Projective closure Is the projective closure of an infinite affine variety (over an algebraically closed field, I only care about the classical case right now) always strictly larger than the affine variety? I know it is an open dense subset of its projective closure, but I don't think it can actually be its own projective closure unless it is finite. 
I guess my intuition has me worried about cases like the plane curve $X^2 + Y^2 - 1$, since the real part is compact, but such a curve must still "escape to infinity" over an algebraically closed field, right?

For your own example: the projectivization of your curve is given by $X^2 + Y^2 - Z^2= 0$. There is the point corresponding to the projective equivalence class of $(1,i,0)$, which does not belong to the affine curve.

For more general varieties given by a single equation, you can always similarly find a projective point that is part of the affine curve. For more general varieties, I should imagine that it is still the case; but for a proper argument a better person in algebraic geometry is needed.
