Algebraic Geometry - J.S. Milne   Top
Course Notes
Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves -- see books.
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups

(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.


  1. Preliminaries from commutative algebra
  2. Algebraic sets
  3. Affine algebraic varieties
  4. Local study
  5. Algebraic varieties
  6. Projective varieties
  7. Complete varieties
  8. Normal varieties; (Quasi-)finite maps; Zariski's main theorem
  9. Regular maps and their fibres
  10. 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.

Title Date Pages      pdf
10 Algebraic schemes: geometry over an arbitrary field ??? pages
11 Surfaces (Intersection theory; Differentials; Riemann-Roch; 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 Riemann-Roch-Grothendieck 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 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.