  Large-scale Monte Carlo simulations are employed to study phase transitions
in the three-dimensional compact abelian Higgs model in adjoint representations
of the matter field, labelled by an integer q, for q=2,3,4,5. We also study
various limiting cases of the model, such as the $Z_q$ lattice gauge theory,
dual to the $3DZ_q$ spin model, and the 3DXY spin model which is dual to the
$Z_q$ lattice gauge theory in the limit $q \to \infty$. We have computed the
first, second, and third moments of the action to locate the phase transition
of the model in the parameter space $(\beta,\kappa)$, where $\beta$ is the
coupling constant of the matter term, and $\kappa$ is the coupling constant of
the gauge term. We have found that for q=3, the three-dimensional compact
abelian Higgs model has a phase-transition line $\beta_{\rm{c}}(\kappa)$ which
is first order for $\kappa$ below a finite {\it tricritical} value
$\kappa_{\rm{tri}}$, and second order above. We have found that the
$\beta=\infty$ first order phase transition persists for finite $\beta$ and
joins the second order phase transition at a tricritical point
$(\beta_{\rm{tri}}, \kappa_{\rm{tri}}) = (1.23 \pm 0.03, 1.73 \pm 0.03)$. For
all other integer $q \geq 2$ we have considered, the entire phase transition
line $\beta_c(\kappa)$ is critical.
