above. If the groups $G$ and $G'$ are equal, then the BHK mirrors $Z_{A^T, G^T}$ and $Z_{(A')^T, (G')^T}$ of these orbifolds are birational.

This theorem drops a hypothesis from Shoemaker’s theorem in that the hypersurfaces $X_A$ and $X_{A'}$ need not be in the same weighted-projective space. It is unclear if this is a strict generalization of Shoemaker’s result as no examples of a common group of phase symmetries $G$ are known when $X_A$ and $X_{A'}$ are hypersurfaces in different weighted-projective spaces; however, the proof we present is novel in that it does not use any toric geometry but rather uses rational maps known as Shioda maps. Using Shioda maps, we show that the Calabi-Yau orbifolds $Z_{A,G}$, $Z_{A',G'}$, $Z_{A^T, G^T}$, and $Z_{(A')^T, (G')^T}$ are all birational to different finite group quotients of a Fermat variety in projective space ${\mathbb{P}}^n$. In the case where the group $G$ equals $G'$, we see that the mirrors $Z_{A^T, G^T}$ and $Z_{(A')^T, (G')^T}$ are birational to the same quotient of the Fermat variety.

Originally, Shioda used these maps to compute Picard numbers of Delsarte surfaces in [@Shi]. These maps entered the multiple mirror literature in [@BvGK] where they were generalized and then used to investigate Picard-Fuchs 