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Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces

Yi Hu·Chien-Hao Liu·Shing-Tung Yau
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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 4-dimensional elliptic Calabi-Yau toric hypersurfaces that appear in a recent work of Braun-Candelas-dlOssa-Grassi. The Maple codes that are employed for the computation are provided. Some directions for future work are listed in the end.

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Yi Hu, Chien-Hao Liu, Shing-Tung Yau. 2004-01-02. Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces. https://arxiv.org/abs/math/0010082

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