A Simple Combinatorial Criterion for Projective Toric Manifolds with Dual Defect

TL;DR

This paper establishes a combinatorial criterion linking lattice polytopes with dual defectiveness in projective toric manifolds, providing new insights into their structure and confirming conjectures in algebraic geometry.

Contribution

It introduces a simple combinatorial condition that characterizes dual defective projective toric manifolds and confirms their equivalence to strict Cayley polytopes in the smooth case.

Findings

Polytope with high codegree defines a dual defective toric manifold.

Such polytopes are Q-normal and are exactly strict Cayley polytopes.

Answers a conjecture related to adjunction theory and toric geometry.

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 a recent paper by Di Rocco, Piene and the first author) and answers partially an adjunction-theoretic conjecture by Beltrametti and Sommese. Also, it follows that smooth lattice polytopes with this property are precisely strict Cayley polytopes, which completes the answer of a question of Batyrev and the second author in the nonsingular case.