Consider the stack of ideal sheaves of l points on a smooth projective variety X. Its good moduli space is the well-known Hilbert scheme of l points Hilb^l(X). In this talk, I will define a certain enlargement U of this stack and prove that it admits a good moduli space. Furthermore, I will describe a different open substack of U that admits a good moduli space, and explain how it yields a surgery diagram via non-GIT wall-crossing. As an application in the toy case l=1, I will explain how this construction helps contract rational curves on higher-dimensional varieties, and use the interpretation of the surgery as a fine moduli of sheaves to prove instances Kawamata’s DK-hypothesis in this setting. This is joint work with Andres Fernandez Herrero.