  We numerically investigate the dynamics of vortex generation in a
two-dimensional, twocomponent Bose-Einstein condensate subjected to an
oscillatory magnetic obstacle. The obstacle creates both repulsive and
attractive Gaussian potentials for the two symmetric spin-$\uparrow$ and
$\downarrow$ components, respectively. We demonstrate that, as the oscillating
frequency f increases, two distinct critical dynamics arise in the generation
of half-quantum vortices (HQVs) with different spin circulations.
Spin-$\uparrow$ vortices are nucleated directly from the moving obstacle at low
f, while spin-$\downarrow$ vortices are created at high f by breaking a spin
wave pulse in front of the obstacle. We find that vortex generation is
suppressed for sufficiently weak obstacles, in agreement with recent
experimental results by Kim et al. [Phys. Rev. Lett. 127, 095302 (2021)]. This
suppression is caused by the finite sweeping distance of the oscillating
obstacle and the reduction in friction in a supersonic regime. Finally, we show
that the characteristic length scale of the HQV generation dynamics is
determined by the spin healing length of the system.
