If a person owns a piece of land, do they own it all the way down to the center of the earth?

not likely. need reference state real property laws , mining , mineral rights laws. attached interesting read should stimulate search further.
http://uclalawreview.org/pdf/55-4-4.pdf

excerpt:
how far below earth’s surface property rights extend? conventional
wisdom landowner holds title between
surface , center of earth. article first legal scholarship
challenge traditional view. demonstrates “center of earth”
theory poetic hyperbole, not binding law. broadly speaking, deeper
disputed region, less courts recognize surface owner’s title.
emergence of new technologies use of deep subsurface—such
heat mining , carbon sequestration, both of may mitigate global
climate change—requires develop new model of subsurface ownership.
accordingly, article proposes , evaluates 4 alternative approaches
subsurface property rights. preferred model recognize surface
owner’s title 1000 feet downward. if adopted, approach
eliminate on 99 percent of supposed real property ownership in
united states.

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