  In this paper, we consider a blow-up solution for the complex-valued
semilinear wave equation with power non-linearity in one space dimension. We
show that the set of non characteristic points $I_0$ is open and that the
blow-up curve is of class $C^{1,\mu_0}$ and the phase $\theta$ is $C^{\mu_0}$
on this set. In order to prove this result, we introduce a Liouville Theorem
for that equation.
