### Metadata

### 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 $(1,1,1,1,t\ne 0)$ we compute the automorphism group of $S$. For $t\ne 1$ it is the semidirect product of the free product ${(\mathbb{Z}/2)}^{*4}$ and the symmetric group ${\U0001d516}_{4}$. In the special case $t=\frac{1}{16}$ we study the action of $Aut\left(S\right)$ 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 $Aut\left(S\right)\to Aut\left(C\right)$ 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.

### References

[Bou81] Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Élements de Mathématique, Masson, 1981 | Zbl

[CD89] Enriques surfaces. I., Progress in Mathematics, Volume 76, Birkhäuser, 1989 | MR | Zbl

[Cob39] Cremona transformations with an invariant rational sextic, Bull. Am. Math. Soc., Volume 45 (1939) no. 4, pp. 285-288 | DOI | MR | Zbl

[Cob82] Algebraic geometry and theta functions, Colloquium Publications, Volume 10, American Mathematical Society, 1982 (reprint of the 1929 edition) | MR

[DO19] A surface with discrete and non-finitely generated automorphism group, Duke Math. J., Volume 168 (2019) no. 6, pp. 941-966 | DOI | Zbl

[Dol08] Reflection groups in algebraic geometry, Bull. Am. Math. Soc., Volume 45 (2008) no. 1, pp. 1-60 | DOI | MR | Zbl

[Dol12] Classical algebraic geometry. A modern view, Cambridge University Press, 2012 | Zbl

[Dol16] A brief introduction to Enriques surfaces, Development of moduli theory, Kyoto 2013 (Advanced Studies in Pure Mathematics) Volume 69 (2016), pp. 1-32 | MR | Zbl

[Dol18] Salem numbers and Enriques surfaces, Exp. Math., Volume 27 (2018) no. 3, pp. 287-301 | DOI | MR | Zbl

[DZ01] Coble rational surfaces, Am. J. Math., Volume 123 (2001) no. 1, pp. 79-114 | DOI | MR | Zbl

[Kon86] Enriques surfaces with finite automorphism groups, Jap. J. Math., Volume 12 (1986) no. 2, pp. 191-282 | DOI | MR | Zbl

[Lei18] A projective variety with discrete, non-finitely generated automorphism group, Invent. Math., Volume 212 (2018) no. 1, pp. 189-211 | DOI | MR | Zbl

[Mas88] Kleinian Groups, Grundlehren der Mathematischen Wissenschaften, Volume 287, Springer, 1988 | MR | 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

[Sch02] Lucy and Lily: A Game of Geometry and Number Theory, Am. Math. Mon., Volume 109 (2002) no. 1, pp. 13-20 | DOI | Zbl

[Ser03] Trees, Springer Monographs in Mathematics, Springer, 2003 | Zbl

[Shi19] On an Enriques surface associated with a quartic Hessian surface, Can. J. Math., Volume 71 (2019) no. 1, pp. 213-246 | DOI | MR | Zbl

[Sil09] The arithmetic of elliptic curves, Graduate Texts in Mathematics, Volume 106, Springer, 2009 | MR | Zbl

[Vin71] Discrete linear groups generated by reflections, Math. USSR, Izv., Volume 5 (1971) no. 5, pp. 1083-1119 | DOI

[Vin72] On the groups of unit elements of certain quadratic forms, Math. USSR, Sb., Volume 16 (1972), pp. 17-35 | DOI | Zbl