On two-elementary K3 surfaces with finite automorphism group

Abstract: We study complex algebraic K3 surfaces of Picard ranks 11,12, and 13 of finite automorphism group that admit a Jacobian elliptic fibration with a section of order two. We prove that the K3 surfaces admit a birational model isomorphic to a projective quartic hypersurface and construct geometrically the frames of all supported Jacobian elliptic fibrations. We determine the dual graphs of all smooth rational curves for these K3 surfaces, the polarizing divisors, and the embedding of the reducible fibers in each frame into the corresponding dual graph.