Pseudo-index of Fano manifolds and smooth blow-ups

Suppose $\pi :X\to Y$ is a smooth blow-up along a submanifold $Z$ of $Y$ between complex Fano manifolds $X$ and $Y$ of pseudo-indices $i_X$ and $i_Y$ respectively (recall that $i_X$ is defined by $i_X := \min {-K_X \cdot C | C {is a rational curve of} X}$)\mathrm{.}Weprovethat$i_X \leq i_Y$if$2\dim (Z)< \dim(Y) + i_Y -1$ and show that this result is optimal by classifying the ``boundary’’ cases. As expected, these results are obtained by studying rational curves on $X$ and $Y$.