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Abstract:
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. Joint work with
Sandra di Rocco and Ragni Piene.
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