  We consider the 1D cubic NLS on $\mathbb R$ and prove a blow-up result for
functions that are of borderline regularity, i.e. $H^s$ for any $s<-\frac 12$
for the Sobolev scale and $\mathcal F L^\infty$ for the Fourier-Lebesgue scale.
This is done by identifying at this regularity a certain functional framework
from which solutions exit in finite time. This functional framework allows,
after using a pseudo-conformal transformation, to reduce the problem to a
large-time study of a periodic Schr\"odinger equation with non-autonomous cubic
nonlinearity. The blow-up result corresponds to an asymptotic completeness
result for the new equation. We prove it using Bourgain's method and exploiting
the oscillatory nature of the coefficients involved in the time-evolution of
the Fourier modes. Finally, as an application we exhibit singular solutions of
the binormal flow. More precisely, we give conditions on the curvature and the
torsion of an initial smooth curve such that the constructed solutions generate
several singularities in finite time.
