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.
[Bou81] Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Élements de Mathématique, Masson, 1981 | Zbl
[Cob82] Algebraic geometry and theta functions, Colloquium Publications, Volume 10, American Mathematical Society, 1982 (reprint of the 1929 edition) | MR
[Dol12] Classical algebraic geometry. A modern view, Cambridge University Press, 2012 | Zbl
[MO15] The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces, Recent advances in algebraic geometry (London Mathematical Society Lecture Note Series) Volume 417, Cambridge University Press, 2015, pp. 307-320 | DOI | MR | Zbl
[Ser03] Trees, Springer Monographs in Mathematics, Springer, 2003 | Zbl
[Vin71] Discrete linear groups generated by reflections, Math. USSR, Izv., Volume 5 (1971) no. 5, pp. 1083-1119 | DOI