Deformation Theory for Finite Cluster Complexes

We study the deformation theory of the Stanley-Reisner rings associated to cluster complexes for skew-symmetrizable cluster algebras of geometric and finite cluster type. In particular, we show that in the skewsymmetric case, these cluster complexes are unobstructed, generalizing a result of Ilten and Christophersen in the An case. We also study the connection between cluster algebras with universal coefficients and cluster complexes. We show that for a full rank positively graded cluster algebra A of geometric and finite cluster type, the cluster algebra A with universal coefficients may be recovered as the universal family over a partial closure of a torus orbit in a multigraded Hilbert scheme. Likewise, we show that under suitable hypotheses, the cluster algebra A may be recovered as the coordinate ring for a certain torus-invariant semiuniversal deformation of the Stanley-Reisner ring of the cluster complex. We apply these results to show that for any cluster algebra A of geometric and finite cluster type, A is Gorenstein, and A is unobstructed if it is skewsymmetric. Moreover, if A has enough frozen variables then it has no nontrivial torus-invariant deformations. We also study the Gröbner theory of the ideal of relations among cluster and frozen variables of A. As a byproduct we generalize previous results in this setting obtained by Bossinger, Mohammadi and Nájera Chávez for Grassmannians of planes and Gr(3, 6).