Unless you take very small cuts, the stock may indeed flex and you might try a follower rest. First try taking a cut of maybe .005 and using a good sulfated cutting oil and see what happens. also make certain you have lubed the dead center.
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Thanks for all the comments.
I was turning a .5 inch diameter steel rod that was about 5 inches
long. One end was in a 3 jaw chuck with the other by a "dead center"
in the tail stock.
The tool might have been at the wrong height, but it was aligned
by eye with the "dead center" which had been aligned with a center in
the head stock. One of the tools I was using was a brand new
carbide with a 7 degree clearance angle. Another was the "diamond
tool", which gave better results.
Which case would cause the tool bit to "dig in" more. The tool above
center or below?