  We study Euclidean M5-branes wrapping vertical divisors in elliptic
Calabi-Yau fourfold compactifications of M/F-theory that admit a Sen limit. We
construct these Calabi-Yau fourfolds as elliptic fibrations over coordinate
flip O3/O7 orientifolds of toric hypersurface Calabi-Yau threefolds. We devise
a method to analyze the Hodge structure (and hence the dimension of the
intermediate Jacobian) of vertical divisors in these fourfolds, using only the
data available from a type IIB compactification on the O3/O7 Calabi-Yau
orientifold. Our method utilizes simple combinatorial formulae (that we prove)
for the equivariant Hodge numbers of the Calabi-Yau orientifolds and their
prime toric divisors, along with a formula for the Euler characteristic of
vertical divisors in the corresponding elliptic Calabi-Yau fourfold. Our
formula for the Euler characteristic includes a conjectured correction term
that accounts for the contributions of pointlike terminal $\mathbb{Z}_2$
singularities corresponding to perturbative O3-planes. We check our conjecture
in a number of explicit examples and find perfect agreement with the results of
direct computations.

High Energy Physics - Theory;Algebraic Geometry