Quenched QCD at zero baryonic chemical potential undergoes a first-order deconfinement phase
transition at a critical temperature $T_c$, which is related to the spontaneous breaking of the global center symmetry.
The center symmetry is broken explicitly by including dynamical quarks, which weaken the first-order phase transition for decreasing quark masses.
At a certain critical quark mass, which corresponds to the $Z(2)$-critical point, the first-order phase transition turns into a smooth crossover.
We investigate the $Z(2)$-critical quark mass for $N_\text{f}=2$ staggered fermions on $N_\tau=8, 10$ lattices, where larger $N_\tau$ correspond to finer lattices.
Monte-Carlo simulations are performed for several quark mass values and aspect ratios in order to extrapolate to the thermodynamic limit.
We present final results for $N_\tau=8$ and preliminary results for $N_\tau=10$ for the critical mass, which are obtained from fitting to a kurtosis finite size scaling formula of the absolute value of the Polyakov loop.