F-GUTs with Mordell-Weil U(1)'s

In this note we study the constraints on F-theory GUTs with extra $U(1)$'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 $P_{112}$ to this Tate’s form we find that the presence of a U(1), already in this local description, is consistent with the exceptional ${\mathcal{E}}_6$ and ${\mathcal{E}}_7$ non-abelian singularities. We briefly comment on a viable ${\mathcal{E}}_6\times U(1)$ effective F-theory model.