 
Summary: Algebraic Geometry I
Lectures 9, 10, and 11
Amod Agashe
October 13, 2008
Housekeeping Note: Dr. Agashe's office hours are now MW 2:153:15.
1 Covering Projective Space by Affine Spaces
In this section, we will develop a natural covering of Pn
k by ndimensional
affine spaces over k. As we will see, this yields homeomorphisms between
projective algebraic sets in Pn and affine algebraic sets in An.
Notation 1.1. We use the abbreviation {[some condition]} to denote the
set of all points that satisfy [some condition]. For example, {x0 = 0} is an
abbreviation for {[x0 : . . . : xn] Pn  x0 = 0}, and if f : X  Y and
c Y , then {f = c} is an abbreviation for {x X  f(x) = c}.
We now define our covering. For i = 0, 1 . . . , n, let Ui = {xi = 0} Pn,
and define i : Ui  An as
i([a0 : . . . : an]) =
a0
ai
, . . . ,
