Schubert decompositions for quiver Grassmannians of tree modules

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Let Q be a quiver, M a representation of Q with an ordered basis and a dimension vector for Q. In this note we extend the methods of L12 to establish Schubert decompositions of quiver Grassmannians (M) into affine spaces to the ramified case, i.e.the canonical morphism F:T→ Q from the coefficient quiver T of M w.r.t. is not necessarily unramified. In particular, we determine the Euler characteristic of (M) as the number of extremal successor closed subsets of T₀, which extends the results of Cerulli Irelli (Cerulli11) and Haupt (Haupt12) (under certain additional assumptions on).