In slide 3 of the Lecture 11, U_1 and U_2 are seen as sub schemes of A^2. However, are not they (naturally) only a sub scheme of A^3?
I understand that we want to "ignore" the data carried by the u_1 and the u_2 coordinates and in some sense project this onto A^2, but I'm not able to see why this will still be a valid Z affine sub scheme of A^2 (probably it is a quasi affine sub scheme of A^2). An explanation of this will be helpful.