  Three-dimensional scalar electrodynamics, with a local U(1) gauge symmetry,
is believed to be dual to a scalar theory with a global U(1) symmetry, near the
phase transition point. The conjectured duality leads to definite predictions
for the scaling exponents of the gauge theory transition in the type II region,
and allows thus to be scrutinized empirically. We review these predictions, and
carry out numerical lattice Monte Carlo measurements to test them: a number of
exponents, characterising the two phases as well as the transition point, are
found to agree with expectations, supporting the conjecture. We explain why
some others, like the exponent characterising the photon correlation length,
appear to disagree with expectations, unless very large system sizes and the
extreme vicinity of the transition point are considered. Finally, we remark
that in the type I region the duality implies an interesting quantitative
relationship between a magnetic flux tube and a 2-dimensional non-topological
soliton.
