  The Landau potentials of $W_3$-algebra models are analyzed with
algebraic-geometric methods. The number of ground states and the number of
independent perturbations of every potential coincide and can be computed. This
number agrees with the structure of ground states obtained in a previous paper,
namely, as the phase structure of the IRF models of Jimbo et al. The
singularities associated to these potentials are identified.
