We study the vortex-wave system for the general surface quasi-geostrophic equations. We establish glocal existence and uniqueness for strong solutions and give a criterion for blow-up. This relies on fine estimates on the commutator structure of the equation and on the behavior of the point-vortex if it collapses with the boundary of the constant part of the vorticity where it lays initially. In a second part of this work, we investigate the question of global existence of weak solutions. In the critical case, we need to introduce a weaker notion of solution to give a meaning to the manipulated quanitites and conclude.