[Paper Review] Cox rings of rational complexity one T-varieties

Klaus Altmann, Lars Petersen|arXiv (Cornell University)|Sep 2, 2010

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper constructs a polyhedral divisor $π_{\operatorname{Cox}}$ on a finite covering of $\mathbb{P}^1$ that describes the Cox ring of a Mori dream space $X$ with a complexity-one torus action. By leveraging $A$-divisors of degree zero on $\mathbb{P}^1$, the authors provide a combinatorial framework to study torus orbits and deformations of $\operatorname{Cox}(X)$, generalizing toric constructions to rational $T$-varieties.

ABSTRACT

Let X be a Mori dream space together with an effective torus action of complexity one. In this note, we construct a polyhedral divisor on a suitable covering of the projective line P^1 which corresponds to the affine spectrum of the Cox ring of X. This description allows for a detailed study of torus orbits and deformations of the latter. Moreover, we present coverings of P^1 together with an action of a finite abelian group A in terms of so-called A-divisors of degree zero on P^1.

Motivation & Objective

• To provide a combinatorial description of the Cox ring of a Mori dream space with a complexity-one torus action.
• To generalize toric constructions—where Cox rings are polynomial rings—to non-toric rational $T$-varieties.
• To study torus orbits and deformations of $\operatorname{Cox}(X)$ via a polyhedral divisor on a branched covering of $\mathbb{P}^1$.
• To introduce $A$-divisors of degree zero on $\mathbb{P}^1$ as a tool for describing finite abelian coverings with group actions.
• To establish a correspondence between the Cox ring of $X$ and a polyhedral divisor $\mathcal{D}_{\operatorname{Cox}}$ on a suitable covering space.

Proposed method

• The authors use the theory of $p$-divisors to describe the total coordinate space $\operatorname{Spec}\operatorname{Cox}(X)$ as an affine $T$-variety.
• They construct a polyhedral divisor $\mathcal{D}_{\operatorname{Cox}}$ on a finite Galois covering $q: \widetilde{Y} \to \mathbb{P}^1$ with group $A$, where $\widetilde{Y}$ is a smooth rational curve.
• The construction involves lifting the divisor class group $\operatorname{Cl}(X)$ via a presentation and using a section $s$ to define polyhedral coefficients $\widetilde{\Delta}_p$ in the tailfan $\sigma$.
• The polyhedral coefficients $\widetilde{\Delta}_p$ are derived from the effective cone and intersection theory on $X$, ensuring compatibility with the $A$-action.
• For varieties with torsion in $\operatorname{Cl}(X)$, the method accounts for the covering via $A$-divisors, where $A \cong \operatorname{Cl}(X)_{\text{tors}}$, and the divisor $\mathcal{D}_{\operatorname{Cox}}$ is defined with multiplicity $|A|$ on preimages of points.
• The resulting $\mathcal{D}_{\operatorname{Cox}}$ is a $\mathbb{Z}$-linear combination of polyhedral coefficients over the points of $\widetilde{Y}$, encoding the Cox ring as a $\operatorname{Cl}(X)$-graded algebra.

Experimental results

Research questions

• RQ1 How can the Cox ring of a rational complexity-one $T$-variety be described combinatorially using polyhedral divisors?
• RQ2 What is the role of finite abelian coverings of $\mathbb{P}^1$ in realizing the total coordinate space of such varieties?
• RQ3 How do torus orbits and deformations of $\operatorname{Cox}(X)$ manifest in the polyhedral divisor framework?
• RQ4 In what way do $A$-divisors of degree zero on $\mathbb{P}^1$ encode group actions and branched coverings in this context?
• RQ5 Can the construction recover known examples such as log del Pezzo surfaces with $E_6$ singularities?

Key findings

• The Cox ring of a rational complexity-one $T$-variety $X$ is realized as the coordinate ring of an affine $T$-variety via a polyhedral divisor $\mathcal{D}_{\operatorname{Cox}}$ on a finite covering of $\mathbb{P}^1$.
• For the $E_6$-type log del Pezzo surface of degree three, the polyhedral divisor $\mathcal{D}_{\operatorname{Cox}}$ has tailcone generated by $(0,1)$ and $(-2,1)$, with coefficients $\widetilde{\Delta}_0 = (0,-1/3)+\sigma$, $\widetilde{\Delta}_1 = (0,1/2)+\sigma$, and $\widetilde{\Delta}_\infty = \overline{(0,0),(-1/3,0)} + \sigma$.
• For the degree-one $E_6$-type surface with $A_2$-singularity, the covering is a 3:1 map $q: \mathbb{P}^1 \to \mathbb{P}^1$ branched over $0$ and $1$, and $\mathcal{D}_{\operatorname{Cox}}$ includes multiplicity three over preimages of each point.
• The compact part $\widetilde{\Delta}^c_0$ of the polyhedral divisor for the del Pezzo surface of degree six is a five-dimensional simplex, reflecting the complexity of the Cox ring structure.
• The method fails to produce a polynomial ring when degenerating the Cox ring of $\mathbb{P}(\Omega_{\mathbb{P}^2})$, as the Minkowski sum of three edges yields a three-dimensional cube, showing the non-toric nature of the total coordinate ring.
• The construction generalizes the toric case: when $X$ is toric, $\mathcal{D}_{\operatorname{Cox}}$ recovers the standard polyhedral resolution of the fan via the positive orthant.