation came from Mirror Symmetry, as this was the generalization of the notion of stability for D-branes fathered by Michael Douglas. Soon afterwards, a whole stream of results flourished around Bridgeland’s original idea, and Bridgeland stability conditions had been seen to find applications in geometric representation theory and in binational geometry. In our talk, we will see how wall-crossing phenomena arising from the structure of the set of stability conditions on an algebraic surface give a new powerful machinery to study and, in some cases, to completely describe the binational geometry of certain moduli spaces of sheaves on the given algebraic surface. In a joint work with Huizenga, Lin, Riedl, Schmidt, Woolf and Zhao, we will apply this machinery to a very special moduli space of sheaves, called Hilbert scheme of points, to progress towards a description of its binational geometry via its Nef cone.