A characterization of virtually free actions via arc spaces and its application to the lower semi-continuity conjecture

In this paper, we study the precise inversion of adjunction (PIA) conjecture and the lower semi-continuity (LSC) conjecture for hyperquotient singularities. Previously known results for these conjectures in this setting required the singularity to be klt, and without this assumption, a counterexample to the PIA conjecture is known to exist. To resolve this obstacle, we introduce a localized notion of virtually free actions and characterize it via the arc spaces of quotient varieties. Utilizing this characterization, we establish a necessary and sufficient condition for the PIA conjecture to hold for arbitrary hyperquotient singularities, thereby clarifying the mechanism of the counterexample. Furthermore, as an application of this insight, we unconditionally establish the LSC conjecture for arbitrary hyperquotient singularities.