  We study circle compactifications of 6d superconformal field theories giving
rise to 5d rank 1 and rank 2 Kaluza-Klein theories. We realise the resulting
theories as M-theory compactifications on local Calabi-Yau 3-folds and match
the prepotentials from geometry and field theory. One novelty in our approach
is that we include explicit dependence on bare gauge couplings and mass
parameters in the description which in turn leads to an accurate
parametrisation of the prepotential including all parameters of the field
theory. We find that the resulting geometries admit "fibre-base" duality which
relates their six-dimensional origin with the purely five-dimensional quantum
field theory interpretation. The fibre-base duality is realised simply by
swapping base and fibre curves of compact surfaces in the local Calabi-Yau
which can be viewed as the total space of the anti-canonical bundle over such
surfaces. Our results show that such swappings precisely occur for surfaces
with a zero self-intersection of the base curve and result in an exchange of
the 6d and 5d pictures.
