The tetrahedron and automorphisms of Enriques and Coble surfaces of Hessian type
Annales Henri Lebesgue, Volume 3 (2020), pp. 1133-1159.


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


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,t0) we compute the automorphism group of S. For t1 it is the semidirect product of the free product (/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.


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