Hi Ken,
I haven't experienced black scurf myself, but according to the RHS it is a fungal disease that results in black growths on potato tuber surfaces that can be rubbed off. The marking is cosmetic and doesn't affect the tuber storage, but the fungus can also cause dieback of shoots and reduced yield.
Apparently the fungus is present in the soil as well as on affected tubers, but avoiding replanting badly marked tubers can help reduce the incidence. As well as chosing clean tubers they suggest crop rotation (plant something else there for a few years), also avoid early planting, plant tubers that have been chitted and plant more shallowly (to speed up shoot emergence from soil and reduce stem canker) and lift tubers as soon as feasible to reduce the amount of fungal spores on them.
There's quite a bit in wikipedia on the causal agent Rhizoctonia solani, which seeems to be responsable for quite a few plant diseases. Apparently it prefers warm wet conditions, so improving drainage may also help. I'm also wondering if a covercrop of mustard might help reduce the presence of the fungal organism in the soil? I've heard that can be a biofumigant against some soil borne diseases.
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