  We give an algebraic proof of a result, due to Bialynicki-Birula and Sommese,
characterizing the invariant open subsets of a normal proper variety equipped
with a $\mathbf{G}_m$-action that admit a proper good quotient. A major
ingredient is the existence result for moduli spaces of algebraic stacks due to
Alper, Halpern-Leistner and Heinloth.
