  The purpose of this paper is to study the validity of global-in-time
Strichartz estimates for the Schr\"odinger equation on $\mathbb{R}^n$, $n\ge3$,
with the negative inverse-square potential $-\sigma|x|^{-2}$ in the critical
case $\sigma=(n-2)^2/4$. It turns out that the situation is different from the
subcritical case $\sigma<(n-2)^2/4$ in which the full range of Strichartz
estimates is known to be hold. More precisely, splitting the solution into the
radial and non-radial parts, we show that (i) the radial part satisfies a
weak-type endpoint estimate, which can be regarded as an extension to higher
dimensions of the endpoint Strichartz estimate with radial data for the
two-dimensional free Schr\"odinger equation; (ii) other endpoint estimates in
Lorentz spaces for the radial part fail in general; (iii) the non-radial part
satisfies the full range of Strichartz estimates.
