We extend the construction of Calabi-Yau manifolds to hypersurfaces
in non-Fano toric varieties, requiring the use of certain Laurent
defining polynomials, and explore the phases of the corresponding gauged
linear sigma models. The associated non-reflexive and non-convex
polytopes provide a generalization of Batyrev’s original work, allowing
us to construct novel pairs of mirror models. We showcase our proposal
for this generalization by examining Calabi-Yau hypersurfaces in
Hirzebruch n-folds, focusing on n=3,4 sequences, and outline the more
general class of so-defined geometries.
