I will talk about the recently proposed $Q$-construction of Ballard-Diemer-Favero as a method to produce kernels for wall-crossing functors for VGIT problems on schemes equipped with a $\mathbb G_m$-action. We study this proposal in the setting of singular schemes and produce kernels for wall-crossing functors for VGIT problems in some cases. Our work relies on taking appropriate cdga resolutions and the language of dg-schemes.

This is joint work with D. Favero.