Are U1 public urls static?

It depends.

Once you've published a file, if you unpublish it and then re-publish it, it will get the same URL the second time as you did the first.

You can move it within a Ubuntu One "volume" (that is, the Ubuntu One folder or any other cloud-synced folders), and it will retain its public URL. You can modify the file and that, also, will keep the URL. However, if you edit it with an editor that renames the file, creates a new one, copies the content in and then deletes the original (as most editors do), and syncdaemon is running, it will see the rename and the delete and your public URL will refer first to the renamed file, and then will disappear.

This is an unfortunate side-effect of the way we're doing public files, and something we need to change, but right now this is what it is.