  The notion of higher order dual varieties of a projective variety is a
natural generalization of the classical notion of projective duality,
introduced by Piene in 1983. In this paper we study higher order dual varieties
of projective toric embeddings. We compute the degree of the second dual
variety of a smooth toric threefold in geometric and combinatorial terms, and
we classify smooth 2-jet spanned projective embeddings of smooth threefolds
whose second dual variety has dimension less than expected. We also describe
the tropicalization of the k-th dual variety of an equivariantly embedded (not
necessarily normal) toric variety.
