  This paper investigates the Poisson geometry associated to a cluster algebra
over the complex numbers, and its relationship to compatible torus actions. We
show, under some assumptions, that each Noetherian cluster algebra has only
finitely many torus invariant Poisson prime ideals and we show how to obtain
using the exchange matrix of an initial seed. In fact, these ideals are
independent of the choice of compatible Poisson structure. In many interesting
cases the ideals can be described more explicitly.
