  Let $\Aa_t$ be the directed quiver of type $\Aa$ with $t$ vertices. For each
dimension vector $d$ there is a dense orbit in the corresponding representation
space. The principal aim of this note is to use just rank conditions to define
the irreducible components in the complement of the dense orbit. Then we
compare this result with already existing ones by Knight and Zelevinsky, and by
Ringel. Moreover, we compare with the fan associated to the quiver $\Aa$ and
derive a new formula for the number of orbits using nilpotent classes. In the
complement of the dense orbit we determine the irreducible components and their
codimension. Finally, we consider several particular examples.
