Proper good quotients for ${\mathbf{G}}_m$-actions

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.