  We compare heterotic string models on orbifolds with supergravity models on
smooth compact spaces, obtained by resolving the orbifold singularities. Our
main focus is on heterotic E8xE8 models on the resolution of the compact
T^6/Z_3 orbifold with Wilson lines. We explain how different gauge fluxes at
various resolved fixed points can be interpreted in blow down as Wilson lines.
Even when such Wilson lines are trivial from the orbifold perspective, they can
still lead to additional symmetry breaking in blow-up. Full agreement is
achieved between orbifold and resolved models, at the level of gauge
interactions, massless spectrum and anomaly cancellation. In this matching the
blow-up modes are of crucial importance: they play the role of model-dependent
axions involved in the cancellation of multiple anomalous U(1)'s on the
resolution. We illustrate various aspects by investigating blow-ups of a Z_3
MSSM model with two Wilson lines: if all its fixed points are resolved
simultaneously, the SM gauge group is necessarily broken. Finally, we explore
in detail the anomaly cancellation on the complex two dimensional resolution of
C^2/Z_2.
