On automorphisms of Enriques surfaces
Let F be an Enriques surface over an algebraically closed field k (char \(k\neq 2)\), and let \(\bar F\) be its K 3-double cover. If \(k={\mathbb{C}}\), the group \(Aut(\bar F)\) of the automorphisms of \(\bar F\) has been studied [via a global Torelli for K 3-surfaces, cf. \textit{A. N. Rudakov} and \textit{I. R. Shafarevich}, J. Sov. Math. 22, 1476-1533 (1983); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 18, 115-207 (1981; Zbl 0518.14015) and this allows to compute also Aut(F), when F is generic, cf. \textit{W. Barth} and \textit{C. Peters}, Invent. Math. 73, 383-411 (1983; Zbl 0518.14023)]. The author studies Aut(F) for an arbitrary F and proves: Let \(H_ F=Pic(F)/Tors\) and \(Aut(F)^*=\rho (Aut(F))\), where \(\rho: Aut(F)\to O(H_ F)\) is the natural representation in the orthogonal group. Let \(W^ n_ F\) be the subgroup of \(O(H_ F)\) generated by reflections into the classes of non rational curves on F. The subgroup G of \(O(H_ F)\) generated by \(Aut(F)^*\) and \(W_ F^ n\) is the semidirect product of \(W_ F^ n\) and \(Aut(F)^*\). Moreover if \(k={\mathbb{C}}\), then G is of finite index in \(O(H_ F)\). - The author also proves: if F does not contain nonsingular rational curves, then Aut(F) is infinite. Finally he furnishes an example of an Enriques surface \(F_ 0\) with finitely many automorphisms such that (if \(k={\mathbb{C}})\) Aut\((\bar F_ 0)\) is infinite. The author conjectures: let F, \(\bar F\) be as above, then \(Aut(\bar F)\) is infinite.
scientific article; zbMATH DE number 3918558
scientific article