Singularities of duals of Grassmannians

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).