[Paper Review] A presentation for the Cox ring of $\overline{M}_{0,6}$

Martha Bernal Guillén, Diane Maclagan|arXiv (Cornell University)|Dec 21, 2017

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper presents a generating set for the Cox ring of the moduli space $​\overline{M}_{0,6}$, the compactified Deligne-Mumford stack of stable curves of genus zero with six marked points. Using order of vanishing computations on boundary divisors and Keel-Vermeire divisors, the authors establish a matrix representation of the relations in the Cox ring, proving that the given matrix $ ilde{R}$ forms a basis for the kernel of the matrix $A$ representing divisor classes, thereby fully describing the ring structure.

ABSTRACT

We compute the relations in the Cox ring of the moduli space $\overline{M}_{0,6}$. This gives a presentation of the Cox ring as a quotient of a polynomial ring with 40 generators by an ideal with 225 generators that come in 5 symmetry classes.

Motivation & Objective

• To determine a complete set of generators and relations for the Cox ring of $​\overline{M}_{0,6}$, the moduli space of stable genus zero curves with six marked points.
• To describe the structure of the Picard group of $​\overline{M}_{0,6}$ via explicit matrix representations of divisor classes.
• To establish that the matrix $ ilde{R}$ forms a basis for the kernel of the matrix $A$ representing boundary and Keel-Vermeire divisors in the Picard group.
• To prove that the relations among the rational functions $z_{ij}$ and $f_ au$ correspond exactly to the kernel of the divisor class matrix $A$.
• To provide a concrete presentation of the Cox ring using order of vanishing data on boundary divisors $\ u_{ij}$ and points $Q_ au$.

Proposed method

• The authors compute the order of vanishing of rational functions $z_{ij}$ and $f_ au$ along boundary divisors $\ u_{ij}$ and points $Q_ au$ in $​\overline{M}_{0,6}$.
• They use the fact that principal divisors have trivial class in the Picard group to equate linear combinations of divisor classes to zero.
• The matrix $A$ is constructed from representatives of boundary divisors and Keel-Vermeire divisors in $\operatorname{Pic}(\overline{M}_{0,6})$.
• The matrix $\tilde{R}$ is shown to be block-structured with identity and zero blocks, and its rows span the kernel of $A$.
• The proof proceeds by verifying four key vanishing conditions: orders of $z_{ij}$ on $\delta_{kl}$, orders of $f_\pi$ on $\delta_{kl}$, orders of $z_{ij}$ on $Q_\pi$, and orders of $f_{\pi'}$ on $Q_\pi$.
• The final step confirms that the matrix $R$ of vanishing orders equals $\tilde{R}$, proving that $\tilde{R}$ spans the kernel of $A$.

Experimental results

Research questions

• RQ1 What is a complete set of generators and relations for the Cox ring of $\overline{M}_{0,6}$?
• RQ2 How do the rational functions $z_{ij}$ and $f_\pi$ behave under restriction to boundary divisors and special points?
• RQ3 What is the structure of the kernel of the matrix $A$ that represents divisor classes in $\operatorname{Pic}(\overline{M}_{0,6})$?
• RQ4 Can the relations among the divisor classes be fully captured by a matrix derived from order of vanishing data?
• RQ5 Is the matrix $\tilde{R}$ a basis for the kernel of $A$, and does it fully describe the Cox ring presentation?

Key findings

• The matrix $\tilde{R}$, which has a block structure with identity and zero blocks, forms a basis for the kernel of the matrix $A$ representing divisor classes in $\operatorname{Pic}(\overline{M}_{0,6})$.
• The order of vanishing of $z_{ij}$ on $\delta_{kl}$ is 1 if $\{i,j\} = \{k,l\}$, and 0 otherwise, confirming their role in generating relations.
• The order of vanishing of $f_\pi$ on $\delta_{kl}$ is 0 for all $\{k,l\} \in \mathcal{E}$, showing that $f_\pi$ does not vanish along boundary divisors.
• The order of vanishing of $z_{ij}$ on $Q_\pi$ is 0 for all $\{i,j\} \in \mathcal{E}$, indicating $z_{ij}$ does not vanish at the points $Q_\pi$.
• The order of vanishing of $f_{\pi'}$ on $Q_\pi$ is 1 if $\pi = \pi'$, and 0 otherwise, confirming $f_\pi$ defines the divisor $Q_\pi$.
• The matrix $R$ of actual vanishing orders equals $\tilde{R}$, proving that $\tilde{R}$ spans the kernel of $A$, and thus fully describes the Cox ring relations.