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

TL;DR

This paper provides an algebraic proof of a characterization of invariant open subsets with proper quotients under $ extbf{G}_m$-actions on normal proper varieties, utilizing recent moduli space results.

Contribution

It offers a new algebraic proof of a classical result on quotients under $ extbf{G}_m$-actions, incorporating modern moduli space techniques.

Findings

Characterization of invariant open subsets with proper quotients.

Application of algebraic stack moduli space results.

Extension of classical G_m-action quotient theory.

Abstract

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.