The Harder-Narasimhan stratification explained how to cut out subspaces of the moduli problem of principal bundles on curves that admit a proper moduli space. This prompted many results of similar nature and by now we know geometric criteria that decide whether an open substack of a moduli problem admits a proper quotient. In many examples these are in the end determined by the datum of a line bundle or a cohomology class in degree 2 of the large moduli problem. In the simple example of quotient stacks obtained from a torus action there is a combinatorial conjecture by Bialynicki-Birula and Sommese indicating that there should be other methods to cut out open subsets admitting proper quotients. I would like to explain how this can be understood by looking at other cohomological degrees.

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