  In this brief note we explore the space of genus one and elliptic fibrations
within CY manifolds, their organizing principles, and how they relate to the
set of all CY manifolds. We provide examples of genus one fibered manifolds
that exhibit different Hodge numbers -- and physically lead to different gauge
groups - than their Jacobian fibrations. We suggest a physical mechanism for
understanding this difference in twisted circle reductions of 6-dimensional
compactifications of F-theory.
