The tetrahedron and automorphisms of Enriques and Coble surfaces of Hessian type

Daniel Allcock,Igor Dolgachev

doi:10.5802/ahl.57

Abstract

Consider a cubic surface satisfying the mild condition that it may be described in Sylvester’s pentahedral form. There is a well-known Enriques or Coble surface S with K3 cover birationally isomorphic to the Hessian surface of this cubic surface. We describe the nef cone and (-2)-curves of S. In the case of pentahedral parameters (1,1,1,1,t≠0) we compute the automorphism group of S. For t≠1 it is the semidirect product of the free product (ℤ/2) *4 and the symmetric group 𝔖 4 . In the special case t=1 16 we study the action of Aut(S) on an invariant smooth rational curve C on the Coble surface S. We describe the action and its image, both geometrically and arithmetically. In particular, we prove that Aut(S)→Aut(C) is injective in characteristic 0 and we identify its image with the subgroup of PGL 2 coming from the isometries of a regular tetrahedron and the reflections across its facets.

Highlights

Let UC[4] be the universal Coxeter group on 4 generators, i.e. a free product of four groups of order 2
1 16 we study the action of Aut(S) on an invariant smooth rational curve C on the Coble surface S
Aut(S) → Aut(C) is injective in characteristic 0 and we identify its image with the subgroup of PGL2 coming from the isometries of a regular tetrahedron and the reflections across its facets

Summary

Introduction

Let UC[4] be the universal Coxeter group on 4 generators, i.e. a free product of four groups of order 2. K has characteristic = 2, 3 and S is the surface obtained by blowing up all ten double points of a certain plane rational curve of degree 6 that admits S4 as its group of projective symmetries (see [Dol, Section 5.4]) This surface is a special case of a rational Coble surface, whose definition we will recall . This allows us to deduce that the restriction homomorphism G → PGL2(k) is faithful when working in characteristic 0 This faithfulness is appealing and suggestive, but does not say anything definite about Coble’s original problem, because UC[4] S4 is much smaller than the automorphism group of a general Coble surface, which is a lattice in SO[9, 1] by [CD89, Theorem 2.10.1].

Enriques and Coble surfaces of Hessian type

The nef cone

Automorphism groups