  We analyze exotic matter representations that arise on singular seven-brane
configurations in F-theory. We develop a general framework for analyzing such
representations, and work out explicit descriptions for models with matter in
the 2-index and 3-index symmetric representations of SU($N$) and SU(2)
respectively, associated with double and triple point singularities in the
seven-brane locus. These matter representations are associated with Weierstrass
models whose discriminants vanish to high order thanks to nontrivial
cancellations possible only in the presence of a non-UFD algebraic structure.
This structure can be described using the normalization of the ring of
intrinsic local functions on a singular divisor. We consider the connection
between geometric constraints on singular curves and corresponding constraints
on the low-energy spectrum of 6D theories, identifying some new examples of
apparent "swampland" theories that cannot be realized in F-theory but have no
apparent low-energy inconsistency.
