Coherent states and rational surfaces

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.