Title: On Fano varieties with large pseudo-index

Abstract:  Consider smooth Fano variety $X$. Let i(X):= min\{C \.cdot 
(-K_{X}) | C is a curve on X\} (pseudo-index). Mukai conjecture states 
that (i(X)-1) \rho(X) \leq n+1. This conjecture is still open when $i(X)$ 
is "small".

In this talk, I will take a baby step towards the proof of Mukai 
conjecture for singular varieties. More precisely, I will bound the picard 
number $\rho(X)$ when $i(X)$ is large and $X$ has only mild singularities.