  In Special Geometry there are two inequivalent notions of "Kodaira type" for
a singular fiber: one associated with its local monodromy and one with its
Hwang-Oguiso characteristic cycle. When the two Kodaira types are not equal the
geometry is subtler and its deformation space gets smaller ("partially frozen"
singularities). The paper analyzes the physical interpretation of the
Hwang-Oguiso invariant in the context of 4d N=2 QFT and describes the
surprising phenomena which appear when it does not coincide with the monodromy
type. The Hwang-Oguiso multiple fibers are in one-to-one correspondence with
the partially frozen singularities in M-theory compactified on a local elliptic
K3: a chain of string dualities relates the two geometric set-ups. Paying
attention to a few subtleties, this correspondence explains in purely geometric
terms how the "same" Kodaira elliptic fiber may have different deformations
spaces. The geometric computation of the number of deformations agrees with the
physical expectations. At the end we briefly outline the implications of the
Hwang-Oguiso invariants for the classification program of 4d N=2 SCFTs.
