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

Allcock, Daniel; Dolgachev, Igor

doi:10.5802/ahl.57

Allcock, Daniel; Dolgachev, Igor

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

Annales Henri Lebesgue, Volume 3 (2020), pp. 1133-1159.

Metadata

DOI10.5802/ahl.57

Keywords Enriques surfaces, Coble surfaces, Automorphism groups, Hyperbolic geometry

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 $(1, 1, 1, 1, t \neq 0)$ we compute the automorphism group of $S$ . For $t \neq 1$ it is the semidirect product of the free product ${(ℤ / 2)}^{* 4}$ and the symmetric group $𝔖_{4}$ . In the special case $t = \frac{1}{16}$ 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 $Aut (S) \to Aut (C)$ 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.

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