Birationally rigid Fano complete intersections

Ten years ago in a paper of the speaker it was proved that a generic Fano complete intersection of index 1 and codimension $k$ in the projective space $\mathbb P^{M+k}$ is birationally superrigid if $M\geq 2k+1$. In my talk, I will show how to improve this result, replacing the latter condition by a much weaker one, $M\geq k+3$. Now the majority of Fano complete intersections of index one are covered.