  The generalized Korteweg-de Vries equations are a class of Hamiltonian
systems in infinite dimension derived from the KdV equation where the quadratic
term is replaced by a higher order power term. These equations have two
conservation laws in the energy space H^1(L^2 norm and energy). We consider in
this paper the {\it critical} generalized KdV equation, which corresponds to
the smallest power of the nonlinearity such that the two conservation laws do
not imply a bound in H^1 uniform in time for all H^1 solutions (and thus global
existence).
  From [15], there do exist for this equation solutions u(t) such that
|u(t)|_{H^1} \to +\infty as T\uparrow T, where T\le +\infty (we call them
blow-up solutions). The question is to describe, in a qualitative way, how blow
up occurs.
  For solutions with L^2 mass close to the minimal mass allowing blow up and
with decay in L^2 at the right, we prove after rescaling and translation which
leave invariant the L^2 norm that the solution converges to a {\it universal}
profile locally in space at the blow-up time T. From the nature of this
profile, we improve the standard lower bound on the blow-up rate for finite
time blow-up solutions.
