Birationality of weighted projective spaces via twisted sectors

Prove that two weighted projective spaces P(w1,…,wm) and P(w′1,…,w′m) with pairwise coprime weight pairs are birational if and only if they share birational twisted sectors.

Background

Twisted sectors in weighted projective spaces are lower-dimensional weighted projective subspaces; the authors seek to leverage Chen–Ruan data to detect birationality.

They formulate a conjectural criterion that equates birationality with agreement of twisted-sector birational types under coprimality assumptions.

References

Conjecture Let $\bb P(w_1,\ldots,w_m)$, $\bb P(w'_1, \ldots, w'_m)$ be two weighted projective spaces with $\gcd(w_i, w_j) = \gcd(w'_i, w'_j) = 1$ for any $i, j \in {1, \ldots, m}$ with $w_i \neq w_j$, $w'_i \neq w'_j$. Then $\bb P(w_1, \ldots, w_m)$ is birational to $\bb P(w'_1, \ldots, w'_m)$ if and only if they share birational twisted sectors.

— A Gromov-Witten approach to $G$-equivariant birational invariants  (2405.07322 - Cavenaghi et al., 2024) in Section 6.2, “Birational invariants for divisorial orbifolds,” Weighted projective spaces