We present a study of phase transitions of the Curie--Weiss Potts model at (inverse) temperature $beta$, in 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 $beta$--$h$ plane as well as the magnitude of the jump of the magnetization (for $q geqslant 3)$. 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 Kertesz line separating (in the $beta$--$h$ plane) the two behaviours. As a result, we get that this line exhibits, as soon as $q geqslant 3$, a very interesting cusp where it separates from the thermodynamic transition line.
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