Abstract

We show that any smooth Q -normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n ⩾ 2 d + 1 . This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill.

Highlights

  • Let P be an n-dimensional lattice polytope in Rn. We represent it as an intersection of half spaces

  • As we shall see in Proposition 2.2, the following definition embodies the polytope version of the notion of nef value for projective varieties

  • An n-dimensional lattice polytope P in Rn is called Q-normal if codegQ(P ) = τ (P )

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Summary

Introduction

Let P be an n-dimensional lattice polytope (i.e., a convex polytope with integer vertices) in Rn. Let P be a smooth n-dimensional lattice polytope and s 1 an integer. As we shall see in Proposition 2.2, the following definition embodies the polytope version of the notion of nef value for projective varieties. Let P ⊂ Rn be an n-dimensional smooth lattice polytope. An n-dimensional lattice polytope P in Rn is called Q-normal if codegQ(P ) = τ (P ). Smooth generalized strict Cayley polytopes are natural examples of Q-normal polytopes. For n-dimensional smooth Q-normal lattice polytope, we can take N (d) = 2d + 1. If P is a smooth Q-normal lattice polytope of dimension n and degree d such that n 2d + 1, P is a strict Cayley polytope. We conjecture that the assumption τ (P ) = codegQ(P ) always holds for smooth lattice polytopes satisfying codeg(P ).

The codegree and the nef value
Generalized Cayley polytopes and toric fibrations
Classifying smooth lattice polytopes with high codegree
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