Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras

We study quiver Grassmannians associated with indecomposable representations (of finite dimension) of the Kronecker quiver. We find a cellular decomposition for them and we compute their Betti numbers. As an application, we find a geometric realization for the atomic basis of cluster algebras of type affine-A1 found by Sherman and Zelevinsky (who called it the canonical basis) and those of type affine-A2 found in an earlier paper of the first author