  We study infinite-distance limits in the complex structure moduli space of
elliptic Calabi-Yau threefolds. In F-theory compactifications to six
dimensions, such limits include infinite-distance trajectories in the
non-perturbative open string moduli space. The limits are described as
degenerations of elliptic threefolds whose central elements exhibit non-minimal
elliptic fibers, in the Kodaira sense, over curves on the base. We show how
these non-crepant singularities can be removed by a systematic sequence of
blow-ups of the base, leading to a union of log Calabi-Yau spaces glued
together along their boundaries. We identify criteria for the blow-ups to give
rise to open chains or more complicated trees of components and analyse the
blow-up geometry. While our results are general and applicable to all
non-minimal degenerations of Calabi-Yau threefolds in codimension one, we
exemplify them in particular for elliptic threefolds over Hirzebruch surface
base spaces. We also explain how to extract the gauge algebra for F-theory
probing such reducible asymptotic geometries. This analysis is the basis for a
detailed F-theory interpretation of the associated infinite-distance limits
that will be provided in a companion paper.
