all around the base of the shoulder. This is done when full length sizing using a Lee die.

Any ideas how to fix this?

you can try cleaning it. you might be able to polish it with a little fritz or semi chrome polish and if you do not have any polishing compound toothpast works in a pinch.
Dave

Is it a scratch, or a shiny ring around the base?

By reading his post I think he as a scratch on the shoulder of the case. I don't think he means around the actual base of the case, which would mean that he better discard it or he may get a case head separation.

I sometimes get a 'shiny ring' around the base of the neck of 303 cases that have been fired in a 'loose' chamber when resizing them. In that instance it's from the shoulder/body junction being pushed back and folded around in the die. This ring looks ‘scratchy’.