The dual varieties of a projective variety can be studied via the conormal varieties lying in the point-hyperplane incidence variety. In the case the variety is a toric embedding, properties of the dual varieties can be translated into properties of the lattice point configuration or polytope. In particular, one can give combinatorial criteria for such a variety to be dual defective or (higher) selfdual. Several examples, including Cayley configurations and rational normal scrolls, will be given. Connections with diophantine problems will be highlighted. This is joint work with Alicia Dickenstein.
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