pdf file for the current version (5.22)

An introductory course. In contrast to most such accounts it studies abstract algebraic varieties, and not just subvarieties of affine and projective space. This approach leads more naturally into scheme theory.

Contents

1. Preliminaries
2. Algebraic sets
3. Affine algebraic varieties
4. Algebaic varieties
5. Local study
6. Projective varieties
7. Complete varieties
8. Finite maps
9. Dimension theory
10. Regular maps and their fibres
11. Algebraic spaces: geometry over an arbitrary field
12. Divisors and intersection theory
13. Coherent sheaves; invertible sheaves
14. Differentials (Outline)
15. Algebraic varieties over the complex numbers (Outline)
16. Descent Theory
17. Lefschetz Pencils (Outline)
Solutions to the Exercises
Annotated Bibliography
Index

Prerequisites

Some familiarity with the basic objects of algebra, namely, rings, modules, fields, and so on, and with the transcendental extensions of fields (Section 8 of my online notes on Fields and Galois Theory).

History

v2.01 (August 24, 1996). First version on the web.
v3.01 (June 13, 1998). Added 5 sections (25 pages) and an index. Minor changes to Sections 0-8. 157pp.
v4.00 (October 30, 2003). Fixed errors; many minor revisions; added exercises; added two sections; 206 pages.
v5.00 (February 20, 2005). Heavily revised; most numbering changed; 227 pages. pdf (old version 5.00)
v5.10 (March 19, 2008). Minor fixes; TeX style changed, so page numbers changed; 241 pages.pdf (old version 5.10)
v5.20 (September 14, 2009). Minor corrections; revised Chapters 1,11,16; 245 pages. pdf (old version 5.20)
v5.21 (March 31, 2011). Minor changes; changed TeX style; 258 pages.
v5.22 (January 13, 2012). Minor fixes; 260 pages.