2008/09/18 by Alicia Dickenstein, Dickenstein, Alicia, Sandra Di Rocco +3
Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.CO
paper · pdf · doi:10.48550/arxiv.0809.3136
Revised version, minor changes. To appear in Advances in Math
arxiv created 2009/04/01 · arxiv updated 2009/12/01
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.