Hello there, and welcome to the board.
I don't know much about paperweights either, and someone should be along to help further soon, but the marks are nothing to worry about. This is just the pontil mark.
See here for more.
And while we're discussing that Wikipedia link... Is it accurate, and if not, would someone volunteer to improve it? Specifically, the third paragraph might not be correct for all collectable glass. (Maybe bottle collectors refer to impressed marks on the base as pontil marks, but it's not standard practice. Or is it?)