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One of the applications of GIT is that it allows to single out parts of moduli problems that admit separated coarse moduli spaces. This is often formulated in terms of stability conditions. Intersingly, numerical stability criteria can be formulated purely in terms of moduli problems (this was formulated independently by Daniel Halpern-Leistner).
One can also give a stack theoretic criterion showing that this often defines separated substacks of moduli stacks admitting coarse moduli spaces. Sometimes this allows to avoid difficult GIT calculations. We apply this to parahoric group schemes on curves. | 
