We know the earth has both gravity and magnetism (which is very weak). Geomagnetism starts at the south pole and ends at the north pole. If I take a large sheet of superconducting material (very thin and several kilometers in area) and place it on the south pole will it levitate? I am not discussing antigravity.
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I can't plug in numbers right now but I doubt it would levitate. As you increase the superconductor volume, you increase its mass and therefore the weight. You should match this with the force from Meissner effect which probably scales with the surface area (since it arises from dissipationles surface currents). So for linear size $L$ gravity will increase as $L^3$ while magnetic forces will increase like $L^2$. Making the size bigger thus works against you.
