Mathematics 256A Fall 2010

Professor: Richard Borcherds

Office hours: Tuesday, Thursday 3:30-5:00 927 Evans Hall

Our class meets in TuTh 11:00-12:30P, 81 EVANS. This is the course home page (address http://math.berkeley.edu/~reb/256A). The course control number is 54515.

Catalogue Description: Mathematics 256A

Three hours of lecture per week. Prerequisites: 250A-250B for 256A; 256A for 256B. Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall.

Textbook and course notes:

The textbook is Algebraic geometry by Hartshorne. We will cover much of chapters 1 (varieties) and 2 (schemes) and maybe some of chapter 3 (cohomology). The goal is that by the end of the course you should have some idea of what a coherent sheaf over a scheme is, and maybe be able to calculate its cohomology groups. There are some draft notes for the course here; these will be expanded and corrected regularly until December.

Background reading

The book Commutative algebra with a view towards algebraic geometry by Eisenbud covers the commutative algebra we need. The older book Introduction to commutative algebra by Atiyah and Macdonald is also fine. The ultimate reference is Grothendieck's FGA, EGA, and SGA. Other online books or notes at about the same level as this course include:

Examinations: None

Homework:

  1. August 31 1.1, 1.2, 1.3, 1.4, 1.5, 1.6
  2. Sept 7 1.7, 1.8, 1.9, 1.11, 1.12
  3. Sept 16 2.1, 2.3, 2.4, 2.5, 2.9
  4. Sept 23 2.12, 2.13, 2.14, 2.15, 2.16, 2.17
  5. Sept 30 3.1, 3.2, 3.4, 3.5, 3.6, 3.8, 3.11, 3.13
  6. Oct 7 3.15, 3.16, 3.17, 3.19, 3.21
  7. Oct 14 4.1, 4.4, 4.5,4.6, 4.10
  8. Oct 21 5.1, 5.2, 5.3, 5.6, chapter V exercise 5.8b
  9. Oct 28 5.8, 5.10, 5.12, 5.14, 5.15
  10. Nov 4 6.1, 6.2, 6.6, 6.7
  11. Nov 9 6.3, 6.4, 6.5, Chapter IV Ex 2.5
  12. Nov 16 7.1, 7.2, 7.3
  13. Nov 23 Chapter II 1.1, 1.3, 1.8, 1.13, 1.18,
  14. Nov 30 2.2, 2.5, 2.6, 2.9, 2.10, 2.11,
The grader for the course is Damien Mondragon. Put homework in his mailbox by Thursday the week after it is assigned.

Study group

VERACITY is an informal study group for algebraic geometry neophytes, based on readings of Ravi Vakil's notes. It meets Tuessdays at 1pm in Room 732.

Links related to the course: