In particular, one knows a smooth point of a variety is a point at which the dimension of the (affine) tangent space is locally constant. I can translate this back into the original vector space and see the dimension, but I don't see how this carries over into projective space.

I suppose I'm looking for a little intuition in this matter...

Also, I'm posting this in the "Linear and Abstract Algebra" forum also. Mods, if this is contrary to any rules, please remove whichever posting is in the wrong place, but since this is an algebraic geometry question, I figured it would be best to get both perspectives.

Thanks in advance.