February 2016, Moscow, Russia

**Title:** Rationally conneted non Fano type vareties

**Abstract:** The class of varieties of Fano type is a singular generalization of Fano varieties which is very well behaved under the MMP, the Cox ring of varietie of Fano type is finitely generated. It is known that all varieties of Fano type are rationally connected. The convers is true in a sense in dimension 2. I will give counterexamples to the converse statement in dimension 3 and higher. I will use the techniques of birational rigidity and the MMP.