Instructor: Nathan Chen Office hours: Wed 2:30-4pm

• Lectures on Mon/Wed 12-1:15pm in Math 310

Email: nathanchen@math.harvard.edu

TA: Rosie Shen TA office hours: Th 3-4pm

Prerequisites: A solid understanding of Hartshorne, Chapters II.1--II.5.

Textbook: We will be using a combination of texts:

• Algebraic geometry by Hartshone

• Algebraic geometry II by Mumford & Oda

The syllabus is here.

First class: January 27, 2025. Last class: April 30, 2025. Here are some class notes (which will be updated).

Lecture 1 (1/27): Review of algebraic geometry I.

Lecture 2 (2/3): Weil and Cartier divisors. Hartshorne II.6

Lecture 3 (2/5): Weil and Cartier divisors, continued.

Lecture 4 (2/10): Invertible sheaves and projective morphisms. Hartshorne II.7

Lecture 5 (2/12): Projective morphisms, global Proj construction. Hartshorne II.7

Lecture 6 (2/19): Projective bundles, blow-ups, universal properties. Hartshorne II.7

Lecture 7 (2/24): Kähler differentials, Euler exact sequence. Mumford & Oda, V.1 and V.2

Lecture 8 (2/26): Regular local rings and Kähler differentials, introduction to cohomology. Mumford & Oda, V.1 and V.2

Lecture 9 (3/3): First definitions of Čech cohomology and properties.

Lecture 10 (3/5): Higher direct image sheaves and the Leray spectral sequence. Mumford & Oda VII.3

Lecture 11 (3/10): Leray spectral sequence, continued. Mumford & Oda VII.3

Lecture 12 (3/12): Cohomology via acyclic complexes. Mumford & Oda VII.4

SPRING BREAK

Lecture 13 (3/24): Cohomology of line bundles on projective space, Serre vanishing. Mumford & Oda VII.5 and Hartshorne III.5

Lecture 14 (3/26): Serre's (cohomological) criterion for ampleness, functorial properties. Mumford & Oda VII.9 and Hartshorne III.5

Lecture 15 (3/31): Ext groups/sheaves, dualizing sheaves. Hartshorne III.6

Lecture 16 (4/2): Serre duality and applications. Hartshorne III.7

Lecture 16 (4/7): Flatness and specialization. Mumford & Oda IV.4

Lecture 16 (4/9): Flat families. Hartshorne III.9

Lecture 16 (4/14): Smooth morphisms. Hartshorne III.10

Lecture 16 (4/16): Semicontinuity of cohomology.

Here is the pdf for the homework. This will be updated every week.

• HW1 due date: Thursday, 2/13 at midnight.

• HW2 due date: Monday, 2/24 at midnight.

• HW3 due date: Thursday, 3/6 at midnight.

• HW4 due date: Monday, 3/17 at midnight.

• HW5 due date: Thursday, 4/3 at midnight.

• HW6 due date: Thursday, 4/17 at midnight.

• HW7 due date: Saturday, 5/3 at midnight.

Final project: here.