$G$-Coregularity of del Pezzo Surfaces

We introduce and study the notion of $G$-coregularity of algebraic varieties endowed with an action of a finite group $G$. We compute the $G$-coregularity of smooth del Pezzo surfaces of degree at least $6$, and give a characterization of groups that can act on conic bundles with $G$-coregularity $0$. We describe the relations between the notions of $G$-coregularity, $G$-log canonical thresholds, $G$-birational rigidity, and exceptional quotient singularities.