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 )
Summary
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 ).
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