  We consider the self-dual Chern-Simons-Schr\"odinger equation (CSS). CSS is
$L^{2}$-critical, admits solitons, and has the pseudoconformal symmetry. In
this work, we consider pseudoconformal blow-up solutions under
$m$-equivariance, $m\geq1$. Our result is threefold. Firstly, we construct a
pseudoconformal blow-up solution $u$ with given asymptotic profile $z^{\ast}$:
\[
\Big[u(t,r)-\frac{1}{|t|}Q\Big(\frac{r}{|t|}\Big)e^{-i\frac{r^{2}}{4|t|}}\Big]e^{im\theta}\to
z^{\ast}\qquad\text{in }H^{1} \] as $t\to0^{-}$, where $Q(r)e^{im\theta}$ is a
static solution. Secondly, we show that such blow-up solutions are unique in a
suitable class. Lastly, we exhibit an instability mechanism of $u$. We
construct a continuous family of solutions $u^{(\eta)}$, $0\leq\eta\ll1$, such
that $u^{(0)}=u$ and for $\eta>0$, $u^{(\eta)}$ is a global scattering solution
exhibiting a rotational instability as $\eta\to0^{+}$: $u^{(\eta)}$ takes an
abrupt spatial rotation by the angle \[ \Big(\frac{m+1}{m}\Big)\pi \] on the
time interval $|t|\lesssim\eta$.
  We are inspired by works in the $L^{2}$-critical NLS. In the seminal work of
Bourgain and Wang (1997), they constructed such pseudoconformal blow-up
solutions. Merle, Rapha\"el, and Szeftel (2013) showed an instability of
Bourgain-Wang solutions. Although CSS shares many features with NLS, there are
essential differences and obstacles over NLS. To name a few, the soliton
profile to CSS shows a slow decay $r^{-(m+2)}$, CSS has nonlocal nonlinearities
responsible for strong long-range interactions, and the instability mechanism
of CSS is completely different from that of NLS. Here, the phase rotation is
the main source of the instability. On the other hand, the self-dual structure
of CSS is our sponsor to overcome these obstacles. We exploited the
self-duality in many places such as the linearization, spectral properties, and
construction of modified profiles.
