Notes on basic algebraic geometry June 16, 2008 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 Griffiths-Harris [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 Cayley-Hamilton theorem . . . . . . . . . . . . . . . . . . . 9 1.5 Affine Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Hilbert's Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 Nilpotent matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Collections: Mathematics