Classifying smooth lattice polytopes via toric fibrations

TL;DR

This paper characterizes smooth Q-normal lattice polytopes, showing they decompose as Cayley sums when their dimension exceeds twice their degree, using toric fibrations and nef value morphisms.

Contribution

It provides a sharp bound for the decomposition of smooth Q-normal lattice polytopes as Cayley sums based on their dimension and degree.

Findings

Any smooth Q-normal lattice polytope of dimension n ≥ 2d+1 is a Cayley sum of equivalent polytopes.

The proof uses the nef value morphism associated with the toric embedding.

The result confirms a conjecture relating polytope decomposition to dimension and degree.

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.