Maybe a better question would be, who is supposed to own the property when a state has been admitted to the Union?
Thats a good point.
The state.
There is a three part Youtube lecture by Stephen Pratt to the Western States Sheriffs Assn that covers that, and much more. It covers Roman Law, Common Law (and their origins and differences), the Constitution and it's writer (Gouverneur Morris, ever heard of him?), Original Intent, Hidalgo and more. Just a great source to start understanding the problem.