(Basic First Course in) Algebraic Geometry
pdf file for the current version (6.00)
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.
- 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
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.
The files dated July 8 or 9, 2015,
are essentially the same as those in v5.22.
||Algebraic schemes: geometry over an arbitrary field
||Divisors and intersection theory
||Coherent sheaves; invertible sheaves
||Algebraic varieties over the complex numbers
||Algebraic schemes and algebraic spaces
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.