  We review the dual relationship between various compact U(1) lattice models
and Abelian Higgs models, the latter being the disorder field theories of
line-like topological excitations in the systems. We point out that the
predicted first-order transitions in the Abelian Higgs models (Coleman-Weinberg
mechanism) are, in three dimensions, in contradiction with direct numerical
investigations in the compact U(1) formulation since these yield continuous
transitions in the major part of the phase diagram. In four dimensions, there
are indications from Monte Carlo data for a similar situation. Concentrating on
the strong-coupling expansion in terms of geometrical objects, surfaces or
lines, with certain statistical weights, we present semi-quantitative arguments
explaining the observed cross-over from first-order to continuous transitions
by the balance between the lowest two weights (``2:1 ratio'') of these
geometrical objects.
