  Using the Born-Oppenheimer approximation, we present a general description of
topological defects dynamics in $p$-atic materials on curved surfaces, and
simplify it in the case of active nematics. We find that activity induces a
geometric contribution to the motility of the $+1/2$ defect. Moreover, in the
case of a cone, the simplest example of a geometry with curvature singularity,
we find that the motility depends on the deficit angle of the cone and changes
sign when the deficit angle is bigger than $\pi$, leading to the change in
active behavior from contractile (extensile) to extensile (contractile)
behavior. Using our analytical framework, we then identify for positively
charged defects the basin of attraction to the cone apex and present
closed-form predictions for defect trajectories near the apex. The analytical
results are quantitatively corroborated against full numerical simulations.
Provided the capture radius is small compared to the cone size, the agreement
is excellent.
