(Basic First Course in) Algebraic Geometry
pdf file for the current version (6.01)
These notes are an introduction to the theory of
algebraic varieties emphasizing the similarities to the theory of manifolds.
In contrast to most such accounts they study abstract algebraic varieties, and
not just subvarieties of affine and projective space. This approach leads more
naturally into scheme theory.
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 
pdf 

10 
Algebraic schemes: geometry over an arbitrary field 
??? 
pages 

11 
Surfaces (Intersection theory; Differentials; RiemannRoch;
Riemann hypothesis for curves) 
??? 
pages 

12 
Divisors and intersection theory 
08.07.15 
7 pages 
pdf 
13 
Coherent sheaves; invertible sheaves 
08.07.15 
7 pages 
pdf 
14 
Differentials (Outline) 
08.07.15 
2 pages 
pdf 
15 
Algebraic varieties over the complex numbers 
08.07.15 
3 pages 
pdf 
16 
Descent theory 
09.07.15 
20 pages 
pdf 
17 
Lefschetz pencils 
09.07.15 
3 pages 
pdf 
18 
Schemes 
??? 
pages 

19 
Cohomology 
??? 
pages 

20 
The RiemannRochGrothendieck theorem 
??? 
pages 

A 
Annotated Bibliography 
00.00.01 
3 pages 
pdf 
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 08. 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.