Math 203B - Algebraic Geometry (Winter 2020)

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 three-quarter 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; Riemann-Roch 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 3-3:50, APM B412. A few lectures will be cancelled and made up at other times (to be agreed upon); see the lecture calendar.

Office hours: Mon 4-5. Appointments can also be made by email.

Textbook: Hartshorne, Algebraic Geometry. UCSD students can get it as a legal free PDF download using SpringerLink. You may also find helpful Ravi Vakil's Math 216 lecture notes. I will occasionally post lecture notes on specific topics.

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, but please ask before the original deadline. Collaboration and outside research is permitted and encouraged; just declare it in as much detail as possible.

Homework:

Problem sets may be submitted until 6pm on the due date in my department mailbox. Typed solutions may also be submitted as email (please send a PDF file).

Homework 1: pdf. Due Wednesday, January 15.
Homework 2: pdf. Due Wednesday, January 22.
Homework 3: pdf. Due Wednesday, January 29.
No homework due February 5.
Homework 4: pdf. Due Wednesday, February 12.
Homework 5: pdf. Due Wednesday, February 19.
Homework 6: pdf. Due Wednesday, February 26.
Homework 7: pdf (updated March 2). Due ~~Wednesday, March 4~~Friday, March 6.
Homework 8: pdf. Due Friday, March 13.

Calendar of lecture topics:

Monday, January 6: motivation for schemes.
Wednesday, January 8: sheaves (Hartshorne II.1).
~~Friday, January 10~~: NO LECTURE.
Monday, January 13: the Zariski prime spectrum of a ring, definition of the structure presheaf (Hartshorne II.2)
Tuesday, January 14 (APM 5829, 2-2:50pm): the structure presheaf is a sheaf (Hartshorne II.2).
Wednesday, January 15: affine schemes (Hartshorne II.2).
~~Friday, January 17~~: NO LECTURE.
~~Monday, January 20~~: NO LECTURE (university holiday).
Tuesday, January 21 (APM 2402, 2-2:50pm): general schemes (Hartshorne II.2).
Wednesday, January 22: the Proj functor; properties of schemes and morphisms (Hartshorne II.2, II.3).
Friday, January 24: properties of schemes and morphisms (Hartshorne II.3); affine communication (pdf).
Monday, January 27 (3-3:30pm): construction of fiber products of schemes (Hartshorne II.3).
Wednesday, January 29: fiber products and fibers, closed immersions, locally of finite type morphisms (Hartshorne II.3).
Friday, January 31: sheaves of modules (Hartshorne II.5).
Monday, February 3: quasicoherent sheaves (Hartshorne II.5).
Tuesday, February 4 (APM 2402, 2-2:50pm): separated morphisms (Hartshorne II.3).
Wednesday, February 5: projective and proper morphisms (pdf).
Friday, February 7: continuation.
Monday, February 10: the Proj construction (Hartshorne II.5).
Wednesday, February 12: graded modules and the Proj construction (Hartshorne II.5).
Friday, February 14: very ample lime bundles (Hartshorne II.5).
~~Monday, February 17~~: NO LECTURE (university holiday).
Wednesday, February 19: very ample line bundles and global generation (Hartshorne II.5).
Friday, February 21: morphisms to projective spaces (Hartshorne II.7).
Monday, February 24: closed immersions into projective spaces (Hartshorne II.7).
Wednesday, February 26: Weil and Cartier divisors (Hartshorne II.6).
Friday, February 28: more on divisors (Hartshorne II.6).
Monday, March 2: ample invertible sheaves, relative Proj, blowing up (Hartshorne II.7).
Wednesday, March 4: more on blowing up (Hartshorne II.7).
Friday, March 6: sheaves differentials (Hartshorne II.8).
Monday, March 9: TBA (lecture by James McKernan).
~~Wednesday, March 11~~: NO LECTURE.
Friday, March 13: TBA.