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.
- Algebraic sets
- Affine algebraic varieties
- Algebaic varieties
- Local study
- Projective varieties
- Complete varieties
- Finite maps
- Dimension theory
- Regular maps and their fibres
- Algebraic spaces: geometry over an arbitrary field
- Divisors and intersection theory
- Coherent sheaves; invertible sheaves
- Differentials (Outline)
- Algebraic varieties over the complex numbers (Outline)
- Descent Theory
- Lefschetz Pencils (Outline)
Solutions to the Exercises
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).
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.