  In previous work, it was argued that the type IIB T^6/Z_2 orientifold with a
choice of flux preserving N=2 supersymmetry is dual to a class of purely
geometric type IIA compactifications on abelian surface (T^4) fibered
Calabi-Yau threefolds. We provide two explicit constructions of the resulting
Calabi-Yau duals. The first is a monodromy based description, analogous to
F-theory encoding of Calabi-Yau geometry via 7-branes and string junctions,
except for T^4 rather than T^2 fibers. The second is an explicit
algebro-geometric construction in which the T^4 fibers arise as the Jacobian
tori of a family of genus-2 curves. This improved description of the duality
map will be a useful tool to extend our understanding of warped
compactifications. We sketch applications to related work to define warped
Kaluza-Klein reduction in toroidal orientifolds, and to check the modified
rules for D-brane instanton zero mode counting due to the presence of flux and
other D-branes. The nontrivial fundamental groups of the Calabi-Yau manifolds
constructed also have potential applications to heterotic model building.
