  Caldero and Zelevinsky studied the geometry of quiver Grassmannians for the
Kronecker quiver and computed their Euler characteristics by examining natural
stratification of quiver Grassmannians. We consider generalized Kronecker
quivers and compute virtual Poincare polynomials of certain varieties which are
the images under projections from strata of quiver Grassmannians to ordinary
Grassmannians. In contrast to the Kronecker quiver case, these polynomials do
not necessarily have positive coefficients. The key ingredient is the explicit
formula for noncommutative cluster variables given by Ralf Schiffler and the
first author.
