Given a Dynkin quiver Q and a Q-representation M, a quiver Grassmannian X = Gre(M) is the projective variety of all subrepresentations of M of dimension vector e. This variety is not smooth in general, and its geometry is rather complicated. In particular it is useful to have a desingularization of X at our disposal. In this talk I will describe how to obtain such a desingularization. This is done by constructing an algebra BQ (which was newto us) with many interesting properties, e.g. it is derived equivalent to the Auslander algebra of kQ. To the Q-representation M it is hence associated a BQ-module ^M whose quiver Grassmannians gives the desired desingularization of X. This is joint work with Evgeny Feigin and Markus Reineke
(arXiv: 1209.3960).