| Summary: | © 2016, Journal of Singularities. All rights reserved. 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 (nonzero 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 demonstrate these results explicitly in the case of bisections, and describe the general framework for multisections of higher degree. We further show that nearly every U(1) gauge symmetry arising in an Ftheory model can be “unHiggsed” to an SU(2) gauge symmetry with adjoint matter, though in certain situations this leads to a model in which a superconformal field theory is coupled to a conventional gauge and gravity theory. The only exceptions are cases in which the attempted unHiggsing leads to a boundary point at an infinite distance from the interior of the moduli space.
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