Synthesis of various carbon nanostructures, including fullerenes, single-walled and multi-walled nanotubes and nanoparticles, by arc discharges relies on ablation of the graphite anode and deposition of synthesized carbonaceous products on the cathode surface and on the reactor chamber walls. For backbone all-carbon system, the cathode deposit plays a critical role in sustaining the arc discharge and thereby, the synthesis processes. This deposit usually exhibits spatially distinct structural variations with three different axially symmetrical morphologies. In particular, a rim of pyrolytic carbon separates the innermost core consisting of multi-walled carbon nanotubes from the outmost ring with powdery amorphous carbon soot. Experiments revealed a strong correlation between the current conducting arc attachment to the cathode deposit and the nanotube forming area in the deposit. Results suggest that particle and heat fluxes from the plasma are responsible for purity of nanotubes in this deposit core area. It appears that a better synthesis selectivity can be obtained in low ablation regime which is characterized by a nearly constant arc current density independent on the anode diameter.