  We prove the local well posedness for the KdV equation in the modulation
space $M^{-1}_{2,1}(\mathbb{R})$. Our method is to substitute the dyadic
decomposition by the uniform decomposition in the discrete Bourgain space. This
wellposedness result enables us to show that the solution map is discontinuous
at the origin with respect to the $H^{s} $-topology as soon as $s<-1$. Making
use of the Miura transform we also deduce a discontinuity result for the $
H^s(\Real) $-topology, $s<0 $, for the solution map associated with the
focussing and defocussing mKdV equations.
