[Paper Review] The automorphism group of the singular K3 surface of discriminant 7

Ujikawa Masashi|arXiv (Cornell University)|Jul 16, 2013

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper provides a complete system of generators for the automorphism group of the singular K3 surface with discriminant 7, using inversion involutions associated with elliptic fibrations of types $A_7$, $D_7$, and $E_7$, together with the group ${\rm PGL}_2(7)$. The key result is that these generators form a split extension, fully describing the automorphism group via the geometry of the Néron–Severi lattice embedded in the $II_{1,25}$ lattice and Conway's fundamental domain.

ABSTRACT

We give a system of generators of the automorphism group of the singular $K3$ surface of discriminant 7. This system of generators consists of the inversion involutions of some elliptic fibrations with a section together with ${ m PGL}_2(7)$.

Motivation & Objective

• To determine a complete system of generators for the automorphism group of the singular K3 surface with discriminant 7, the next smallest after 3 and 4.
• To extend the method used for discriminants 3 and 4 to a higher discriminant case using lattice embeddings and reflection groups.
• To describe the automorphism group via a finite set of geometrically meaningful transformations, specifically inversion involutions of elliptic fibrations and a finite group action.
• To establish that the automorphism group is a split extension of ${\rm PGL}_2(7)$ by the normal subgroup generated by the inversion involutions.

Proposed method

• Embed the Néron–Severi lattice $S_X$ of the singular K3 surface of discriminant 7 into the even unimodular lattice $II_{1,25}$ as the orthogonal complement of an $A_6$ root lattice.
• Utilize Conway's fundamental domain $D$ for the reflection group of $II_{1,25}$, whose faces correspond to Leech lattice vectors.
• Restrict $D$ to $S_X \otimes \mathbb{R}$ to obtain a polyhedral cone $D'$ with finitely many faces, which lies within the ample cone $D(S_X)$.
• Identify the faces of $D'$ with root systems: $A_6 \oplus A_1$, $A_7$, $D_7$, and $E_7$, each corresponding to a geometric structure on the K3 surface.
• Construct inversion involutions $\iota_R$ for each face type ($R = A_7, D_7, E_7$) that act as symmetries preserving the ample cone.
• Prove that the group generated by these involutions and ${\rm PGL}_2(7)$ acts as the full automorphism group of the ample cone, hence of the K3 surface.

Experimental results

Research questions

• RQ1 What is a complete system of generators for the automorphism group of the singular K3 surface of discriminant 7?
• RQ2 How can the automorphism group be described using geometric structures like elliptic fibrations and root systems?
• RQ3 Can the automorphism group be realized as a split extension involving ${\rm PGL}_2(7)$ and inversion involutions?
• RQ4 To what extent do the faces of the restricted Conway fundamental domain $D'$ correspond to geometric features of the K3 surface?
• RQ5 Is the group generated by the inversion involutions and ${\rm PGL}_2(7)$ sufficient to generate the full automorphism group of the surface?

Key findings

• The automorphism group of the singular K3 surface of discriminant 7 is generated by the inversion involutions associated with elliptic fibrations of types $A_7$, $D_7$, and $E_7$, together with the group ${\rm PGL}_2(7)$.
• The restriction of Conway's fundamental domain $D$ to $S_X \otimes \mathbb{R}$ yields a polyhedral cone $D'$ with finitely many faces, enabling explicit computation of the automorphism group.
• The faces of $D'$ correspond to root systems $A_6 \oplus A_1$, $A_7$, $D_7$, and $E_7$, each linked to a geometric structure on the K3 surface.
• The inversion involutions $\iota_R$ for $R = A_7, D_7, E_7$ act as symmetries on the ample cone and preserve the lattice structure.
• The automorphism group is a split extension of the finite group ${\rm PGL}_2(7)$ by the normal subgroup generated by the inversion involutions.
• The proof establishes that any automorphism of the surface can be composed with an element of the generated group to lie in the automorphism group of the ample cone $D(S_X)$, confirming completeness.