Stability Conditions for Multigraded Rings

Abstract

Let $D$ be a finitely generated abelian group and $S$ a $D$-graded ring. We introduce a geometric semistability condition for points $x \in \text{Spec}(S)$, characterized by maximal-dimensional orbit cones $σ(x)$. This set of geometrically semistable points $X^{\mathrm{gss}}$ yields a new framework for the $D$-graded Proj construction, which is equivalently given as the geometric quotient of $D(S_+) = \text{Spec}(S) \setminus V(S_+)$ by the torus $\text{Spec}(S_0[D])$, where $S_+ \unlhd S$ is the ideal generated by all relevant elements. We show that orbit cones are unions of relevant cones $\mathcal C_D(f)$. This yields a chamber decomposition of the weight space $σ(S) = \overline{\text{Cone}}(d \in D \mid S_d \neq 0)$, determined entirely by relevant elements. In particular, we obtain $\text{Proj}^D(S) = X^{\mathrm{gss}} / \text{Spec}(S_0[D])$. As an application, for a simplicial toric (pre-)variety $X$ with full-dimensional convex support and $S = \text{Cox}(X)$, this chamber decomposition of its weight space recovers the secondary fan of $X$. Consequently, when $d \in D = \text{Cl}(X)$, the space $\text{Proj}^D(S)$ is exactly the direct limit of all GIT quotients $\mathbb A^n //_{χ^d} \text{Spec}(S_0[D])$ of $X$.