  We study the $A$-discriminant of toric varieties. We reduce its computation
to the case of irreducible configurations and describe its behavior under
specialization of some of the variables to zero. We prove a Gale dual
characterization of dual defect toric varieties and deduce from it the
classsification of such varieties in codimension less than or equal to four.
This classification motivates a decomposition theorem which yields a sufficient
condition for a toric variety to be dual defect. For codimension less than or
equal to four, this condition is also necessary and we expect this to be the
case in general.
