Pseudo-index of Fano manifolds and smooth blow-ups

Abstract

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 $i_X:=\min \{-K_X\cdot C|CisarationalcurveofX\}$). We prove that $i_X\le i_Y$ 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 $Y$.