Sections, multisections, and U(1) fields in F-theory

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We show that genus-one fibrations lacking a global section fit naturally into the geometric moduli space of Weierstrass models. Elliptic fibrations with multiple sections (nontrivial Mordell-Weil rank), which give rise in F-theory to abelian U(1) fields, arise as a subspace of the set of genus-one fibrations with multisections. Higgsing of certain matter multiplets charged under abelian gauge fields in the corresponding supergravity theories break the U(1) gauge symmetry to a discrete gauge symmetry group. We further show that in six dimensions every U(1) gauge symmetry arising in an F-theory model can be found by Higgsing an SU(2) gauge symmetry with adjoint matter, and that a similar structure holds for F-theory geometries giving 4D supergravity theories.