Abstract
We present a study of phase transitions of the mean-field Potts model at (inverse) temperature β, in the presence of an external field h. Both thermodynamic and topological aspects of these transitions are considered. For the first aspect we complement previous results and give an explicit equation of the thermodynamic transition line in the β-h plane as well as the magnitude of the jump of the magnetization (for 
). The signature of the latter aspect is characterized here by the presence or not of a giant component in the clusters of a Fortuin-Kasteleyn type representation of the model. We give the equation of the Kertész line separating (in the β-h plane) the two behaviours. As a result, we get that this line exhibits, as soon as 
, a very interesting cusp where it separates from the thermodynamic transition line.