The tetrahedron and automorphisms of Enriques and Coble surfaces of Hessian type

Allcock, Daniel; Dolgachev, Igor

Annales Henri Lebesgue, Volume 3 (2020), pp. 1133-1159.

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 ${𝔖}_4$. 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.

Donnons-nous une surface cubique et faisons l’hypothèse, faible, que cette surface peut être décrite sous forme pentaèdrique de Sylvester. Il est bien connu que l’on peut trouver une surface de Enriques ou de Coble $S$ dont un revêtement double est une surface K3 birationnellement isomorphe à la hessienne de cette surface cubique. Nous décrivons le cône nef et les $-2$-courbes de $S$. Si les paramètres pentaèdriques sont $(1,1,1,1,t\ne 0)$ nous calculons le groupe d’automorphismes de $S$. Lorsque $t\ne 1$, c’est le produit semi-direct du produit libre ${(\mathbb{Z}/2)}^{*4}$ et du groupe symétrique ${𝔖}_4$. Dans le cas particulier $t=\frac{1}{16}$ nous étudions l’action de $Aut\left(S\right)$ sur une courbe rationnelle, lisse et invariante $C$ de la surface de Coble $S$. Nous décrivons l’action et son image, de manière géométrique et arithmétique à la fois. En particulier, nous montrons que l’homomorphisme $Aut\left(S\right)\to Aut\left(C\right)$ est injectif en caractéristique $0$ et nous identifions son image au sous-groupe de ${PGL}_2$ associé aux isométries d’un tétraèdre régulier et aux réflexions le long de ses faces.

Classification: 14J50, 20F65, 20F67
Keywords: Enriques surfaces, Coble surfaces, Automorphism groups, Hyperbolic geometry

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