Abstract: Good moduli space maps are generally far from separated. However for many purposes they behave as if they are proper maps. In this talk I will explain two recent results in this direction (joint with Elmanto, Satriano and Inchiostro, Satriano respectively) 1) that a universal family exists over a proper generically finite covering of the good moduli space, and 2) that the good moduli space map satisfies a strong version of the existence part of the valuative criterion of properness. Along the way I will explain several structural results about good moduli spaces and especially gerbes.
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