[Paper Review] Cox rings of du Val singularities

Laura Facchini, Víctor González‐Alonso+1 more|arXiv (Cornell University)|Feb 3, 2015

Coding theory and cryptography2 references3 citations

TL;DR

This paper introduces and computes the Cox ring for du Val singularities—specifically $\mathbb{D}_n$, $\mathbb{E}_6$, $\mathbb{E}_7$, and $\mathbb{E}_8$—by defining the Cox ring of a surface singularity via the relative Picard group of its minimal resolution. The key contribution is explicit presentations of these rings as quotients of polynomial rings, with relations derived from the geometry of $(-2)$-curves in the exceptional divisor.

ABSTRACT

In this note we introduce Cox rings of singularities and explicitly compute them in the case of du Val singularities $\mathbb{D}_n,\mathbb{E}_6,\mathbb{E}_7$ and $\mathbb{E}_8$.

Motivation & Objective

• To define and formalize the concept of the Cox ring for normal surface singularities, extending the classical construction from smooth varieties to singular ones.
• To compute the Cox ring explicitly for du Val singularities, which are rational surface singularities with a well-understood resolution geometry.
• To provide a systematic method for computing Cox rings in cases where the relative Picard group is free and finitely generated, using the structure of $(-2)$-curves in the exceptional divisor.
• To lay a foundation for studying Cox rings of more general surface singularities by starting with the simplest and most symmetric class: du Val singularities.

Proposed method

• Define the Cox ring of a surface singularity $(X,O)$ as $\operatorname{Cox}(X) = \bigoplus_{L \in \operatorname{Pic}(\widetilde{X}/X)} H^0(\widetilde{X}, L)$, where $\widetilde{X}$ is the minimal resolution and $\operatorname{Pic}(\widetilde{X}/X)$ is the relative Picard group.
• Use the isomorphism $\operatorname{Pic}(\widetilde{X}/X) \cong \mathbb{Z}^n$ induced by the $(-2)$-curves $E_i$ in the exceptional divisor $E = \pi^{-1}(O)$, with multidegree $\delta_i(L) = \deg(L|_{E_i})$.
• Construct a torus action on $\operatorname{Cox}(X)$ via the grading by $\operatorname{Pic}(\widetilde{X}/X)$, enabling the use of Geometric Invariant Theory (GIT) for constructing the singularity as a quotient.
• For $\mathbb{D}_n$, explicitly compute the Cox ring using a recursive reduction of line bundles via multiplication by sections $y_i$ corresponding to $E_i$, leading to a presentation in terms of monomials.
• For $\mathbb{E}_n$ singularities, derive the Cox ring as a quotient of a polynomial ring by a single relation: $y_1x_1^2 + y_2y_3^2x_3^3 + \cdots + y_{n-1}^{n-4}x_{n-1}^{n-3} = 0$, after eliminating redundant variables via reduction lemmas.
• Use the structure of the dual graph of the resolution to guide variable assignment and relation construction, ensuring consistency with the multidegree grading.

Experimental results

Research questions

• RQ1 How can the Cox ring be generalized from smooth varieties to normal surface singularities with a free relative Picard group?
• RQ2 What is the explicit structure of the Cox ring for du Val singularities $\mathbb{D}_n$, $\mathbb{E}_6$, $\mathbb{E}_7$, and $\mathbb{E}_8$?
• RQ3 Can the Cox ring of an $\mathbb{E}_n$ singularity be presented as a quotient of a polynomial ring with a single defining relation, and if so, what is the form of that relation?
• RQ4 Under what conditions do reduction lemmas (e.g., Lemma 4.6) fail in more general configurations of $(-2)$-curves, and how does this affect the Cox ring structure?
• RQ5 To what extent can the method used for du Val singularities be extended to other surface singularities, such as Hirzebruch-Jung or non-Dynkin configurations?

Key findings

• The Cox ring of the $\mathbb{D}_n$ singularity is isomorphic to $\mathbb{C}[x_1, x_2, x_3, y_0, y_1, \dots, y_n] / (y_1x_1^2 + y_2y_3^2x_3^3 + \cdots + y_{n-1}^{n-4}x_{n-1}^{n-3})$, with explicit monomial generators and relations.
• For $\mathbb{E}_6$, the Cox ring is $\mathbb{C}[Z_1, Z_2, Z_3, Z_4] / (Z_2^2 - Z_3Z_4)$, with $Z_i$ expressed as monomials in $x_i$ and $y_i$, and the singularity realized as the intersection of this hypersurface with a linear section.
• The Cox ring of $\mathbb{E}_7$ is $\mathbb{C}[Z_1, Z_2, Z_3, Z_4] / (Z_3^2 - Z_2Z_4)$, with the singularity obtained as the intersection with a cubic hypersurface $H_7$.
• The Cox ring of $\mathbb{E}_8$ is $\mathbb{C}[Z_1, Z_2, Z_3] / (Z_2^5 + Z_3^3 + Z_1^2)$, corresponding to the $E_8$-configuration.
• The general form of the defining relation for $\mathbb{E}_n$ singularities is $y_1x_1^2 + y_2y_3^2x_3^3 + \cdots + y_{n-1}^{n-4}x_{n-1}^{n-3} = 0$, which matches the pattern seen in lower-rank cases.
• The reduction lemmas used in $\mathbb{D}_n$ fail in more complex configurations (e.g., trivalent nodes with no length-1 branches), showing that the method is sensitive to the dual graph structure.