When I was in Florida last month, Guncle Kevin had a trick to show me. He uses "Peek Polish Paste" to eliminate scratches from china and pottery. In this photo, the left side has been Peeked, and the right has not.
He quickly, very quickly, showed me how to use it. Rub it on. Rub it off, do a little scrubble and the lines disappear.
Again, left side polished, right side all scratched up. Nice, huh?
This guy is not just handsome, he is pretty handy too!