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Caldero and Zelevinsky studied the geometry of quiver Grassmannians for theKronecker quiver and computed their Euler characteristics by examining naturalstratification of quiver Grassmannians. We consider generalized Kroneckerquivers and compute virtual Poincare polynomials of certain varieties which arethe images under projections from strata of quiver Grassmannians to ordinaryGrassmannians. In contrast to the Kronecker quiver case, these polynomials donot necessarily have positive coefficients. The key ingredient is the explicitformula for noncommutative cluster variables given by Ralf Schiffler and thefirst author.

On natural maps from strata of quiver Grassmannians to ordinary Grassmannians