F-GUTs with Mordell–Weil U(1)U(1)'s

In this note we study the constraints on F-theory GUTs with extra U(1)U(1)'s in the context of elliptic fibrations with rational sections. We consider the simplest case of one abelian factor (Mordell–Weil rank one) and investigate the conditions that are induced on the coefficients of its Tate form. Converting the equation representing the generic hypersurface P112P112 to this Tate's form we find that the presence of a U(1)U(1), already in this local description, is consistent with the exceptional E6E6 and E7E7 non-abelian singularities. We briefly comment on a viable E6×U(1)E6×U(1) effective F-theory model.