This paper contains the material discussed in the series of three lectures that I gave during the workshop of the ICRA 2018 in Prague. I will introduce the reader to some of the techniques used in the study of the geometry of quiver Grassmannians. The notes are quite elementary and thought for phd students or young researchers. I assume that the reader is familiar with the representation theory of quivers.
Quiver Grassmannians are projective varieties parametrizing subrepresentations
of given dimension in a quiver representation. We define a class of quiver Grassmannians
generalizing those which realize degenerate flag varieties. We show that each irreducible
component of the quiver Grassmannians in question is isomorphic to a Schubert variety.We
give an explicit description of the set of irreducible components, identify all the Schubert
varieties arising, and compute the Poincar´e polynomials of these quiver Grassmannians.
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