Saturated blowups and canonical reduction of stabilizers.

Abstract: We introduce a construction call the {\em saturated blowup} of an Artin stack with good moduli space. The saturated blowup is a birational map of stacks which induces a proper birational map on good moduli spaces. Given an Artin stack ${\mathcal X}$ with good moduli space $X$, there is a canonical sequence of saturated blowups which produces a stack whose rigidification is a DM stack. When the stack is smooth, all of the stacks in the sequence of saturated blowups are also smooth. This construction generalizes earlier work of Kirwan and Reichstein in geometric invariant theory and the talk is based on joint work with David Rydh.