pdf file for the current version (6.10)
This is a basic first course in algebraic geometry. 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 while not ignoring the intuition provided by differential geometry.
Contents
- Preliminaries from commutative algebra
- Algebraic sets
- Affine algebraic varieties
- Local study
- Algebraic varieties
- Projective varieties
- Complete varieties
- Normal varieties; (Quasi-)finite maps; Zariski’s main theorem
- Regular maps and their fibres Solutions to the Exercises
- Index
Prerequisites
Some familiarity with the basic objects of algebra, namely, rings, modules, fields, and so on, as usually covered in advanced undergraduate or beginning graduate courses.
(Topics in) Algebraic Geometry
These chapters discuss a few more advanced topics. They can be read in almost any order, except that some assume the first.
| Title | Date | Pages | |||
|---|---|---|---|---|---|
| 10 | Algebraic schemes: geometry over an arbitrary field | 04.11.24 | 41 pages | ||
| 11 | Surfaces (Intersection theory; Differentials; Riemann-Roch; Riemann hypothesis for curves) | 04.11.24 | 38 pages | ||
| 12 | Divisors and intersection theory | 04.11.24 | 9 pages | ||
| 13 | Coherent sheaves and vector bundles | 04.11.24 | 8 pages | ||
| 14 | Differentials (Outline) | 04.11.24 | 3 pages | ||
| 15 | Algebraic varieties over the complex numbers | 04.11.24 | 3 pages | ||
| 16 | Descent theory (see Articles) | pages | |||
| 17 | Lefschetz pencils | 04.11.24 | 3 pages | ||
| 18 | Schemes | pages | |||
| 19 | Cohomology | pages | |||
| 20 | The Riemann-Roch-Grothendieck theorem | pages | |||
| A | Annotated Bibliography | 00.00.01 | 3 pages |
History of the first 9/10 chapters.
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. pdf (old version 5.22)
v6.00 (August 24, 2014). Heavily revised. Split off the basic first course from the topics; 223+ pages.
v6.01 (August 23, 2015). Minor fixes; 226 pages.
v6.02 (March 19, 2017). Minor fixes; 221 pages.pdf (old version)
v6.03 (November 2, 2023). Minor fixes; 223 pages.pdf (old version)
v6.10 (November 11, 2024). Minor fixes; 231 pages.
