Equivalences of families of stacky toric Calabi-Yau hypersurfaces

Given the same anti-canonical linear system on two distinct toric varieties, we provide a derived equivalence between partial crepant resolutions of the corresponding stacky hypersurfaces. The applications include: a derived unification of toric mirror constructions, calculations of Picard lattices for linear systems of quartics in $\mathbb{P}^3$, and a birational reduction of Reid’s list to 81 families.The first author was supported by NSERC, PIMS, and a McCalla professorship at the University of Alberta. The second author was supported by NSERC through a Discovery Grant and as a Canada Research Chair. The third author was supported in part by NSF Grant # DMS-1401446 and EPSRC Grant EP/N004922/1