  The state spaces of generalised coherent states associated with special
unitary groups are shown to form rational curves and surfaces in the space of
pure states. These curves and surfaces are generated by the various Veronese
embeddings of the underlying state space into higher-dimensional state spaces.
This construction is applied to the parameterisation of generalised coherent
states, which is useful for practical calculations and provides an elementary
combinatorial approach to the geometry of the coherent state space. The results
are extended to Hilbert spaces with indefinite inner products, leading to the
introduction of a new kind of generalised coherent states.
