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).