Cluster complexes are a certain class of simplicial complexes that naturally arise in the theory of cluster algebras. They codify a wealth of fundamental information about cluster algebras. The purpose of this talk is to elaborate on a geometric relationship between cluster algebras and cluster complexes. In vague words this relationship is the following: cluster algebras of finite cluster type with universal coefficients may be obtained via a torus action on a Hilbert scheme. In particular, we will discuss the deformation theory of the Stanley-Reisner ring associated to a finite cluster complex and present some applications related to the Gröbner theory of the ideal of relations among cluster and frozen variables of a cluster algebra of finite cluster type. Time permitting I will elaborate on how to generalize this approach to the context of tau-tilting finite algebras. This is based on a joint project with Nathan Ilten and Hipolito Treffinger whose first outcome is the preprint arXiv:2111.02566.