A Simple Combinatorial Criterion for Projective Toric Manifolds with Dual Defect

Abstract

We show that any smooth lattice polytope P with codegree greater or equal than (dim(P) + 3)/2 (or equivalently, with degree smaller than dim(P)/2), defines a dual defective projective toric manifold. This implies that P is Q-normal (in the terminology of [11]) and answers partially an adjunction-theoretic conjecture by BeltramettiSommese (see [5],[4],[11]). Also, it follows from [24] that smooth lattice polytopes with this property are precisely strict Cayley polytopes, which completes the answer in [11] of a question in [1] for smooth polytopes.