  We explore 6-dimensional compactifications of F-theory exhibiting (2,0)
superconformal theories coupled to gravity that include discretely charged
superconformal matter. Beginning with F-theory geometries with Abelian gauge
fields and superconformal sectors, we provide examples of Higgsing transitions
which break the $U(1)$ gauge symmetry to a discrete remnant in which the matter
fields are also non-trivially coupled to a (2,0) SCFT. In the compactification
background this corresponds to a geometric transition linking two fibered
Calabi-Yau geometries defined over a singular base complex surface. An
elliptically fibered Calabi-Yau threefold with non-zero Mordell-Weil rank can
be connected to a smooth non-simply connected genus one fibered geometry
constructed as a Calabi-Yau quotient. These hyperconifold transitions exhibit
multiple fibers in co-dimension 2 over the base.
