On the quiver Grassmannian in the acyclic case

Let AA be the path algebra of a quiver QQ with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given AA-module MM. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when MM is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras.