Automorphism groups of rational elliptic surfaces with section and constant \(J\)-map (Q403204)

The aim of the paper is the classification of automorphisms on a rational elliptic surface. Motivation for this is due to the article of \textit{V. Bouchard} and \textit{R. Donagi} [Commun. Number Theory Phys. 2, No. 1, 1--61 (2008; Zbl 1165.14032)], where the authors construct Calabi-Yau threefolds starting from pairs of rational elliptic surfaces with suitable pairs of automorphisms. This construction motivates the search for a general classification of automorphisms on rational elliptic surfaces. In the paper under review, the author considers a rational surface \(B\) with an elliptic fibration admitting a section \(\sigma\), called the zero section. He focuses on the case of constant \(J\)-map, since the case of non-constant \(J\) map is already treated in [\textit{T. Karayayla}, Adv. Math. 230, No. 1, 1--54 (2012; Zbl 1237.14044)]. It is known that the automorphism group of \(B\) is the semi-direct product of the group of sections of the fibration \(MW(B)\) and the subgroup \(\mathrm{Aut}_{\sigma}(B)\) of automorphisms preserving the zero section \(\sigma\). The possibilities for \(MW(B)\) are classified by Oguiso and Shioda; in this paper the author shows all possibilities for \(\mathrm{Aut}_{\sigma}(B)\) in all possible configurations of singular fibers that can occur on a rational elliptic surface. Existence of these groups is proven as well, with a geometric construction.