Tailstock to headstock alignment, too much pressure on the mandrel/mandrel not straight with the axis, live center itself not precision made, bad taper angle or alignment in either tailstock or headstock, all kinds of things.
If you have a dial indicator and base, check to see if it is running true at both ends (or turn it on slow and watch the mandrel turn without the live center engaged. That would give a starting point for further diagnoses otherwise I suppose hit and miss is the only thing I would try. If you have the capability of making an accurately center drilled piece of known round material (such as drill rod) or know someone that can make yo one -very easy on a metal lathe, maybe not possible withoutone- you could put that between centers and just feel (or measure) whether or not there is and wobble at the tailstock that is not felt at the headstock. Perfectly smooth turning at both ends would eliminate most lathe problems and put it more likely on the mandrel or the way it is being used.
TBC eliminates mandrel problems, once I tried it I haven't used a mandrel again. (I still consider myself a beginner at wood lathes and hand turning stuff, so take what I say under consideration as such, but I have a good deal of metal lathe experience on small metal lathes)
