  We analyse several explicit toric examples of compact K3-fibred Calabi-Yau
three-folds which can be used for the study of string dualities and are crucial
ingredients for the construction of LARGE Volume type IIB vacua with promising
applications to cosmology and particle phenomenology. In order to build a
phenomenologically viable model, on top of the two moduli corresponding to the
base and the K3 fibre, we demand also the existence of two additional rigid
divisors: the first supporting the non-perturbative effects needed to achieve
moduli stabilisation, and the second allowing the presence of chiral matter on
wrapped D-branes. We clarify the topology of these rigid divisors by discussing
the interplay between a diagonal structure of the Calabi-Yau volume and
D-terms. Del Pezzo divisors appearing in the volume form in a completely
diagonal way are natural candidates for supporting non-perturbative effects and
for quiver constructions, while `non-diagonal' del Pezzo and rigid but not del
Pezzo divisors are particularly interesting for model building in the geometric
regime. Searching through the existing list of four dimensional reflexive
lattice polytopes, we find 158 examples admitting a Calabi-Yau hypersurface
which is a K3 fibration with four K\"ahler moduli where at least one of them is
a `diagonal' del Pezzo. We work out explicitly the topological details of a few
examples showing how, in the case of simplicial polytopes, all the del Pezzo
divisors are `diagonal', while `non-diagonal' ones appear only in the case of
non-simplicial polytopes. A companion paper will use these results in the study
of moduli stabilisation for globally consistent explicit Calabi-Yau
compactifications with the local presence of chirality.
