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

Theoretically, that would be the case.

We are accustomed to thinking of land as a flat surface portrayed on a land survey of parcels with square corners, ninety degree angles, all laid out on a grid. However, we forget that on a larger scale the Earth is a sphere. Theoretically, where private ownership of land is allowed, each landowner would own wedge-shaped chunks of earth, some irregularly shaped, that taper toward the center of the sphere as they sink below the surface since the earth is round.

In the United States, generally, A landowner owns what is beneath their land in the form of mineral rights unless they were reserved, leased, sold by a prior owner or alienated in some other way. However, that ownership of the land is subject to all manner of modern laws and restrictions. The owner in fee of a parcel of land owns it absolutely. She can sell it, build on it, and leave it to her heirs in her will. If she dies without a will the laws of intestacy will see to it that the heirs get the property legally through probate.

In many other countries the government has reserved the mineral rights to land located within their borders, even in privately owned land.