Well, no, I argue that it doesn't move at all under you, for the same reason that a ball thrown upward isn't deflected above you: The Earth is moving you with it, and when you jump, you're only changing the outward radial component of your velocity. Because the acceleration due to gravity is radially inward, it only affects the radial component of your velocity, and thus the horizontal component is unaffected.
Though, I suppose if you were to jump really really really really high, your moment of inertia would change enough that the Earth might movie underneath you a very small amount, now that I think about it.
So I suppose we're both sort of right. It's funny, I spend all day at school studying complicated atomic/molecular physics, and I have to actually think about it to answer questions about simple kinematics. It's amazing how quickly you forget that stuff.