Math 203B  Algebraic Geometry (Winter 2016)
Course description:
This course provides an introduction to algebraic geometry. Algebraic geometry
is a central subject in modern mathematics, and an active area of research.
It has connections with number theory, differential geometry, symplectic
geometry, mathematical physics, string theory, representation theory,
combinatorics and others.
Math 203 is a threequarter sequence. Math 203A covered affine and projective varieties.
Math 203B will focus more heavily on sheaves, schemes, and the modern
language of algebraic geometry.
Topics to be covered: sheaves and schemes; quasicoherent sheaves;
sheaf cohomology; RiemannRoch theorem for curves and applications;
properties of morphisms of schemes (separated, proper, smooth, etale).
Additional topics will be covered in Math 203C.
(For some idea of my plans, see my past course web sites.)
Instructor: Kiran Kedlaya,
kedlaya [at] math [etcetera], APM 7202.
Lectures: MWF 1011, APM B412. Some lectures will be rescheduled to Tue 1011, APM 5829; see the lecture calendar.
Office hours: To be announced week by week. Appointments can also be made by email.
Textbook: No required text. I will provide my own notes.
It may be helpful to have access to a copy of Hartshorne, Algebraic Geometry
but UCSD students can get it as a legal free ebook download using SpringerLink.
You may also find helpful Ravi Vakil's Math 216 lecture notes.
Initially the notes by Gathmann used in Math 203A may be helpful, but I will diverge from them fairly early on.
The ultimate technical reference for the theory of schemes is Grothendieck's EGA Johan de Jong's Stacks Project. Do not try to read it cover to cover! Instead, feel free to search through
it for individual topics (including homework problems).
Prerequisites: Math 203A, preferably taken last quarter. If you do not
meet this prerequisite, please contact me as soon as possible!
Grading: 100% homework (no final exam). Problem sets will be assigned weekly (see below); please do them! It is effectively impossible to learn this subject passively. Some flexibility with due dates is available as long as you ask before the original deadline. Collaboration and outside research is permitted; just declare it in as much detail as possible.
Announcements:

Week 1 office hours: Tuesday, January 5, 23.

Week 2 office hours: Tuesday, January 12, 2:303:30.

Week 3 office hours: Wednesday, January 20, 1112.

Week 4 office hours: Tuesday, January 26, 2:303:30 (note change).

Week 5 office hours: Tuesday, February 2, 23.

Week 6 office hours: Tuesday, February 9, 23.

Week 7 office hours: Wednesday, February 17, 1112.

Week 8 office hours: Tuesday, February 23, 23.

Week 9 office hours: none.

Week 10 office hours: Tuesday, March 8, 23.
Homework:
Problem sets may be submitted until 5pm on the due date in my department mailbox, or if you type your solutions you may submit via
SageMathCloud. To set this up, create an account on SMC (a free account will suffice), then email me your username.

Homework 1: PDF, solutions. Due Wednesday, January 13.

Homework 2: PDF, solutions. Due Wednesday, January 20.

Homework 3: PDF, solutions. Due Wednesday, January 27.

Homework 4: PDF, solutions. Due Wednesday, February 3.

Homework 5: PDF, solutions. Due Wednesday, February 10.

Homework 6: PDF. Due Wednesday, February 17.

Homework 7: PDF. Due Friday, February 26 in class (note unusual date).

Homework 8: PDF. Due Wednesday, March 9. (No homework due the week of March 2.)
Calendar of lecture topics:

Monday, January 4: motivation for schemes.

Wednesday, January 6: categories and sheaves.

Friday, January 8: NO LECTURE.

Monday, January 11: affine schemes.

Tuesday, January 12 (makeup): schemes.

Wednesday, January 13: continuation

Friday, January 15: NO LECTURE.

Monday, January 18: NO LECTURE (university holiday).

Wednesday, January 20: the Proj construction.

Friday, January 22: modules over schemes.

Monday, January 25: fiber products.

Tuesday, January 26 (makeup): affine communication,
closed subschemes.

Wednesday, January 27: Projective schemes.

Friday, January 29: Sections of quasicoherent sheaves.

Monday, February 1: Derived functors.

Tuesday, February 2 (makeup): continuation

Wednesday, February 3: Overview of sheaf cohomology.

Friday, February 5: Cohomology of projective space.

Monday, February 8: continuation

Tuesday, February 9 (makeup): Euler characteristics and Hilbert polynomials.

Wednesday, February 10: Differentials and smoothness.

Friday, February 12: NO LECTURE.

Monday, February 15: NO LECTURE (university holiday).

Wednesday, February 17: Line bundles on curves.

Friday, February 19: RiemannRoch theorem and applications.

Monday, February 22: More on genera of curves.

Tuesday, February 23 (makeup): cancelled due to low attendance.

Wednesday, February 24: continuation of Monday.

Friday, February 26: Serre duality (Hartshorne III.7).

Monday, February 29: NO LECTURE.

Wednesday, March 2: NO LECTURE.

Friday, March 4: NO LECTURE.

Monday, March 7: Separated morphisms (Hartshorne II.4).

Tuesday, March 8 (makeup): continuation

Wednesday, March 9: proper morphisms.

Friday, March 11: TBA.