  In this paper, we will consider the $L^2$-critical fractional Schr\"odinger
equation $iu_t-|D|^{\beta}u+|u|^{2\beta}u=0$ with initial data $u_0\in
H^{\beta/2}(\mathbb{R})$ and $\beta$ close to $2$. We will show that the
solution blows up in finite time if the initial data has negative energy and
slightly supercritical mass. We will also give a specific description for the
blow-up dynamics. This is an extension of the work of F. Merle and P. Rapha\"el
for $L^2$-critical Schr\"odinger equations but the nonlocal structure of this
equation and the lack of some symmetries make the analysis more complicated,
hence some new strategies are required.
