Stability Conditions for Multigraded Rings

TL;DR

This paper introduces a geometric semistability condition for multigraded rings, leading to a new framework for the $D$-graded Proj construction and a chamber decomposition of the weight space, connecting to GIT quotients and secondary fans.

Contribution

It develops a novel geometric semistability criterion for points in multigraded spectra, establishing a new $D$-graded Proj framework and linking it to chamber decompositions and GIT quotients.

Findings

Orbit cones are unions of relevant cones.

The chamber decomposition is determined by relevant elements.

The construction recovers the secondary fan of a toric variety.

Abstract

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