  We consider the following nonlinear Schr\"{o}dinger equation with an inverse
potential: \[ i\frac{\partial u}{\partial t}+\Delta
u+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2\sigma}}u=0 \] in $\mathbb{R}^N$. From
the classical argument, the solution with subcritical mass ($\|u\|_2<\|Q\|_2$)
is global and bounded in $H^1(\mathbb{R}^N)$. Here, $Q$ is the ground state of
the mass-critical problem. Therefore, we are interested in the existence and
behaviour of blow-up solutions for the threshold
($\left\|u_0\right\|_2=\left\|Q\right\|_2$). Previous studies investigate the
existence and behaviour of the critical-mass blow-up solution when the
potential is smooth or unbounded but algebraically tractable. There exist no
results when classical methods can not be used, such as the inverse power type
potential. However, we construct a critical-mass initial value for which the
corresponding solution blows up in finite time. Moreover, we show that the
corresponding blow-up solution converges to a certain blow-up profile in virial
space.
