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 with K3 cover birationally isomorphic to the Hessian surface of this cubic surface. We describe the nef cone and -curves of . In the case of pentahedral parameters we compute the automorphism group of . For it is the semidirect product of the free product and the symmetric group . In the special case we study the action of on an invariant smooth rational curve on the Coble surface . We describe the action and its image, both geometrically and arithmetically. In particular, we prove that is injective in characteristic and we identify its image with the subgroup of coming from the isometries of a regular tetrahedron and the reflections across its facets.
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