Computing isolated orbifolds in weighted flag varieties

Given a weighted flag variety wΣ(μ,u) corresponding to chosen fixed parameters μ and u, we present an algorithm to compute lists of all possible projectively Gorenstein n-folds, having canonical weight k and isolated orbifold points, appearing as weighted complete intersections in wΣ(μ,u) or some projective cone(s) over wΣ(μ,u). We apply our algorithm to compute lists of interesting classes of polarized 3-folds with isolated orbifold points in the codimension 8 weighted G₂ variety. We also show the existence of some families of log-terminal Q-Fano 3-folds in codimension 8 by explicitly constructing them as quasilinear sections of a weighted G₂-variety.