Classifying smooth lattice polytopes via toric fibrations

Details Classifying smooth lattice polytopes via toric fibrations Classifying smooth lattice polytopes via toric fibrations

2008-09-18

arxiv.org/abs/0809.3136v2

Mathematics

Algebraic Geometry

Revised version, minor changes. To appear in Advances in Math

Scientific paper

We define Q-normal lattice polytopes. Natural examples of such polytopes are Cayley sums of strictly combinatorially equivalent lattice polytopes, which correspond to particularly nice toric fibrations, namely toric projective bundles. In a recent paper Batyrev and Nill have suggested that there should be a bound, N(d), such that every lattice polytope of degree d and dimension at least N(d) decomposes as a Cayley sum. We give a sharp answer to this question for smooth Q-normal polytopes. We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a Cayley sum of strictly combinatorially equivalent polytopes if n is greater than or equal to 2d+1. The proof relies on the study of the nef value morphism associated to the corresponding toric embedding.

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