Deformation Theory for Finite Cluster Complexes

TL;DR

This paper investigates the deformation theory of cluster complexes associated with skew-symmetrizable cluster algebras, showing unobstructedness in certain cases and connecting cluster algebras with universal coefficients to geometric deformation spaces.

Contribution

It generalizes known results on unobstructedness of cluster complexes and links universal coefficient cluster algebras to universal families over deformation spaces.

Findings

Cluster complexes are unobstructed in the skew-symmetric case.

Universal coefficient cluster algebras can be recovered from deformation spaces.

Cluster algebras of finite type are Gorenstein and unobstructed if skew-symmetric.

Abstract

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 skew-symmetric case, these cluster complexes are unobstructed, generalizing a result of Ilten and Christophersen in the $A_n$ 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 $\mathcal{A}$ of geometric and finite cluster type, the cluster algebra $\mathcal{A}^{\mathrm{univ}}$ 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 $\mathcal{A}^{\mathrm{univ}}$ 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 $\mathcal{A}$ of geometric and finite cluster type, $\mathcal{A}$ is Gorenstein, and $\mathcal{A}$ is unobstructed if it is skew-symmetric. Moreover, if $\mathcal{A}$ has enough frozen variables then it has no non-trivial torus-invariant deformations. We also study the Gr\"obner theory of the ideal of relations among cluster and frozen variables of $\mathcal{A}$. As a byproduct we generalize previous results in this setting obtained by Bossinger, Mohammadi and N\'ajera Ch\'avez for Grassmannians of planes and $\text{Gr}(3,6)$.