Computing isolated orbifolds in weighted flag varieties

Given a weighted flag variety $w\Sigma(\mu,u)$ corresponding to chosen fixed parameters $\mu$ 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 $w\Sigma(\mu,u) $ or some projective cone(s) over $w\Sigma(\mu,u)$. We apply our algorithm to compute lists of interesting classes of polarized 3-folds with isolated orbifold points in the codimension 8 weighted $G_2$ variety. We also show the existence of some families of log-terminal $\mathbb Q$-Fano 3-folds in codimension 8 by explicitly constructing them as quasilinear sections of a weighted $G_2$-variety.

💡 Research Summary

The paper introduces an algorithmic framework for enumerating all possible projectively Gorenstein n‑dimensional varieties with isolated orbifold points that arise as weighted complete intersections (or as sections of projective cones) inside a fixed weighted flag variety wΣ(µ,u). The authors start by recalling the necessary background on cyclic quotient singularities, baskets, graded rings, and Hilbert series. They emphasize the decomposition of the Hilbert series of an isolated orbifold, due to Buckley–Reid–Zhou (BRZ), into an “initial term” PI(t) representing the smooth contribution and a sum of “orbifold terms” PQi(t) corresponding to each singular point in the basket. The initial term is a rational function with numerator A(t), a Gorenstein‑symmetric polynomial of degree equal to the co‑index c = k + n + 1, while each PQi(t) has denominator (1‑t)^n (1‑t^r) and a numerator B(t) obtained as the inverse modulo of the product ∏(1‑t^{a_i}) with respect to (1‑t^r).

The algorithm proceeds in four concrete steps:

1. Hilbert series and canonical class of the ambient weighted flag variety.
   For a given pair (µ,u) the weighted flag variety wΣ(µ,u) is embedded into a weighted projective space P