### Abstract

Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. We then turn to the study of the restriction of a toric morphism to a toric hypersurface. The details of this can be understood by the various restrictions of a line bundle with a section that defines the hypersurface. These general toric geometry discussions give rise to a computational scheme for the details of a toric morphism and the induced fibration of toric hypersurfaces therein. We apply this scheme to study the family of complex 4-dimensional elliptic Calabi-Yau toric hypersurfaces that appear in a recent work of Braun-Candelas-dlOssa-Grassi. Some directions for future work are listed in the end.

Original language | English (US) |
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Pages (from-to) | 457-505 |

Number of pages | 49 |

Journal | Advances in Theoretical and Mathematical Physics |

Volume | 6 |

Issue number | 3 |

State | Published - May 1 2002 |

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### Keywords

- Elliptic calabi-yau manifold
- Fibration
- Fibred calabi-yau manifold
- Heterotic-string/F-theory duality
- Primitive cone
- Relative star
- Toric calabi-yau hypersurface
- Toric morphism

### ASJC Scopus subject areas

- Mathematics(all)
- Physics and Astronomy(all)

### Cite this

*Advances in Theoretical and Mathematical Physics*,

*6*(3), 457-505.