This is a joint work with Francesco Esposito. We study some projective
varieties which are closed subvarieties of partial flag varieties and called
quiver Grassmannians. In the last years many authors have noticed the
importance of quiver Grassmannians in the theory of cluster algebras but many
geometric aspects, like the existence of a cellular decomposition, the
computation of Betti numbers and Euler characteristic are still open. We make
the first step in this direction by studying the "smallest" interesting quiver
Grassmannians. Surprisingly this approach has an interesting counterpart in the
theory of cluster algebras.

The seminar is thought for general audience, in particular we will try to
explain everything without using quivers and cluster algebras.