  In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber
ceases to be flat over compact Riemann surfaces of genus $g$ in the base. These
non-flat fibers contribute Kaehler moduli to the four-fold but also add to the
three-form cohomology for $g>0$. In F-/M-theory these sectors are to be
interpreted as compactifications of six/five dimensional $\mathcal{N}=(1,0)$
superconformal matter theories. The three-form cohomology leads to additional
chiral singlets proportional to the dimension of five dimensional Coulomb
branch of those sectors. We construct explicit examples for E-string theories
as well as higher rank cases. For the E-string theories we further investigate
conifold transitions that remove those non-flat fibers. First, we show how
non-flat fibers can be deformed from curves down to isolated points in the
base. This removes the chiral singlet of the three-forms and leads to
non-perturbative four-point couplings among matter fields which can be
understood as remnants of the former E-string. Alternatively, the non-flat
fibers can be avoided by performing birational base changes, analogous to 6D
tensor branches. For compact bases these transitions alternate all Hodge
numbers but leave the Euler number invariant.
