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.