Extremality of principal quiver Grassmannians

We study families of quiver Grassmannians, parametrizing subrepresentations of representations of Dyn\-kin quivers, which are naturally associated to a projective and an injective representation.

Our main result is an explicit description of the flat irreducible locus of such a family.

More precisely, we prove that a member of this family is irreducible and of the expected dimension if and only if the ambient quiver representation degenerates to the direct sum of the defining projective and injective representations.