I am going to talk about a recent paper with Alicia Dickenstein, which extends work of Sandra Di Rocco, Ragni Piene and Alicia. We give an alternative way how to check whether a projective toric manifold associated to a smooth lattice polytope has dual defect. This involves a combinatorial invariant, called the codegree, which is of independent interest in the theory of lattice polytopes.

If time permits, I will explain how our result fits in nicely with a general conjecture of Beltrametti and Sommese in the adjunction theory of polarized manifolds.