  Spin foams are candidate state-sum models for transition amplitudes in
quantum gravity. An active research subject is to identify the possible
divergences of spin foam models, or alternatively to show that models are
finite. We will discuss in detail the (non--occurrence of) divergences in the
Barrett-Crane model, formulated as an integral of delta function weights only.
We will furthermore present a simple method to estimate the divergence degree
of the so-called bubbles for general spin foam models. Divergences in spin
foams are expected to be related to the existence of gauge symmetries
(diffeomorphisms). Thus we have to conclude that such gauge symmetries are not
(fully) present in the model we consider. But we will identify a class of gauge
symmetries which occur at special solutions of the equations imposed by the
delta function weights. This situation is surprisingly similar to the case of
broken diffeomorphism symmetries in discrete gravity, which are present around
flat solutions. We introduce a method to derive (Ward-identity-like) equations
for the vertex amplitude of the model in the case of broken gauge symmetries.
