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:
- August 31 1.1, 1.2, 1.3, 1.4, 1.5, 1.6
- Sept 7 1.7, 1.8, 1.9, 1.11, 1.12
- Sept 16 2.1, 2.3, 2.4, 2.5, 2.9
- Sept 23 2.12, 2.13, 2.14, 2.15, 2.16, 2.17
- Sept 30 3.1, 3.2, 3.4, 3.5, 3.6, 3.8, 3.11, 3.13
- Oct 7 3.15, 3.16, 3.17, 3.19, 3.21
- Oct 14 4.1, 4.4, 4.5,4.6, 4.10
- Oct 21 5.1, 5.2, 5.3, 5.6, chapter V exercise 5.8b
- Oct 28 5.8, 5.10, 5.12, 5.14, 5.15
- Nov 4 6.1, 6.2, 6.6, 6.7
- Nov 9 6.3, 6.4, 6.5, Chapter IV Ex 2.5
- Nov 16 7.1, 7.2, 7.3
- Nov 23 Chapter II 1.1, 1.3, 1.8, 1.13, 1.18,
- 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:
-
The
Macaulay 2 ,
CoCoA,
and
SINGULAR computer packages for commutative algebra.
-
GAGA by Serre,English translation
- Algebraic geometry on wikipedia
-
FAC by Serre is an excellent introduction to sheaf cohomology for coherent sheaves.
English translation
- MathOverflow is a good place to ask technical questions about algebraic geometry: it seems to be overrun with people who know EGA off by heart.
- Mathematicians:
Castelnuovo,
Cayley,
Euclid,
Gordan,
Gorenstein,
Grothendieck,
Hensel,
Hilbert,
Hironaka,
Krull,
Kummer,
Lasker,
Macaulay,
Emmy Noether,
Serre,
Sylvester,
Weil,
Weyl,
Zariski
- Tex files of solutions to some of the problems in
Hartshorne's book on algebraic geometry.
1.1
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
4.1.
4.3.