  Resolutions of certain toroidal orbifolds, like T6/Z2xZ2, are far from
unique, due to triangulation dependence of their resolved local singularities.
This leads to an explosion of the number of topologically distinct smooth
geometries associated to a single orbifold. By introducing a parameterisation
to keep track of the triangulations used at all resolved singularities
simultaneously, (self-)intersection numbers and integrated Chern classes can be
determined for any triangulation configuration. Using this method the
consistency conditions of line bundle models and the resulting chiral spectra
can be worked out for any choice of triangulation. Moreover, by superimposing
the Bianchi identities for all triangulation options much simpler though
stronger conditions are uncovered. When these are satisfied, flop--transitions
between all different triangulations are admissible. Various methods are
exemplified by a number of concrete models on resolutions of the T6/Z2xZ2
orbifold.
