  We study the dispersive blow-up phenomena for the Schr\"odinger-Korteweg-de
Vries (S-KdV) system. Roughly, dispersive blow-up has being called to the
development of point singularities due to the focussing of short or long waves.
In mathematical terms, we show that the existence of this kind of singularities
is provided by the linear dispersive solution by proving that the Duhamel term
is smoother. It seems that this result is the first regarding systems of
nonlinear dispersive equations. To obtain our results we use, in addition to
smoothing properties, persistence properties for solutions of the IVP in
fractional weighted Sobolev spaces which we establish here.
