  We consider a variant of the charge-Q compact Abelian-Higgs model, in which
an Nf-dimensional complex vector is coupled with an Abelian Z_q gauge field.
For Nf=2 and Q=1 we observe several transition lines that belong to the O(4),
O(3), and O(2) vector universality classes, depending on the symmetry breaking
pattern at the transition. The universality class is independent of $q$ as long
as q>=3. The universality class of the transition is uniquely determined by the
behavior of the scalar fields; gauge fields do not play any role. We also
investigate the system for Nf=15 and Q=2. In the presence of U(1) gauge fields,
the system undergoes transitions associated with charged fixed points of the
Abelian-Higgs field theory. These continuous transitions turn into first-order
ones when the U(1) gauge fields are replaced by the discrete Z_q fields: in the
present compact model charged transitions appear to be very sensitive to the
nature of the gauge fields
