On singular Fano varieties with a divisor of Picard number one

In this talk we will study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties in order to obtain explicit examples stressing the difference with the smooth case. Finally, we treat the lifting of extremal contractions to universal coverings spaces in codimension 1 (which are related with the first fundamental group of the smooth locus).