Gröbner cones for finite-type cluster algebras

Let \mathcal{A} be a cluster algebra of finite cluster type. We study the Gröbner cone \mathcal{C}_{\mathcal{A}} parametrizing term orders inducing an initial degeneration of the ideal I_{\mathcal{A}} of relations among the cluster variables of \mathcal{A} to the ideal generated by products of incompatible cluster variables. We show that for any cluster variable v , the weight induced by taking compatibility degrees with v belongs to \mathcal{C}_{\mathcal{A}} . This allows us to construct an explicit circular term order and prove a conjecture of Ilten, Nájera Chávez, and Treffinger. Furthermore, we give explicit descriptions of the rays and lineality spaces of \mathcal{C}_{\mathcal{A}} in terms of combinatorial models for cluster algebras of types A_{n} , B_{n} , C_{n} , D_{n} with a special choice of frozen variables, and in the case of no frozen variables.