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.