Professor: Mark Haiman
Office: 771 Evans
Office hours: WF 11-12 or by appointment
Phone: (510) 642-4318

This course is a two-semester introduction to the modern language ("schemes") and technical machinery of algebraic geometry, supplemented with examples and applications chosen according to my particular taste.

The foundations of modern algebraic geometry were laid down by Alexander Grothendieck and his collaborators in the 1960's. My intention is to base the course fairly directly on Grothendieck's treatise "Éléments de géométrie algébrique" (generally known as EGA), and perhaps parts of "Séminaire de Géometrie Algébrique" (SGA). Since no English translation is available, we will use as a supplementary text "Algebraic Geometry and Arithmetic Curves," by Qing Liu. As we go along, I will prepare some summaries of definitions and results, with outlines of proofs or pointers to them in the Liu textbook. In the lectures we'll further discuss proofs of the basic theorems, especially where key ideas are involved, and illustrate the abstract concepts with examples, something which EGA and SGA generally omit.

Prerequisites: Math 250A or equivalent; knowledge of some material from Math 250B also would be helpful. The fall semester (256A) is a necessary prerequisite for the spring semester (256B).

Textbook:

• Qing Liu, Algebraic Geometry and Arithmetic Curves, Oxford University Press, 2002.

Recommended reading:

• A. Grothendieck, Éléments de géométrie algébrique, I-IV, Publications Mathématiques de l'IHES, vols. 4, 8, 11, 17, 20, 24, 28, 32 (1960-1967). EGA I also has a later edition by Grothendieck and Dieudonné, Grundlehren der mathematischen Wissenschaften, Vol. 166, Springer-Verlag (1971).
• A. Grothendieck and collaborators, Séminaire de Géométrie Algébrique, 1-7.

Syllabus: The course topics fall into two parts, with the lectures to oscillate between them. Foundations covers the basic definitions and technical machinery. Here I'll follow EGA quite closely. Applications consists of constructions and examples, which I'll introduce in class to supplement the EGA stuff. The topics under this heading are sort of a wish list—some could be the subject of an entire course, so we'll only touch the surface; others we may not get to at all.

As we proceed, I'll prepare outlines of relevant parts of EGA, with pointers to corresponding parts of Liu where applicable. Links will appear alongside the associated topics on the syllabus below.

Complete table of contents of EGA. The 1971 edition of EGA I, indicated on the syllabus by I^*, is somewhat different.

Foundations

1. Introduction: the idea of schemes
2. Sheaves (EGA 0:3,4—revised 9/19)
3. Rings of fractions (EGA 0:1)
4. Affine schemes (EGA 0:2, EGA I:1.1-3)
5. Coherent and quasi-coherent sheaves on affine schemes (EGA 0:5, EGA I:1.4-6)
6. Morphisms of affine schemes; Spec as an adjoint functor (EGA I^*:1.6)
7. Schemes, morphisms, gluing, schemes over a base, local schemes (EGA I:2)
8. Morphisms of finite type, algebraic schemes (EGA I:6)
9. Jacobson schemes and Hilbert's Nullstellensatz (EGA IV:10.1-4)
10. Noetherian and Artinian schemes (I:6.1-2)
11. Products, base change (EGA I:3)
12. Functor of points; geometric points (I:3.4)
13. Surjective and injective morphisms; fibers (I:3.5-7)
14. Immersions, preimage and image of morphisms (EGA I:4—revised 11/9; EGA I:9.5)
15. Reduced schemes (EGA I:5)
16. Separated morphisms and schemes (I:5.3-5)
17. Direct and inverse images of quasi-coherent sheaves (I:1.6, EGA I:9.1-2,9.6)
18. Quasi-coherent sheaves of algebras, affine morphisms (I:9.6, EGA II:1)
19. Projective schemes, Proj of a graded algebra (EGA II:2.1-4)
20. Sheaves on Proj (EGA II:2.5-9)
21. Proj of a sheaf of graded algebras (EGA II:3)
22. Projective space bundles, Segre morphism (EGA II:4.1-4—revised 2/16)
23. Ample line bundles (EGA II:4.5-6, I:9-3.4—revised 4/23)
24. Divisors (EGA IV:20.1, 21.1-3, EGA IV:20.2, 21.4,6)
25. Dimension (EGA 0:14,16.1-3—revised 3/11, EGA IV:5.1-6)
26. Derived category and derived functors (Notes, 1-5—revised 4/17).
    Daniel Murfet, a student of Amnon Neeman, has also written detailed notes on these and other topics.
27. Sheaf cohomology (after EGA III, but using the derived category)
28. Quasi-affine and quasi-projective morphisms, Serre criterion (EGA II:5)
29. Proper and projective morphisms (II:5.4-5)
30. Chow's lemma (II:5.6)
31. Blowups, projective cones, projective closure (II:8.1-8.7)
32. Regular rings and regular schemes (EGA 0:17, IV:2.1,3, 6.1,6.5-8)
33. Flatness and dimension (IV:2.1,3, 6.1)
34. Regular and smooth morphisms (IV:6.5-8)
35. Associated points (IV:3.1-3)
36. Finite morphisms, integral closure and normality (II:6.1-3)
37. Differentials (0:20, IV:16.1-5)
38. Smooth, unramified and étale morphisms (0:19, 22, IV:16.10, 12, 17.1-12)
39. Cohomology and singularities; Grothendieck duality

Applications

1. Schemes of finite type over a field
2. Curves, surfaces, hypersurfaces
3. Tangent cones, tangent and normal bundles
4. Algebraic groups, group actions, quotients
5. Toric varieties
6. Grassmann and flag varieties
7. Nilpotent orbits, Springer varieties, Hessenberg varieties
8. Hilbert schemes
9. Chow varieties
10. Divisors
11. Blowing up
12. Intersection theory
13. Riemann-Roch theorem (Grothendieck version)
14. Rationality, unirationality and rational connectedness
15. Resolution of singularities
16. Cohen-Macaulay and Gorenstein schemes, duality

Homework (fall): Problem Set 1 (#1 changed 9/9/05) due 9/19, Problem Set 2 due 9/26, Problem Set 3 (#7, #8 changed 10/3/05) due 10/3, Problem Set 4 due 10/10, Problem Set 5 (#1-3 changed 10/12) due 10/17, Problem Set 6 (#2 changed 10/21) due 10/31, Problem Set 7 due 11/7, Problem Set 8 due 11/14, Problem Set 9 (#9 added 11/18, #2 changed 11/21, #8 changed 11/23) due 11/28. Problem Set 10 (#6 changed 12/9) due Friday 12/9. Extra problems for winter break.

Homework (spring): Problem Set 1 due 2/3, Problem Set 2 (#5 changed 2/10) due 2/10, Problem Set 3 (#2, #5 changed 2/17, #7 changed 2/22) due 2/24, Problem Set 4 due 3/3, Problem Set 5 (#4 changed 3/13) due 3/17, Problem Set 6 (#2 changed 3/21) due 3/24, Problem Set 7 (#4(d),5,6 added 4/7, #3 changed 4/12) due 4/14, Problem Set 8 due 5/5 (#7-#11 added 4/27, #12-#14 added 4/28, #15 added 5/1).

Grading: I'll pose more homework problems than I actually expect you to do, and grade based on your doing a reasonable fraction of them.
The GSI for spring semester in Ron Fertig. Grading policy will be similar to fall: Ron will choose a subset of the problems to correct fully and return with comments, and record which others you solved or made a decent attempt at.