We initiate the systematic investigation of non-flat resolutions of
non-minimal singularities in elliptically fibered Calabi--Yau threefolds.
Compactification of M-theory on these geometries provides an alternative
approach to studying phases of five-dimensional superconformal field theories
(5d SCFTs). We argue that such resolutions capture non-trivial holonomies in
the circle reduction of the 6d conformal matter theory that is the F-theory
interpretation of the singular fibration. As these holonomies become mass
deformations in the 5d theory, non-flat resolutions furnish a novel method in
the attempt to classify 5d SCFTs through 6d SCFTs on a circle. A particularly
pleasant aspect of this proposal is the explicit embedding of the 5d SCFT's
enhanced flavor group inside that of the parent 6d SCFT, which can be read off
from the geometry. We demonstrate these features in toric examples which
realize 5d theories up to rank four.Comment: 49 pages + appendix and references, 10 figures; v2: discussion of
prepotentials improved, references added, typos corrected; v3+v4: typos
fixed, accepted for publication in JHE
