A simple combinatorial criterion for projective toric manifolds with dual defect

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 \cite{DDP09}) and answers partially an adjunction-theoretic conjecture by Beltrametti-Sommese (see \cite{BSW92},\cite{BS95},\cite{DDP09}). Also, it follows from \cite{DR06} that smooth lattice polytopes with this property are precisely strict Cayley polytopes, which completes the answer in \cite{DDP09} of a question in \cite{BN07} for smooth polytopes.

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Published 1 January 2010