  We investigate the phase diagram and the nature of the phase transitions in a
three-dimensional model characterized by a global SU($N$) symmetry, a local
U(1) symmetry, and the absence of monopoles. It represents a natural
generalization of the gauge monopole-free (MF) CP$^{N-1}$ model, in which the
fixed-length constraint (London limit) is relaxed. We have performed Monte
Carlo simulations for $N=2$ and 25, observing a finite-temperature transition
in both cases, related to the condensation of a local gauge-invariant order
parameter. For $N=2$ results for the MF model are consistent with a weak
first-order transition. A continuous transition would be possible only if
scaling corrections were anomalously large. For $N=25$ the results in the
general MF model are also consistent with a first-order transition, that
becomes weaker as the size of the field-length fluctuations decreases.
