  Let $X$ be a smooth irreducible nondegenerate projective variety and let
$X^*$ denote its dual variety. It is well known that $\sigma_2(X)^*$, the dual
of the 2-secant variety of $X$, is a component of the singular locus of $X^*$.
The locus of bitangent hyperplanes, i.e. hyperplanes tangent to at least two
points of $X$, is a component of the sigular locus of $X^*$. In this paper we
provide a sufficient condition for this component to be of maximal dimension
and show how it can be used to determine which dual varieties of Grassmannians
are normal. That last part may be compared to what has been done for
hyperdeterminants by J. Weyman and A. Zelevinski (1996).
