Euler characteristic of quiver Grassmannians and Ringel-Hall algebras of string algebras

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We compute the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and prove their positivity. This generalizes some results by G.C. Irelli [arXiv:0910.2592]. As an application we consider the Ringel-Hall algebra C(A) of some string algebras A and compute in combinatorial terms the products of arbitrary functions in C(A).