Hirzebruch Surfaces, Tyurin Degenerations and Toric Mirrors: Bridging Generalized Calabi-Yau Constructions

There is a large number of different ways of constructing Calabi-Yau manifolds, as well as related non-geometric formulations, relevant in string compactifications. Showcasing this diversity, we discuss explicit deformation families of discretely distinct Hirzebruch hypersurfaces in $\mathbb{P}^n \times \mathbb{P}^1$ and identify their toric counterparts in detail. This precise isomorphism is then used to investigate some of their special divisors of interest, and in particular the secondary deformation family of their Calabi-Yau subspaces. In particular, most of the above so called Hirzebruch scrolls are non-Fano, and their (regular) Calabi-Yau hypersurfaces are Tyurin-degenerate, but admit novel (Laurent) deformations by special rational sections as well as a sweeping generalization of the transposition construction of mirror models. This bi-projective embedding also reveals a novel deformation connection between distinct toric spaces, and so also the various divisors of interest including their Calabi-Yau subspaces.