[Paper Review] A 1-dimensional family of Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1

Toshiyuki Katsura, Shigeyuki Kondō|arXiv (Cornell University)|Nov 12, 2014

Algebraic Geometry and Number Theory8 references8 citations

TL;DR

This paper constructs a 1-dimensional family of classical and supersingular Enriques surfaces in characteristic 2, whose canonical covers are the unique supersingular K3 surface with Artin invariant 1. It identifies 30 nonsingular rational curves and ten non-effective $(-2)$-divisors on these surfaces, showing that their reflection group has finite index in the orthogonal group of the Néron-Severi lattice modulo torsion, with the automorphism group of the associated polyhedron isomorphic to $\mathfrak{S}_6 \cdot \mathbb{Z}/2\mathbb{Z}$. The construction relies on Frobenius base change and rational vector fields on an elliptic K3 surface with $I_6$ fibers.

ABSTRACT

We give a 1-dimensional family of classical and supersingular Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1. Moreover we show that there exist 30 nonsingular rational curves and ten non-effective (-2)-divisors on these Enriques surfaces whose reflection group is of finite index in the orthogonal group of the Neron-Severi lattice modulo torsion.

Motivation & Objective

• To construct a 1-dimensional family of Enriques surfaces in characteristic 2 whose canonical covers are the supersingular K3 surface with Artin invariant 1.
• To analyze the geometry of the Néron-Severi lattice modulo torsion of these Enriques surfaces.
• To identify and study 30 nonsingular rational curves and ten non-effective $(-2)$-divisors on the Enriques surfaces.
• To determine the structure of the reflection group generated by these $(-2)$-divisors and its relation to the orthogonal group of the Néron-Severi lattice.
• To investigate the automorphism group of the Enriques surface via the action of the symmetric group $\mathfrak{S}_6$ and outer automorphisms.

Proposed method

• Construct an elliptic surface over $\mathbb{P}^1$ defined by the equation $y^2 + y + x^3 + s x (y^2 + y + 1) = 0$, which has four $I_3$ fibers over $s = 1, \omega, \omega^2, \infty$.
• Apply Frobenius base change $s = t^2$ to obtain a new elliptic surface with 12 rational double points of type $A_1$, leading to a supersingular K3 surface $Y$ with Artin invariant 1.
• Define a rational vector field $D$ on $Y$ using parameters $a, b \in k$ with $a + b = ab$ and $a^3 \neq 1$, and take the quotient by the action of $D$ to obtain the Enriques surface $X = X_{a,b}$.
• Identify 42 rational curves on $Y$ as components of four $I_6$ fibers and 18 sections, and descend them to 30 rational curves on $X$ via the quotient map.
• Construct a graph $\Gamma$ from 15 duads (transpositions), 15 synthemes (products of three disjoint transpositions), and 10 $(-2)$-divisors in $\text{Num}(X)$, and analyze its parabolic subdiagrams.
• Use Vinberg's theory of hyperbolic reflection groups to show that the reflection group $W(\Gamma)$ has finite index in $\text{O}(\text{Num}(X))$, and compute the automorphism group of the associated polyhedron.

Experimental results

Research questions

• RQ1 What is the structure of the Néron-Severi lattice modulo torsion for Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1?
• RQ2 How many nonsingular rational curves and non-effective $(-2)$-divisors exist on such Enriques surfaces, and what is their geometric configuration?
• RQ3 What is the index of the reflection group generated by these $(-2)$-divisors inside the orthogonal group of the Néron-Severi lattice?
• RQ4 Can the automorphism group of the Enriques surface be described in terms of symmetric group actions and outer automorphisms?
• RQ5 Do the Cremona transformations on the covering K3 surface descend to automorphisms of the Enriques surface?

Key findings

• The paper constructs a 1-dimensional family of Enriques surfaces parametrized by $a, b \in k$ with $a + b = ab$ and $a^3 \neq 1$, where $a = b = 0$ gives a supersingular Enriques surface and otherwise gives classical Enriques surfaces.
• There exist exactly 30 nonsingular rational curves and 10 non-effective $(-2)$-divisors on the Enriques surface $X_{a,b}$, whose dual graph $\Gamma$ corresponds to the incidence structure of duads, synthemes, and permutations in $\mathfrak{S}_6$.
• The reflection group $W(\Gamma)$ generated by reflections in the 40 $(-2)$-divisors is of finite index in $\text{O}(\text{Num}(X))$, the orthogonal group of the Néron-Severi lattice modulo torsion.
• The automorphism group of the polyhedron $D(\Gamma)$ is isomorphic to $\mathfrak{S}_6 \cdot \mathbb{Z}/2\mathbb{Z}$, where the $\mathbb{Z}/2\mathbb{Z}$ factor arises from the outer automorphism of $\mathfrak{S}_6$.
• The group $W(\Gamma)$ is generated by reflections in the 40 $(-2)$-divisors, and its action realizes the full orthogonal group up to sign and finite index.
• Under the conjecture that certain Cremona transformations on the K3 cover descend to $X$, the automorphism group $\text{Aut}(X)$ is generated by $\mathfrak{S}_6 \cdot \mathbb{Z}/2\mathbb{Z}$ and a group $G$ of reflections in the 10 non-effective $(-2)$-divisors, up to finite groups.