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Cracked polytopes and Fano toric complete intersections. (English) Zbl 1469.14104

The context of the paper under review is the construction and classification of Fano varieties via mirror symmetry.
Stemming from the Laurent inversion construction in [T. Coates et al., Pure Appl. Math. Q. 15, No. 4, 1135–1179 (2019; Zbl 1439.14122)], the author introduces cracked polytopes. These, by definition, are such that their intersection with the minimal cones of a complete fan (shape) is a set of unimodular polytopes. The paper connects the notion of cracked polytopes to embeddings of toric varieties \(X_P\) into possibly smooth ambient toric varieties \(Y\). Along with extra conditions, the (local) smoothness of \(Y\) is equivalent to requiring that \(P\) is cracked along a shape.
The author mainly examines the case in which \(X_P\) embeds in \(Y\) as a complete intersection, making use of the scaffolding technique. A particular attention is given to Fano polytopes, as cracked polytopes well describe toric degenerations of Fano varieties with very ample anticanonical divisor, especially in dimension 3 and 4.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14J45 Fano varieties
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Citations:

Zbl 1439.14122
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Full Text: DOI arXiv

References:

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