[Paper Review] Luna-Vust invariants of Cox rings and skeletons of spherical varieties

Giuliano Gagliardi|arXiv (Cornell University)|Aug 29, 2016

Advanced Algebra and Geometry4 citations

TL;DR

This paper determines the Luna-Vust invariants of the Cox ring of a spherical variety from those of the original variety, explicitly describing the divisor class group of the Cox ring and proving that every spherical variety admits iteration of Cox rings. A key result is a combinatorial proof that the Cox ring is determined by the spherical skeleton, confirming a known result via new methods.

ABSTRACT

Given the Luna-Vust invariants of a spherical variety, we determine the Luna-Vust invariants of the spectrum of its Cox ring. In particular, we deduce an explicit description of the divisor class group of the Cox ring. It follows that every spherical variety admits iteration of Cox rings. Moreover, we obtain a combinatorial proof of the fact that the Cox ring is determined by the spherical skeleton, which is a known result following from the description of the Cox ring due to Brion. Finally, we show that a conjectural combinatorial smoothness criterion can be reduced to the case of a factorial affine spherical variety with a fixed point.

Motivation & Objective

• To determine the Luna-Vust invariants of the spectrum of the Cox ring from those of the original spherical variety.
• To provide an explicit description of the divisor class group of the Cox ring of a spherical variety.
• To prove that every spherical variety admits iteration of Cox rings.
• To give a combinatorial proof that the Cox ring is determined by the spherical skeleton, independently of Brion's geometric construction.
• To reduce a conjectural smoothness criterion for spherical varieties to the case of factorial affine spherical varieties with a fixed point.

Proposed method

• Use the Luna-Vust theory of spherical embeddings to analyze the combinatorial invariants (ρ, σ, and V) of the Cox ring's spectrum from those of the original variety.
• Construct the divisor class group of the Cox ring via explicit computation from the Luna-Vust invariants of the original spherical variety.
• Apply the theory of spherical skeletons and spherical roots to analyze the structure of the Cox ring and its iterated construction.
• Use the notion of a complete spherical skeleton and the associated polyhedral cone structure to analyze the behavior of the Cox ring under iteration.
• Reduce the conjectural smoothness criterion by analyzing the structure of the spherical root system and the behavior of the spherical roots under modifications.
• Employ case analysis on the Dynkin diagram types and root systems (e.g., double edges, short roots) to construct modified spherical skeletons and compare their invariants.

Experimental results

Research questions

• RQ1 How can the Luna-Vust invariants of the Cox ring of a spherical variety be explicitly computed from those of the original variety?
• RQ2 What is the precise structure of the divisor class group of the Cox ring of a spherical variety?
• RQ3 Under what conditions is the Cox ring of a spherical variety factorial?
• RQ4 Can the combinatorial determination of the Cox ring via the spherical skeleton be proven independently of geometric constructions?
• RQ5 To what extent can a conjectural smoothness criterion for spherical varieties be reduced to the case of factorial affine spherical varieties with a fixed point?

Key findings

• The Luna-Vust invariants of the spectrum of the Cox ring are explicitly determined from those of the original spherical variety.
• The divisor class group of the Cox ring is described explicitly, and it is shown that the Cox ring is factorial if and only if the spherical variety satisfies a specific condition on its invariants.
• Every spherical variety admits iteration of the Cox ring construction, meaning the process can be applied repeatedly to obtain a sequence of Cox rings.
• A combinatorial proof is given that the Cox ring is determined by the spherical skeleton, providing an alternative to Brion's geometric construction.
• The conjectural smoothness criterion for spherical varieties can be reduced to the case of a factorial affine spherical variety with a fixed point.
• The proof involves constructing modified spherical skeletons via changes in the root system and color structure, showing that the invariant wp(ℛ) is non-decreasing under such modifications.