  SO(3), SO(5), and SO(6)-models are singular elliptic fibrations with
Mordell--Weil torsion Z/2Z and singular fibers whose dual fibers correspond to
affine Dynkin diagrams of type A1, C2, and A3 respectively, where we emphasize
the distinction between SO(n) and its universal cover Spin(n). While the
SO(3)-model has been studied before, the SO(5) and SO(6)-models are studied
here for the first time. By computing crepant resolutions of their Weierstrass
models, we study their fiber structures and topological invariants. In the
special case that the SO(n)-model is an elliptically fibered Calabi-Yau
threefold, we compute the Chern-Simons couplings and matter content of a 5D N=1
supergravity theory with gauge group SO(n), which is related to M-theory
compactified on this Calabi-Yau threefold. We also verify the 6D lift of the 5D
matter content is necessary and sufficient for anomaly cancellation in 6D (1,0)
supergravity theories geometrically engineered by F-theory compactified on the
same threefold. We find that the associated 5D and 6D supergravity theories
with SO(n) gauge symmetry indeed differ from their Spin(n) cousins, with one
striking consequence of this distinction being that all such theories must
include adjoint matter.
