Stability Conditions for Multigraded Rings

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 $\sigma (x)$. This set of geometrically semistable points $X^{\mathrm{g}\mathrm{s}\mathrm{s}}$ 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 $S_{+}⊴S$ is the ideal generated by all relevant elements. We show that orbit cones are unions of relevant cones ${\text{CC}}_D(f)$. This yields a chamber decomposition of the weight space $\sigma (S)=\overline{\text{Cone}}(d\in D\mid S_d\mathrm{\ne }0)$, determined entirely by relevant elements. In particular, we obtain ${\text{Proj}}^D(S)=X^{\mathrm{g}\mathrm{s}\mathrm{s}}\text{sslash}\text{Spec}(S_0[D])$. 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 ${\text{BA}}^n{\text{sslash}}_{{\chi }^d}\text{Spec}(S_0[D])$ of $X$.