 
Summary: Notes on basic algebraic geometry
June 16, 2008
These are my notes for an introductory course in algebraic geometry. I have
trodden lightly through the theory and concentrated more on examples. Some
examples are handled on the computer using Macaulay2, although I use this as
only a tool and won't really dwell on the computational issues.
Of course, any serious student of the subject should go on to learn about
schemes and cohomology, and (at least from my point of view) some of the
analytic theory as well. Hartshorne [Ht] has become the canonical introduction
to the first topic, and GriffithsHarris [GH] the second.
1
Contents
1 Affine Geometry 3
1.1 Algebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Weak Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 The CayleyHamilton theorem . . . . . . . . . . . . . . . . . . . 9
1.5 Affine Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Hilbert's Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Nilpotent matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 12
