Let be a quiver, a representation of with an ordered basis and a dimension vector for . In this note we extend the methods of Lorscheid (2014) to establish Schubert decompositions of quiver Grassmannians Gr into affine spaces to the ramified case, i.e., the canonical morphism from the coefficient quiver of w.r.t. is not necessarily unramified.
In particular, we determine the Euler characteristic of Gr as the number of extremal successor closed subsets of , which extends the results of Cerulli Irelli (2011) and Haupt (2012) (under certain additional assumptions on ).
Oliver Lorscheid. "Schubert decompositions for quiver Grassmannians of tree modules." Algebra Number Theory 9 (6) 1337 - 1362, 2015. https://doi.org/10.2140/ant.2015.9.1337