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

KeywordsEnriques 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 $\left(-2\right)$-curves of $S$. In the case of pentahedral parameters $\left(1,1,1,1,t\ne 0\right)$ we compute the automorphism group of $S$. For $t\ne 1$ it is the semidirect product of the free product ${\left(ℤ/2\right)}^{*4}$ and the symmetric group ${𝔖}_{4}$. 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] Bourbaki, Nicolas Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Élements de Mathématique, Masson, 1981 | Zbl 0483.22001

[CD89] Cossec, François; Dolgachev, Igor V. Enriques surfaces. I., Progress in Mathematics, Volume 76, Birkhäuser, 1989 | MR 986969 | Zbl 0665.14017

[Cob39] Coble, Arthur B. Cremona transformations with an invariant rational sextic, Bull. Am. Math. Soc., Volume 45 (1939) no. 4, pp. 285-288 | Article | MR 1563967 | Zbl 0021.15304

[Cob82] Coble, Arthur B. Algebraic geometry and theta functions, Colloquium Publications, Volume 10, American Mathematical Society, 1982 (reprint of the 1929 edition) | MR 733252

[DO19] Dinh, Tien-Cuong; Oguiso, Keiji A surface with discrete and non-finitely generated automorphism group, Duke Math. J., Volume 168 (2019) no. 6, pp. 941-966 | Article | Zbl 1427.14085

[Dol08] Dolgachev, Igor V. Reflection groups in algebraic geometry, Bull. Am. Math. Soc., Volume 45 (2008) no. 1, pp. 1-60 | Article | MR 2358376 | Zbl 1278.14001

[Dol12] Dolgachev, Igor V. Classical algebraic geometry. A modern view, Cambridge University Press, 2012 | Zbl 1252.14001

[Dol16] Dolgachev, Igor V. A brief introduction to Enriques surfaces, Development of moduli theory, Kyoto 2013 (Advanced Studies in Pure Mathematics) Volume 69 (2016), pp. 1-32 | MR 3586505 | Zbl 1369.14048

[Dol18] Dolgachev, Igor V. Salem numbers and Enriques surfaces, Exp. Math., Volume 27 (2018) no. 3, pp. 287-301 | Article | MR 3857665 | Zbl 1404.14043

[DZ01] Dolgachev, Igor V.; Zhang, De-Qi Coble rational surfaces, Am. J. Math., Volume 123 (2001) no. 1, pp. 79-114 | Article | MR 1827278 | Zbl 1056.14054

[Kon86] Kondō, Shigeyuki Enriques surfaces with finite automorphism groups, Jap. J. Math., Volume 12 (1986) no. 2, pp. 191-282 | Article | MR 914299 | Zbl 0616.14031

[Lei18] Leisieutre, John A projective variety with discrete, non-finitely generated automorphism group, Invent. Math., Volume 212 (2018) no. 1, pp. 189-211 | Article | MR 3773792 | Zbl 1393.14012

[Mas88] Maskit, Bernard Kleinian Groups, Grundlehren der Mathematischen Wissenschaften, Volume 287, Springer, 1988 | MR 959135 | Zbl 0627.30039

[MO15] Mukai, Shigeru; Ohashi, Hisanori 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 | Article | MR 3380455 | Zbl 1326.14097

[Sch02] Schwartz, Richard Evan Lucy and Lily: A Game of Geometry and Number Theory, Am. Math. Mon., Volume 109 (2002) no. 1, pp. 13-20 | Article | Zbl 1081.00501

[Ser03] Serre, Jean-Pierre Trees, Springer Monographs in Mathematics, Springer, 2003 | Zbl 1013.20001

[Shi19] Shimada, Ichiro On an Enriques surface associated with a quartic Hessian surface, Can. J. Math., Volume 71 (2019) no. 1, pp. 213-246 | Article | MR 3928261 | Zbl 1409.14068

[Sil09] Silverman, Joseph Hillel The arithmetic of elliptic curves, Graduate Texts in Mathematics, Volume 106, Springer, 2009 | MR 2514094 | Zbl 1194.11005

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

[Vin72] Vinberg, Èrnest Borisovich On the groups of unit elements of certain quadratic forms, Math. USSR, Sb., Volume 16 (1972), pp. 17-35 | Article | Zbl 0252.20054