Automorphism groups of rational elliptic surfaces with section and constant J-map

摘要

In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is C. The automorphism group of such a surface β : B →P~1, denoted by Aut(B), consists of all biholomorphic maps on the complex manifold B. The group Aut(B)is isomorphic to the semi-direct productMW(B)>△Aut_α(B)of the Mordell-Weil groupMW(B)(the group of sections of B), and the subgroup Aut_α(B)of the automorphisms preserving a fixed section α of Bwhich is called the zero section on B. The Mordell-Weil group MW(B)is determined by the configuration of singular fibers on the elliptic surface Bdue to Oguiso and Shioda [9]. In this work, the subgroup Aut_α(B)is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.