  We investigate the geometrical structure of Vaidya's spacetime in the case of
a white hole with decreasing mass, stabilising to a black hole in finite or
infinite time or evaporating completely. Our approach relies on a detailed
analysis of the ordinary differential equation describing the incoming
principal null geodesics, among which are the generators of the past horizon.
We devote special attention to the case of a complete evaporation in infinite
time and establish the existence of an asymptotic light-like singularity of the
conformal curvature, touching both the past space-like singularity and future
time-like infinity. This singularity is present independently of the decay rate
of the mass. We derive an explicit formula that relates directly the strength
of this null singularity to the asymptotic behaviour of the mass function.
